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Theorem oml3 452
Description: Orthomodular law. Kalmbach 83 p. 22.
Hypotheses
Ref Expression
oml3.1 a =< b
oml3.2 (b ^ a') = 0
Assertion
Ref Expression
oml3 a = b
Proof of Theorem oml3
StepHypRef Expression
1 oml3.2 . . . . 5 (b ^ a') = 0
21ax-r1 35 . . . 4 0 = (b ^ a')
3 ancom 74 . . . 4 (b ^ a') = (a' ^ b)
42, 3ax-r2 36 . . 3 0 = (a' ^ b)
54lor 70 . 2 (a v 0) = (a v (a' ^ b))
6 or0 102 . 2 (a v 0) = a
7 oml3.1 . . 3 a =< b
87oml2 451 . 2 (a v (a' ^ b)) = b
95, 6, 83tr2 64 1 a = b
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v wo 6   ^ wa 7  0wf 9
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  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-le2 131
This theorem is referenced by:  fh1  469  fh2  470  mh  879
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