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Theorem test2 803
Description: Part of an attempt to crack a potential Kalmbach axiom.
Assertion
Ref Expression
test2 (a v b) =< ((a == b)' v ((c v (a ^ b)) ^ (c' v (a ^ b))))
Proof of Theorem test2
StepHypRef Expression
1 dfnb 95 . . . . 5 (a == b)' = ((a v b) ^ (a' v b'))
2 anidm 111 . . . . 5 ((a ^ b) ^ (a ^ b)) = (a ^ b)
31, 22or 72 . . . 4 ((a == b)' v ((a ^ b) ^ (a ^ b))) = (((a v b) ^ (a' v b')) v (a ^ b))
4 comor1 461 . . . . . . 7 (a v b) C a
5 comor2 462 . . . . . . 7 (a v b) C b
64, 5com2an 484 . . . . . 6 (a v b) C (a ^ b)
74comcom2 183 . . . . . . 7 (a v b) C a'
85comcom2 183 . . . . . . 7 (a v b) C b'
97, 8com2or 483 . . . . . 6 (a v b) C (a' v b')
106, 9fh4r 476 . . . . 5 (((a v b) ^ (a' v b')) v (a ^ b)) = (((a v b) v (a ^ b)) ^ ((a' v b') v (a ^ b)))
11 ax-a2 31 . . . . . . . 8 ((a v b) v (a ^ b)) = ((a ^ b) v (a v b))
12 lea 160 . . . . . . . . . 10 (a ^ b) =< a
13 leo 158 . . . . . . . . . 10 a =< (a v b)
1412, 13letr 137 . . . . . . . . 9 (a ^ b) =< (a v b)
1514df-le2 131 . . . . . . . 8 ((a ^ b) v (a v b)) = (a v b)
1611, 15ax-r2 36 . . . . . . 7 ((a v b) v (a ^ b)) = (a v b)
17 df-a 40 . . . . . . . . 9 (a ^ b) = (a' v b')'
1817lor 70 . . . . . . . 8 ((a' v b') v (a ^ b)) = ((a' v b') v (a' v b')')
19 df-t 41 . . . . . . . . 9 1 = ((a' v b') v (a' v b')')
2019ax-r1 35 . . . . . . . 8 ((a' v b') v (a' v b')') = 1
2118, 20ax-r2 36 . . . . . . 7 ((a' v b') v (a ^ b)) = 1
2216, 212an 79 . . . . . 6 (((a v b) v (a ^ b)) ^ ((a' v b') v (a ^ b))) = ((a v b) ^ 1)
23 an1 106 . . . . . 6 ((a v b) ^ 1) = (a v b)
2422, 23ax-r2 36 . . . . 5 (((a v b) v (a ^ b)) ^ ((a' v b') v (a ^ b))) = (a v b)
2510, 24ax-r2 36 . . . 4 (((a v b) ^ (a' v b')) v (a ^ b)) = (a v b)
263, 25ax-r2 36 . . 3 ((a == b)' v ((a ^ b) ^ (a ^ b))) = (a v b)
2726ax-r1 35 . 2 (a v b) = ((a == b)' v ((a ^ b) ^ (a ^ b)))
28 leor 159 . . . 4 (a ^ b) =< (c v (a ^ b))
29 leor 159 . . . 4 (a ^ b) =< (c' v (a ^ b))
3028, 29le2an 169 . . 3 ((a ^ b) ^ (a ^ b)) =< ((c v (a ^ b)) ^ (c' v (a ^ b)))
3130lelor 166 . 2 ((a == b)' v ((a ^ b) ^ (a ^ b))) =< ((a == b)' v ((c v (a ^ b)) ^ (c' v (a ^ b))))
3227, 31bltr 138 1 (a v b) =< ((a == b)' v ((c v (a ^ b)) ^ (c' v (a ^ b))))
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   == tb 5   v wo 6   ^ wa 7  1wt 8
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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