:: LMOD_6 semantic presentation
theorem Th1: :: LMOD_6:1
theorem Th2: :: LMOD_6:2
theorem Th3: :: LMOD_6:3
theorem Th4: :: LMOD_6:4
:: deftheorem Def1 LMOD_6:def 1 :
canceled;
:: deftheorem Def2 defines trivial LMOD_6:def 2 :
theorem Th5: :: LMOD_6:5
theorem Th6: :: LMOD_6:6
theorem Th7: :: LMOD_6:7
:: deftheorem Def3 defines @ LMOD_6:def 3 :
theorem Th8: :: LMOD_6:8
canceled;
theorem Th9: :: LMOD_6:9
:: deftheorem Def4 LMOD_6:def 4 :
canceled;
:: deftheorem Def5 defines @ LMOD_6:def 5 :
theorem Th10: :: LMOD_6:10
theorem Th11: :: LMOD_6:11
theorem Th12: :: LMOD_6:12
theorem Th13: :: LMOD_6:13
theorem Th14: :: LMOD_6:14
canceled;
theorem Th15: :: LMOD_6:15
:: deftheorem Def6 defines <: LMOD_6:def 6 :
:: deftheorem Def7 defines c= LMOD_6:def 7 :
theorem Th16: :: LMOD_6:16
for b
1 being
set for b
2 being
Ringfor b
3, b
4 being
LeftMod of b
2 holds
( b
3 c= b
4 implies ( ( b
1 in b
3 implies b
1 in b
4 ) & ( b
1 is
Vector of b
3 implies b
1 is
Vector of b
4 ) ) )
theorem Th17: :: LMOD_6:17
for b
1 being
Ringfor b
2 being
Scalar of b
1for b
3, b
4 being
LeftMod of b
1for b
5, b
6, b
7 being
Vector of b
3for b
8, b
9, b
10 being
Vector of b
4 holds
( b
3 c= b
4 implies (
0. b
3 = 0. b
4 & ( b
5 = b
8 & b
6 = b
9 implies b
5 + b
6 = b
8 + b
9 ) & ( b
7 = b
10 implies b
2 * b
7 = b
2 * b
10 ) & ( b
7 = b
10 implies
- b
10 = - b
7 ) & ( b
5 = b
8 & b
6 = b
9 implies b
5 - b
6 = b
8 - b
9 ) &
0. b
4 in b
3 &
0. b
3 in b
4 & ( b
8 in b
3 & b
9 in b
3 implies b
8 + b
9 in b
3 ) & ( b
10 in b
3 implies b
2 * b
10 in b
3 ) & ( b
10 in b
3 implies
- b
10 in b
3 ) & ( b
8 in b
3 & b
9 in b
3 implies b
8 - b
9 in b
3 ) ) )
theorem Th18: :: LMOD_6:18
theorem Th19: :: LMOD_6:19
canceled;
theorem Th20: :: LMOD_6:20
canceled;
theorem Th21: :: LMOD_6:21
theorem Th22: :: LMOD_6:22
for b
1 being
Ringfor b
2, b
3, b
4 being
LeftMod of b
1 holds
( b
2 c= b
3 & b
3 c= b
4 implies b
2 c= b
4 )
theorem Th23: :: LMOD_6:23
theorem Th24: :: LMOD_6:24
theorem Th25: :: LMOD_6:25
theorem Th26: :: LMOD_6:26
theorem Th27: :: LMOD_6:27
theorem Th28: :: LMOD_6:28
theorem Th29: :: LMOD_6:29
theorem Th30: :: LMOD_6:30
theorem Th31: :: LMOD_6:31
theorem Th32: :: LMOD_6:32
theorem Th33: :: LMOD_6:33
theorem Th34: :: LMOD_6:34
theorem Th35: :: LMOD_6:35
for b
1 being
Ringfor b
2 being
LeftMod of b
1for b
3, b
4, b
5 being
Subspace of b
2 holds
( b
3 c= b
4 implies b
4 + (b3 /\ b5) = (b3 + b4) /\ (b4 + b5) )
theorem Th36: :: LMOD_6:36
theorem Th37: :: LMOD_6:37
for b
1 being
Ringfor b
2 being
LeftMod of b
1for b
3, b
4, b
5 being
Subspace of b
2 holds
( b
3 c= b
4 & b
5 c= b
4 implies b
3 + b
5 c= b
4 )
theorem Th38: :: LMOD_6:38
theorem Th39: :: LMOD_6:39
theorem Th40: :: LMOD_6:40
theorem Th41: :: LMOD_6:41
for b
1 being
Ringfor b
2 being
LeftMod of b
1for b
3, b
4 being
Subspace of b
2 holds
( b
3 c= b
4 iff for b
5 being
Vector of b
2 holds
( b
5 in b
3 implies b
5 in b
4 ) )
theorem Th42: :: LMOD_6:42
theorem Th43: :: LMOD_6:43
theorem Th44: :: LMOD_6:44