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Theorem lem4.6.7 1101
Description: Equation 4.15 of [MegPav2000] p. 23. (Contributed by Roy F. Longton, 3-Jul-05.)
Hypothesis
Ref Expression
lem4.6.7.1 a' =< b
Assertion
Ref Expression
lem4.6.7 b =< (a ->1 b)
Proof of Theorem lem4.6.7
StepHypRef Expression
1 leid 148 . . . . . . 7 a =< a
21sklem 230 . . . . . 6 (a' v a) = 1
32ax-r1 35 . . . . 5 1 = (a' v a)
4 lem4.6.7.1 . . . . . . 7 a' =< b
54df-le2 131 . . . . . 6 (a' v b) = b
65ax-r1 35 . . . . 5 b = (a' v b)
73, 62an 79 . . . 4 (1 ^ b) = ((a' v a) ^ (a' v b))
8 ax-a3 32 . . . . . 6 ((b v a') v (a ^ b)) = (b v (a' v (a ^ b)))
98ax-r1 35 . . . . 5 (b v (a' v (a ^ b))) = ((b v a') v (a ^ b))
10 le1 146 . . . . . . . . 9 b =< 1
11 leid 148 . . . . . . . . 9 b =< b
1210, 11ler2an 173 . . . . . . . 8 b =< (1 ^ b)
13 le1 146 . . . . . . . . 9 a' =< 1
1413, 4ler2an 173 . . . . . . . 8 a' =< (1 ^ b)
1512, 14lel2or 170 . . . . . . 7 (b v a') =< (1 ^ b)
16 le1 146 . . . . . . . 8 a =< 1
1716leran 153 . . . . . . 7 (a ^ b) =< (1 ^ b)
1815, 17lel2or 170 . . . . . 6 ((b v a') v (a ^ b)) =< (1 ^ b)
19 leao2 163 . . . . . . 7 (1 ^ b) =< (b v a')
2019ler 149 . . . . . 6 (1 ^ b) =< ((b v a') v (a ^ b))
2118, 20lebi 145 . . . . 5 ((b v a') v (a ^ b)) = (1 ^ b)
229, 21ax-r2 36 . . . 4 (b v (a' v (a ^ b))) = (1 ^ b)
23 comid 187 . . . . . 6 a C a
2423comcom3 454 . . . . 5 a' C a
254lecom 180 . . . . 5 a' C b
2624, 25fh3 471 . . . 4 (a' v (a ^ b)) = ((a' v a) ^ (a' v b))
277, 22, 263tr1 63 . . 3 (b v (a' v (a ^ b))) = (a' v (a ^ b))
2827df-le1 130 . 2 b =< (a' v (a ^ b))
29 df-i1 44 . . 3 (a ->1 b) = (a' v (a ^ b))
3029ax-r1 35 . 2 (a' v (a ^ b)) = (a ->1 b)
3128, 30lbtr 139 1 b =< (a ->1 b)
Colors of variables: term
Syntax hints:   =< 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-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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by: (None)
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