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Theorem u2lemle2 716
Description: Dishkant implication to l.e.
Hypothesis
Ref Expression
u2lemle2.1 (a ->2 b) = 1
Assertion
Ref Expression
u2lemle2 a =< b
Proof of Theorem u2lemle2
StepHypRef Expression
1 ax-a2 31 . . . . . . 7 (b v (a' ^ b')) = ((a' ^ b') v b)
21lan 77 . . . . . 6 (a ^ (b v (a' ^ b'))) = (a ^ ((a' ^ b') v b))
3 coman1 185 . . . . . . . . 9 (a' ^ b') C a'
43comcom7 460 . . . . . . . 8 (a' ^ b') C a
5 coman2 186 . . . . . . . . 9 (a' ^ b') C b'
65comcom7 460 . . . . . . . 8 (a' ^ b') C b
74, 6fh2 470 . . . . . . 7 (a ^ ((a' ^ b') v b)) = ((a ^ (a' ^ b')) v (a ^ b))
8 ancom 74 . . . . . . . . . 10 ((a ^ a') ^ b') = (b' ^ (a ^ a'))
9 anass 76 . . . . . . . . . 10 ((a ^ a') ^ b') = (a ^ (a' ^ b'))
10 dff 101 . . . . . . . . . . . . 13 0 = (a ^ a')
1110ax-r1 35 . . . . . . . . . . . 12 (a ^ a') = 0
1211lan 77 . . . . . . . . . . 11 (b' ^ (a ^ a')) = (b' ^ 0)
13 an0 108 . . . . . . . . . . 11 (b' ^ 0) = 0
1412, 13ax-r2 36 . . . . . . . . . 10 (b' ^ (a ^ a')) = 0
158, 9, 143tr2 64 . . . . . . . . 9 (a ^ (a' ^ b')) = 0
1615ax-r5 38 . . . . . . . 8 ((a ^ (a' ^ b')) v (a ^ b)) = (0 v (a ^ b))
17 ax-a2 31 . . . . . . . 8 (0 v (a ^ b)) = ((a ^ b) v 0)
1816, 17ax-r2 36 . . . . . . 7 ((a ^ (a' ^ b')) v (a ^ b)) = ((a ^ b) v 0)
197, 18ax-r2 36 . . . . . 6 (a ^ ((a' ^ b') v b)) = ((a ^ b) v 0)
202, 19ax-r2 36 . . . . 5 (a ^ (b v (a' ^ b'))) = ((a ^ b) v 0)
2120ax-r1 35 . . . 4 ((a ^ b) v 0) = (a ^ (b v (a' ^ b')))
22 df-i2 45 . . . . . . 7 (a ->2 b) = (b v (a' ^ b'))
2322ax-r1 35 . . . . . 6 (b v (a' ^ b')) = (a ->2 b)
24 u2lemle2.1 . . . . . 6 (a ->2 b) = 1
2523, 24ax-r2 36 . . . . 5 (b v (a' ^ b')) = 1
2625lan 77 . . . 4 (a ^ (b v (a' ^ b'))) = (a ^ 1)
2721, 26ax-r2 36 . . 3 ((a ^ b) v 0) = (a ^ 1)
28 or0 102 . . 3 ((a ^ b) v 0) = (a ^ b)
29 an1 106 . . 3 (a ^ 1) = a
3027, 28, 293tr2 64 . 2 (a ^ b) = a
3130df2le1 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   ->2 wi2 13
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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  3vroa  831  imp3  841
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