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Theorem oml4 487
Description: Orthomodular law.
Assertion
Ref Expression
oml4 ((a == b) ^ a) =< b
Proof of Theorem oml4
StepHypRef Expression
1 ancom 74 . . 3 ((a == b) ^ a) = (a ^ (a == b))
2 dfb 94 . . . . 5 (a == b) = ((a ^ b) v (a' ^ b'))
32lan 77 . . . 4 (a ^ (a == b)) = (a ^ ((a ^ b) v (a' ^ b')))
4 coman1 185 . . . . . . 7 (a ^ b) C a
54comcom 453 . . . . . 6 a C (a ^ b)
6 coman1 185 . . . . . . . . 9 (a' ^ b') C a'
76comcom 453 . . . . . . . 8 a' C (a' ^ b')
87comcom2 183 . . . . . . 7 a' C (a' ^ b')'
98comcom5 458 . . . . . 6 a C (a' ^ b')
105, 9fh1 469 . . . . 5 (a ^ ((a ^ b) v (a' ^ b'))) = ((a ^ (a ^ b)) v (a ^ (a' ^ b')))
11 or0 102 . . . . . 6 ((a ^ b) v 0) = (a ^ b)
12 anidm 111 . . . . . . . . . 10 (a ^ a) = a
1312ran 78 . . . . . . . . 9 ((a ^ a) ^ b) = (a ^ b)
1413ax-r1 35 . . . . . . . 8 (a ^ b) = ((a ^ a) ^ b)
15 anass 76 . . . . . . . 8 ((a ^ a) ^ b) = (a ^ (a ^ b))
1614, 15ax-r2 36 . . . . . . 7 (a ^ b) = (a ^ (a ^ b))
17 ancom 74 . . . . . . . . 9 (b' ^ 0) = (0 ^ b')
18 an0 108 . . . . . . . . 9 (b' ^ 0) = 0
19 dff 101 . . . . . . . . . 10 0 = (a ^ a')
2019ran 78 . . . . . . . . 9 (0 ^ b') = ((a ^ a') ^ b')
2117, 18, 203tr2 64 . . . . . . . 8 0 = ((a ^ a') ^ b')
22 anass 76 . . . . . . . 8 ((a ^ a') ^ b') = (a ^ (a' ^ b'))
2321, 22ax-r2 36 . . . . . . 7 0 = (a ^ (a' ^ b'))
2416, 232or 72 . . . . . 6 ((a ^ b) v 0) = ((a ^ (a ^ b)) v (a ^ (a' ^ b')))
25 ancom 74 . . . . . 6 (a ^ b) = (b ^ a)
2611, 24, 253tr2 64 . . . . 5 ((a ^ (a ^ b)) v (a ^ (a' ^ b'))) = (b ^ a)
2710, 26ax-r2 36 . . . 4 (a ^ ((a ^ b) v (a' ^ b'))) = (b ^ a)
283, 27ax-r2 36 . . 3 (a ^ (a == b)) = (b ^ a)
291, 28ax-r2 36 . 2 ((a == b) ^ a) = (b ^ a)
30 lea 160 . 2 (b ^ a) =< b
3129, 30bltr 138 1 ((a == b) ^ a) =< b
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   == tb 5   v wo 6   ^ wa 7  0wf 9
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by: (None)
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