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Theorem comi1 709
Description: Commutation expressed with ->1.
Hypothesis
Ref Expression
comi1.1 a C b
Assertion
Ref Expression
comi1 b =< (a ->1 b)
Proof of Theorem comi1
StepHypRef Expression
1 ancom 74 . . . . 5 (b ^ a) = (a ^ b)
21ax-r5 38 . . . 4 ((b ^ a) v (b ^ a')) = ((a ^ b) v (b ^ a'))
3 ax-a2 31 . . . 4 ((a ^ b) v (b ^ a')) = ((b ^ a') v (a ^ b))
42, 3ax-r2 36 . . 3 ((b ^ a) v (b ^ a')) = ((b ^ a') v (a ^ b))
5 lear 161 . . . 4 (b ^ a') =< a'
65leror 152 . . 3 ((b ^ a') v (a ^ b)) =< (a' v (a ^ b))
74, 6bltr 138 . 2 ((b ^ a) v (b ^ a')) =< (a' v (a ^ b))
8 comi1.1 . . . 4 a C b
98comcom 453 . . 3 b C a
109df-c2 133 . 2 b = ((b ^ a) v (b ^ a'))
11 df-i1 44 . 2 (a ->1 b) = (a' v (a ^ b))
127, 10, 11le3tr1 140 1 b =< (a ->1 b)
Colors of variables: term
Syntax hints:   =< wle 2   C wc 3  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12
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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by: (None)
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