:: RLVECT_3 semantic presentation
Lemma52:
for V being RealLinearSpace
for F, G being FinSequence of the carrier of V
for f being Function of the carrier of V, REAL holds f (#) (F ^ G) = (f (#) F) ^ (f (#) G)
theorem Th1: :: RLVECT_3:1
theorem Th2: :: RLVECT_3:2
theorem Th3: :: RLVECT_3:3
theorem Th4: :: RLVECT_3:4
:: deftheorem Def1 defines linearly-independent RLVECT_3:def 1 :
theorem Th5: :: RLVECT_3:5
canceled;
theorem Th6: :: RLVECT_3:6
theorem Th7: :: RLVECT_3:7
theorem Th8: :: RLVECT_3:8
theorem Th9: :: RLVECT_3:9
theorem Th10: :: RLVECT_3:10
theorem Th11: :: RLVECT_3:11
theorem Th12: :: RLVECT_3:12
theorem Th13: :: RLVECT_3:13
theorem Th14: :: RLVECT_3:14
:: deftheorem Def2 defines Lin RLVECT_3:def 2 :
theorem Th15: :: RLVECT_3:15
canceled;
theorem Th16: :: RLVECT_3:16
canceled;
theorem Th17: :: RLVECT_3:17
theorem Th18: :: RLVECT_3:18
Lemma182:
for x being set
for V being RealLinearSpace holds
( x in (0). V iff x = 0. V )
theorem Th19: :: RLVECT_3:19
theorem Th20: :: RLVECT_3:20
theorem Th21: :: RLVECT_3:21
theorem Th22: :: RLVECT_3:22
Lemma186:
for V being RealLinearSpace
for W1, W3, W2 being Subspace of V st W1 is Subspace of W3 holds
W1 /\ W2 is Subspace of W3
Lemma187:
for V being RealLinearSpace
for W1, W2, W3 being Subspace of V st W1 is Subspace of W2 & W1 is Subspace of W3 holds
W1 is Subspace of W2 /\ W3
Lemma188:
for V being RealLinearSpace
for W1, W2, W3 being Subspace of V st W1 is Subspace of W2 holds
W1 is Subspace of W2 + W3
Lemma189:
for V being RealLinearSpace
for W1, W3, W2 being Subspace of V st W1 is Subspace of W3 & W2 is Subspace of W3 holds
W1 + W2 is Subspace of W3
theorem Th23: :: RLVECT_3:23
theorem Th24: :: RLVECT_3:24
theorem Th25: :: RLVECT_3:25
theorem Th26: :: RLVECT_3:26
Lemma192:
for M being non empty set
for CF being Choice_Function of M st not {} in M holds
dom CF = M
theorem Th27: :: RLVECT_3:27
theorem Th28: :: RLVECT_3:28
:: deftheorem Def3 defines Basis RLVECT_3:def 3 :
theorem Th29: :: RLVECT_3:29
canceled;
theorem Th30: :: RLVECT_3:30
canceled;
theorem Th31: :: RLVECT_3:31
canceled;
theorem Th32: :: RLVECT_3:32
theorem Th33: :: RLVECT_3:33
theorem Th34: :: RLVECT_3:34
canceled;
theorem Th35: :: RLVECT_3:35
theorem Th36: :: RLVECT_3:36
theorem Th37: :: RLVECT_3:37
theorem Th38: :: RLVECT_3:38
theorem Th39: :: RLVECT_3:39
theorem Th40: :: RLVECT_3:40
theorem Th41: :: RLVECT_3:41