:: WAYBEL_7 semantic presentation
theorem Th1: :: WAYBEL_7:1
canceled;
theorem Th2: :: WAYBEL_7:2
canceled;
theorem Th3: :: WAYBEL_7:3
theorem Th4: :: WAYBEL_7:4
theorem Th5: :: WAYBEL_7:5
theorem Th6: :: WAYBEL_7:6
canceled;
theorem Th7: :: WAYBEL_7:7
canceled;
theorem Th8: :: WAYBEL_7:8
theorem Th9: :: WAYBEL_7:9
theorem Th10: :: WAYBEL_7:10
theorem Th11: :: WAYBEL_7:11
theorem Th12: :: WAYBEL_7:12
theorem Th13: :: WAYBEL_7:13
theorem Th14: :: WAYBEL_7:14
theorem Th15: :: WAYBEL_7:15
:: deftheorem Def1 defines prime WAYBEL_7:def 1 :
theorem Th16: :: WAYBEL_7:16
theorem Th17: :: WAYBEL_7:17
:: deftheorem Def2 defines prime WAYBEL_7:def 2 :
theorem Th18: :: WAYBEL_7:18
theorem Th19: :: WAYBEL_7:19
theorem Th20: :: WAYBEL_7:20
theorem Th21: :: WAYBEL_7:21
theorem Th22: :: WAYBEL_7:22
theorem Th23: :: WAYBEL_7:23
theorem Th24: :: WAYBEL_7:24
theorem Th25: :: WAYBEL_7:25
:: deftheorem Def3 defines ultra WAYBEL_7:def 3 :
Lemma78:
for L being with_infima Poset
for F being Filter of L
for X being non empty finite Subset of L
for x being Element of L st x in uparrow (fininfs (F \/ X)) holds
ex a being Element of L st
( a in F & x >= a "/\" (inf X) )
theorem Th26: :: WAYBEL_7:26
Lemma85:
for L being with_suprema Poset
for I being Ideal of L
for X being non empty finite Subset of L
for x being Element of L st x in downarrow (finsups (I \/ X)) holds
ex i being Element of L st
( i in I & x <= i "\/" (sup X) )
theorem Th27: :: WAYBEL_7:27
theorem Th28: :: WAYBEL_7:28
theorem Th29: :: WAYBEL_7:29
theorem Th30: :: WAYBEL_7:30
:: deftheorem Def4 defines is_a_cluster_point_of WAYBEL_7:def 4 :
:: deftheorem Def5 defines is_a_convergence_point_of WAYBEL_7:def 5 :
theorem Th31: :: WAYBEL_7:31
theorem Th32: :: WAYBEL_7:32
theorem Th33: :: WAYBEL_7:33
theorem Th34: :: WAYBEL_7:34
theorem Th35: :: WAYBEL_7:35
theorem Th36: :: WAYBEL_7:36
theorem Th37: :: WAYBEL_7:37
:: deftheorem Def6 defines pseudoprime WAYBEL_7:def 6 :
theorem Th38: :: WAYBEL_7:38
theorem Th39: :: WAYBEL_7:39
theorem Th40: :: WAYBEL_7:40
theorem Th41: :: WAYBEL_7:41
theorem Th42: :: WAYBEL_7:42
theorem Th43: :: WAYBEL_7:43
:: deftheorem Def7 defines multiplicative WAYBEL_7:def 7 :
theorem Th44: :: WAYBEL_7:44
theorem Th45: :: WAYBEL_7:45
theorem Th46: :: WAYBEL_7:46
theorem Th47: :: WAYBEL_7:47
E149:
now
let L be
lower-bounded continuous LATTICE;
let p be
Element of
L;
assume that E33:
L -waybelow is
multiplicative
and E42:
for
a,
b being
Element of
L holds
( not
a "/\" b << p or
a <= p or
b <= p )
and E43:
not
p is
prime
;
consider x being
Element of
L,
y being
Element of
L such that E44:
(
x "/\" y <= p & not
x <= p & not
y <= p )
by , WAYBEL_6:def 6;
E45:
for
a being
Element of
L holds
( not
waybelow a is
empty &
waybelow a is
directed )
;
then consider u being
Element of
L such that E56:
(
u << x & not
u <= p )
by Th9, WAYBEL_3:24;
consider v being
Element of
L such that E57:
(
v << y & not
v <= p )
by Th9, , WAYBEL_3:24;
(
[u,x] in L -waybelow &
[v,y] in L -waybelow )
by E42, E43, WAYBEL_4:def 2;
then
[(u "/\" v),(x "/\" y)] in L -waybelow
by , ;
then
u "/\" v << x "/\" y
by WAYBEL_4:def 2;
then
u "/\" v << p
by Th9, WAYBEL_3:2;
hence
contradiction
by , E42, E43;
end;
theorem Th48: :: WAYBEL_7:48
theorem Th49: :: WAYBEL_7:49
theorem Th50: :: WAYBEL_7:50