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Theorem wwoml3 213
Description: Weak orthomodular law.
Hypotheses
Ref Expression
wwoml3.1 a =< b
wwoml3.2 (b ^ a') = 0
Assertion
Ref Expression
wwoml3 (a == b) = 1
Proof of Theorem wwoml3
StepHypRef Expression
1 wwoml3.2 . . . . . 6 (b ^ a') = 0
21ax-r1 35 . . . . 5 0 = (b ^ a')
3 ancom 74 . . . . 5 (b ^ a') = (a' ^ b)
42, 3ax-r2 36 . . . 4 0 = (a' ^ b)
54lor 70 . . 3 (a v 0) = (a v (a' ^ b))
65rbi 98 . 2 ((a v 0) == b) = ((a v (a' ^ b)) == b)
7 or0 102 . . 3 (a v 0) = a
87rbi 98 . 2 ((a v 0) == b) = (a == b)
9 wwoml3.1 . . 3 a =< b
109wwoml2 212 . 2 ((a v (a' ^ b)) == b) = 1
116, 8, 103tr2 64 1 (a == b) = 1
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   == tb 5   v wo 6   ^ wa 7  1wt 8  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
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:  wwfh1  216  wwfh2  217
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