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Theorem lem3.3.7i4e1 1069
Description: Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to 4, and this is the first part of the equation. (Contributed by Roy F. Longton, 3-Jul-05.)
Assertion
Ref Expression
lem3.3.7i4e1 (a ->4 (a ^ b)) = (a ==4 (a ^ b))
Proof of Theorem lem3.3.7i4e1
StepHypRef Expression
1 lear 161 . . . . . 6 (a ^ (a ^ b)) =< (a ^ b)
2 lea 160 . . . . . . 7 (a ^ b) =< a
3 leid 148 . . . . . . 7 (a ^ b) =< (a ^ b)
42, 3ler2an 173 . . . . . 6 (a ^ b) =< (a ^ (a ^ b))
51, 4lebi 145 . . . . 5 (a ^ (a ^ b)) = (a ^ b)
65ax-r5 38 . . . 4 ((a ^ (a ^ b)) v (a' ^ (a ^ b))) = ((a ^ b) v (a' ^ (a ^ b)))
76ax-r5 38 . . 3 (((a ^ (a ^ b)) v (a' ^ (a ^ b))) v ((a' v (a ^ b)) ^ (a ^ b)')) = (((a ^ b) v (a' ^ (a ^ b))) v ((a' v (a ^ b)) ^ (a ^ b)'))
82lecon 154 . . . . . . . 8 a' =< (a ^ b)'
98ortha 438 . . . . . . 7 (a' ^ (a ^ b)) = 0
109lor 70 . . . . . 6 ((a ^ b) v (a' ^ (a ^ b))) = ((a ^ b) v 0)
1110ax-r5 38 . . . . 5 (((a ^ b) v (a' ^ (a ^ b))) v ((a' v (a ^ b)) ^ (a ^ b)')) = (((a ^ b) v 0) v ((a' v (a ^ b)) ^ (a ^ b)'))
12 or0 102 . . . . . 6 ((a ^ b) v 0) = (a ^ b)
1312ax-r5 38 . . . . 5 (((a ^ b) v 0) v ((a' v (a ^ b)) ^ (a ^ b)')) = ((a ^ b) v ((a' v (a ^ b)) ^ (a ^ b)'))
14 leor 159 . . . . . . 7 (a ^ b) =< (a' v (a ^ b))
15 lea 160 . . . . . . 7 ((a' v (a ^ b)) ^ (a ^ b)') =< (a' v (a ^ b))
1614, 15lel2or 170 . . . . . 6 ((a ^ b) v ((a' v (a ^ b)) ^ (a ^ b)')) =< (a' v (a ^ b))
17 leo 158 . . . . . . . . 9 a' =< (a' v (a ^ b))
1817, 8ler2an 173 . . . . . . . 8 a' =< ((a' v (a ^ b)) ^ (a ^ b)')
1918lerr 150 . . . . . . 7 a' =< ((a ^ b) v ((a' v (a ^ b)) ^ (a ^ b)'))
20 leo 158 . . . . . . 7 (a ^ b) =< ((a ^ b) v ((a' v (a ^ b)) ^ (a ^ b)'))
2119, 20lel2or 170 . . . . . 6 (a' v (a ^ b)) =< ((a ^ b) v ((a' v (a ^ b)) ^ (a ^ b)'))
2216, 21lebi 145 . . . . 5 ((a ^ b) v ((a' v (a ^ b)) ^ (a ^ b)')) = (a' v (a ^ b))
2311, 13, 223tr 65 . . . 4 (((a ^ b) v (a' ^ (a ^ b))) v ((a' v (a ^ b)) ^ (a ^ b)')) = (a' v (a ^ b))
24 an1 106 . . . . 5 ((a' v (a ^ b)) ^ 1) = (a' v (a ^ b))
2524ax-r1 35 . . . 4 (a' v (a ^ b)) = ((a' v (a ^ b)) ^ 1)
263sklem 230 . . . . . 6 ((a ^ b)' v (a ^ b)) = 1
2726ax-r1 35 . . . . 5 1 = ((a ^ b)' v (a ^ b))
2827lan 77 . . . 4 ((a' v (a ^ b)) ^ 1) = ((a' v (a ^ b)) ^ ((a ^ b)' v (a ^ b)))
2923, 25, 283tr 65 . . 3 (((a ^ b) v (a' ^ (a ^ b))) v ((a' v (a ^ b)) ^ (a ^ b)')) = ((a' v (a ^ b)) ^ ((a ^ b)' v (a ^ b)))
304, 1lebi 145 . . . . 5 (a ^ b) = (a ^ (a ^ b))
3130lor 70 . . . 4 ((a ^ b)' v (a ^ b)) = ((a ^ b)' v (a ^ (a ^ b)))
3231lan 77 . . 3 ((a' v (a ^ b)) ^ ((a ^ b)' v (a ^ b))) = ((a' v (a ^ b)) ^ ((a ^ b)' v (a ^ (a ^ b))))
337, 29, 323tr 65 . 2 (((a ^ (a ^ b)) v (a' ^ (a ^ b))) v ((a' v (a ^ b)) ^ (a ^ b)')) = ((a' v (a ^ b)) ^ ((a ^ b)' v (a ^ (a ^ b))))
34 df-i4 47 . 2 (a ->4 (a ^ b)) = (((a ^ (a ^ b)) v (a' ^ (a ^ b))) v ((a' v (a ^ b)) ^ (a ^ b)'))
35 df-id4 53 . 2 (a ==4 (a ^ b)) = ((a' v (a ^ b)) ^ ((a ^ b)' v (a ^ (a ^ b))))
3633, 34, 353tr1 63 1 (a ->4 (a ^ b)) = (a ==4 (a ^ b))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8  0wf 9   ->4 wi4 15   ==4 wid4 21
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
This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i4 47  df-id4 53  df-le1 130  df-le2 131
This theorem is referenced by: (None)
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