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Theorem u5lemle2 719
Description: Relevance implication to l.e.
Hypothesis
Ref Expression
u5lemle2.1 (a ->5 b) = 1
Assertion
Ref Expression
u5lemle2 a =< b
Proof of Theorem u5lemle2
StepHypRef Expression
1 df-i5 48 . . . . . 6 (a ->5 b) = (((a ^ b) v (a' ^ b)) v (a' ^ b'))
21ax-r1 35 . . . . 5 (((a ^ b) v (a' ^ b)) v (a' ^ b')) = (a ->5 b)
3 u5lemle2.1 . . . . 5 (a ->5 b) = 1
42, 3ax-r2 36 . . . 4 (((a ^ b) v (a' ^ b)) v (a' ^ b')) = 1
54lan 77 . . 3 (a ^ (((a ^ b) v (a' ^ b)) v (a' ^ b'))) = (a ^ 1)
6 comanr1 464 . . . . . 6 a C (a ^ b)
7 comanr1 464 . . . . . . 7 a' C (a' ^ b)
87comcom6 459 . . . . . 6 a C (a' ^ b)
96, 8com2or 483 . . . . 5 a C ((a ^ b) v (a' ^ b))
10 comanr1 464 . . . . . 6 a' C (a' ^ b')
1110comcom6 459 . . . . 5 a C (a' ^ b')
129, 11fh1 469 . . . 4 (a ^ (((a ^ b) v (a' ^ b)) v (a' ^ b'))) = ((a ^ ((a ^ b) v (a' ^ b))) v (a ^ (a' ^ b')))
136, 8fh1 469 . . . . . . 7 (a ^ ((a ^ b) v (a' ^ b))) = ((a ^ (a ^ b)) v (a ^ (a' ^ b)))
14 anass 76 . . . . . . . . . . 11 ((a ^ a) ^ b) = (a ^ (a ^ b))
1514ax-r1 35 . . . . . . . . . 10 (a ^ (a ^ b)) = ((a ^ a) ^ b)
16 anidm 111 . . . . . . . . . . 11 (a ^ a) = a
1716ran 78 . . . . . . . . . 10 ((a ^ a) ^ b) = (a ^ b)
1815, 17ax-r2 36 . . . . . . . . 9 (a ^ (a ^ b)) = (a ^ b)
19 ancom 74 . . . . . . . . . 10 ((a ^ a') ^ b) = (b ^ (a ^ a'))
20 anass 76 . . . . . . . . . 10 ((a ^ a') ^ b) = (a ^ (a' ^ b))
21 dff 101 . . . . . . . . . . . . 13 0 = (a ^ a')
2221ax-r1 35 . . . . . . . . . . . 12 (a ^ a') = 0
2322lan 77 . . . . . . . . . . 11 (b ^ (a ^ a')) = (b ^ 0)
24 an0 108 . . . . . . . . . . 11 (b ^ 0) = 0
2523, 24ax-r2 36 . . . . . . . . . 10 (b ^ (a ^ a')) = 0
2619, 20, 253tr2 64 . . . . . . . . 9 (a ^ (a' ^ b)) = 0
2718, 262or 72 . . . . . . . 8 ((a ^ (a ^ b)) v (a ^ (a' ^ b))) = ((a ^ b) v 0)
28 or0 102 . . . . . . . 8 ((a ^ b) v 0) = (a ^ b)
2927, 28ax-r2 36 . . . . . . 7 ((a ^ (a ^ b)) v (a ^ (a' ^ b))) = (a ^ b)
3013, 29ax-r2 36 . . . . . 6 (a ^ ((a ^ b) v (a' ^ b))) = (a ^ b)
31 ancom 74 . . . . . . 7 ((a ^ a') ^ b') = (b' ^ (a ^ a'))
32 anass 76 . . . . . . 7 ((a ^ a') ^ b') = (a ^ (a' ^ b'))
3321lan 77 . . . . . . . . 9 (b' ^ 0) = (b' ^ (a ^ a'))
3433ax-r1 35 . . . . . . . 8 (b' ^ (a ^ a')) = (b' ^ 0)
35 an0 108 . . . . . . . 8 (b' ^ 0) = 0
3634, 35ax-r2 36 . . . . . . 7 (b' ^ (a ^ a')) = 0
3731, 32, 363tr2 64 . . . . . 6 (a ^ (a' ^ b')) = 0
3830, 372or 72 . . . . 5 ((a ^ ((a ^ b) v (a' ^ b))) v (a ^ (a' ^ b'))) = ((a ^ b) v 0)
3938, 28ax-r2 36 . . . 4 ((a ^ ((a ^ b) v (a' ^ b))) v (a ^ (a' ^ b'))) = (a ^ b)
4012, 39ax-r2 36 . . 3 (a ^ (((a ^ b) v (a' ^ b)) v (a' ^ b'))) = (a ^ b)
41 an1 106 . . 3 (a ^ 1) = a
425, 40, 413tr2 64 . 2 (a ^ b) = a
4342df2le1 135 1 a =< b
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v wo 6   ^ wa 7  1wt 8  0wf 9   ->5 wi5 16
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-i5 48  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by: (None)
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