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Theorem oaidlem2 931
Description: Lemma for identity-like OA law.
Hypothesis
Ref Expression
oaidlem2.1 ((d v ((a ->1 c) ^ (b ->1 c)))' v ((a ->1 c) ->1 (b ->1 c))) = 1
Assertion
Ref Expression
oaidlem2 ((a ->1 c) ^ (d v ((a ->1 c) ^ (b ->1 c)))) =< (b ->1 c)
Proof of Theorem oaidlem2
StepHypRef Expression
1 anidm 111 . . . . . . . . . 10 ((a ->1 c) ^ (a ->1 c)) = (a ->1 c)
21ax-r1 35 . . . . . . . . 9 (a ->1 c) = ((a ->1 c) ^ (a ->1 c))
32ran 78 . . . . . . . 8 ((a ->1 c) ^ (b ->1 c)) = (((a ->1 c) ^ (a ->1 c)) ^ (b ->1 c))
4 anass 76 . . . . . . . 8 (((a ->1 c) ^ (a ->1 c)) ^ (b ->1 c)) = ((a ->1 c) ^ ((a ->1 c) ^ (b ->1 c)))
53, 4ax-r2 36 . . . . . . 7 ((a ->1 c) ^ (b ->1 c)) = ((a ->1 c) ^ ((a ->1 c) ^ (b ->1 c)))
6 leor 159 . . . . . . . 8 ((a ->1 c) ^ (b ->1 c)) =< (d v ((a ->1 c) ^ (b ->1 c)))
76lelan 167 . . . . . . 7 ((a ->1 c) ^ ((a ->1 c) ^ (b ->1 c))) =< ((a ->1 c) ^ (d v ((a ->1 c) ^ (b ->1 c))))
85, 7bltr 138 . . . . . 6 ((a ->1 c) ^ (b ->1 c)) =< ((a ->1 c) ^ (d v ((a ->1 c) ^ (b ->1 c))))
98df-le2 131 . . . . 5 (((a ->1 c) ^ (b ->1 c)) v ((a ->1 c) ^ (d v ((a ->1 c) ^ (b ->1 c))))) = ((a ->1 c) ^ (d v ((a ->1 c) ^ (b ->1 c))))
10 ax-a3 32 . . . . . 6 (((d v ((a ->1 c) ^ (b ->1 c)))' v (a ->1 c)') v ((a ->1 c) ^ (b ->1 c))) = ((d v ((a ->1 c) ^ (b ->1 c)))' v ((a ->1 c)' v ((a ->1 c) ^ (b ->1 c))))
11 ax-a2 31 . . . . . . . 8 ((d v ((a ->1 c) ^ (b ->1 c)))' v (a ->1 c)') = ((a ->1 c)' v (d v ((a ->1 c) ^ (b ->1 c)))')
12 oran3 93 . . . . . . . 8 ((a ->1 c)' v (d v ((a ->1 c) ^ (b ->1 c)))') = ((a ->1 c) ^ (d v ((a ->1 c) ^ (b ->1 c))))'
1311, 12ax-r2 36 . . . . . . 7 ((d v ((a ->1 c) ^ (b ->1 c)))' v (a ->1 c)') = ((a ->1 c) ^ (d v ((a ->1 c) ^ (b ->1 c))))'
1413ax-r5 38 . . . . . 6 (((d v ((a ->1 c) ^ (b ->1 c)))' v (a ->1 c)') v ((a ->1 c) ^ (b ->1 c))) = (((a ->1 c) ^ (d v ((a ->1 c) ^ (b ->1 c))))' v ((a ->1 c) ^ (b ->1 c)))
15 df-i1 44 . . . . . . . . 9 ((a ->1 c) ->1 (b ->1 c)) = ((a ->1 c)' v ((a ->1 c) ^ (b ->1 c)))
1615lor 70 . . . . . . . 8 ((d v ((a ->1 c) ^ (b ->1 c)))' v ((a ->1 c) ->1 (b ->1 c))) = ((d v ((a ->1 c) ^ (b ->1 c)))' v ((a ->1 c)' v ((a ->1 c) ^ (b ->1 c))))
1716ax-r1 35 . . . . . . 7 ((d v ((a ->1 c) ^ (b ->1 c)))' v ((a ->1 c)' v ((a ->1 c) ^ (b ->1 c)))) = ((d v ((a ->1 c) ^ (b ->1 c)))' v ((a ->1 c) ->1 (b ->1 c)))
18 oaidlem2.1 . . . . . . 7 ((d v ((a ->1 c) ^ (b ->1 c)))' v ((a ->1 c) ->1 (b ->1 c))) = 1
1917, 18ax-r2 36 . . . . . 6 ((d v ((a ->1 c) ^ (b ->1 c)))' v ((a ->1 c)' v ((a ->1 c) ^ (b ->1 c)))) = 1
2010, 14, 193tr2 64 . . . . 5 (((a ->1 c) ^ (d v ((a ->1 c) ^ (b ->1 c))))' v ((a ->1 c) ^ (b ->1 c))) = 1
219, 20lem3.1 443 . . . 4 ((a ->1 c) ^ (b ->1 c)) = ((a ->1 c) ^ (d v ((a ->1 c) ^ (b ->1 c))))
2221ax-r1 35 . . 3 ((a ->1 c) ^ (d v ((a ->1 c) ^ (b ->1 c)))) = ((a ->1 c) ^ (b ->1 c))
2322bile 142 . 2 ((a ->1 c) ^ (d v ((a ->1 c) ^ (b ->1 c)))) =< ((a ->1 c) ^ (b ->1 c))
24 lear 161 . 2 ((a ->1 c) ^ (b ->1 c)) =< (b ->1 c)
2523, 24letr 137 1 ((a ->1 c) ^ (d v ((a ->1 c) ^ (b ->1 c)))) =< (b ->1 c)
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v wo 6   ^ wa 7  1wt 8   ->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
This theorem is referenced by: (None)
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