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Theorem mlalem 832
Description: Lemma for Mladen's OML.
Assertion
Ref Expression
mlalem ((a == b) ^ (b ->1 c)) =< (a ->1 c)
Proof of Theorem mlalem
StepHypRef Expression
1 comanr2 465 . . . . . 6 b C (a ^ b)
21comcom3 454 . . . . 5 b' C (a ^ b)
3 comanr1 464 . . . . . 6 b C (b ^ c)
43comcom3 454 . . . . 5 b' C (b ^ c)
52, 4fh2 470 . . . 4 ((a ^ b) ^ (b' v (b ^ c))) = (((a ^ b) ^ b') v ((a ^ b) ^ (b ^ c)))
6 anass 76 . . . . . . 7 ((a ^ b) ^ b') = (a ^ (b ^ b'))
7 dff 101 . . . . . . . . 9 0 = (b ^ b')
87ax-r1 35 . . . . . . . 8 (b ^ b') = 0
98lan 77 . . . . . . 7 (a ^ (b ^ b')) = (a ^ 0)
10 an0 108 . . . . . . 7 (a ^ 0) = 0
116, 9, 103tr 65 . . . . . 6 ((a ^ b) ^ b') = 0
12 le0 147 . . . . . 6 0 =< (a' v (a ^ c))
1311, 12bltr 138 . . . . 5 ((a ^ b) ^ b') =< (a' v (a ^ c))
14 anass 76 . . . . . . 7 ((a ^ b) ^ (b ^ c)) = (a ^ (b ^ (b ^ c)))
15 an12 81 . . . . . . 7 (a ^ (b ^ (b ^ c))) = (b ^ (a ^ (b ^ c)))
16 anass 76 . . . . . . . . 9 ((b ^ a) ^ (b ^ c)) = (b ^ (a ^ (b ^ c)))
1716ax-r1 35 . . . . . . . 8 (b ^ (a ^ (b ^ c))) = ((b ^ a) ^ (b ^ c))
18 an4 86 . . . . . . . 8 ((b ^ a) ^ (b ^ c)) = ((b ^ b) ^ (a ^ c))
1917, 18ax-r2 36 . . . . . . 7 (b ^ (a ^ (b ^ c))) = ((b ^ b) ^ (a ^ c))
2014, 15, 193tr 65 . . . . . 6 ((a ^ b) ^ (b ^ c)) = ((b ^ b) ^ (a ^ c))
21 lear 161 . . . . . . 7 ((b ^ b) ^ (a ^ c)) =< (a ^ c)
22 leor 159 . . . . . . 7 (a ^ c) =< (a' v (a ^ c))
2321, 22letr 137 . . . . . 6 ((b ^ b) ^ (a ^ c)) =< (a' v (a ^ c))
2420, 23bltr 138 . . . . 5 ((a ^ b) ^ (b ^ c)) =< (a' v (a ^ c))
2513, 24lel2or 170 . . . 4 (((a ^ b) ^ b') v ((a ^ b) ^ (b ^ c))) =< (a' v (a ^ c))
265, 25bltr 138 . . 3 ((a ^ b) ^ (b' v (b ^ c))) =< (a' v (a ^ c))
27 anass 76 . . . 4 ((a' ^ b') ^ (b' v (b ^ c))) = (a' ^ (b' ^ (b' v (b ^ c))))
28 lea 160 . . . . 5 (a' ^ (b' ^ (b' v (b ^ c)))) =< a'
29 leo 158 . . . . 5 a' =< (a' v (a ^ c))
3028, 29letr 137 . . . 4 (a' ^ (b' ^ (b' v (b ^ c)))) =< (a' v (a ^ c))
3127, 30bltr 138 . . 3 ((a' ^ b') ^ (b' v (b ^ c))) =< (a' v (a ^ c))
3226, 31lel2or 170 . 2 (((a ^ b) ^ (b' v (b ^ c))) v ((a' ^ b') ^ (b' v (b ^ c)))) =< (a' v (a ^ c))
33 dfb 94 . . . 4 (a == b) = ((a ^ b) v (a' ^ b'))
34 df-i1 44 . . . 4 (b ->1 c) = (b' v (b ^ c))
3533, 342an 79 . . 3 ((a == b) ^ (b ->1 c)) = (((a ^ b) v (a' ^ b')) ^ (b' v (b ^ c)))
36 lear 161 . . . . . 6 (a' ^ b') =< b'
37 leo 158 . . . . . 6 b' =< (b' v (b ^ c))
3836, 37letr 137 . . . . 5 (a' ^ b') =< (b' v (b ^ c))
3938lecom 180 . . . 4 (a' ^ b') C (b' v (b ^ c))
40 coman1 185 . . . . . . 7 (a' ^ b') C a'
41 coman2 186 . . . . . . 7 (a' ^ b') C b'
4240, 41com2or 483 . . . . . 6 (a' ^ b') C (a' v b')
43 oran3 93 . . . . . 6 (a' v b') = (a ^ b)'
4442, 43cbtr 182 . . . . 5 (a' ^ b') C (a ^ b)'
4544comcom7 460 . . . 4 (a' ^ b') C (a ^ b)
4639, 45fh2rc 480 . . 3 (((a ^ b) v (a' ^ b')) ^ (b' v (b ^ c))) = (((a ^ b) ^ (b' v (b ^ c))) v ((a' ^ b') ^ (b' v (b ^ c))))
4735, 46ax-r2 36 . 2 ((a == b) ^ (b ->1 c)) = (((a ^ b) ^ (b' v (b ^ c))) v ((a' ^ b') ^ (b' v (b ^ c))))
48 df-i1 44 . 2 (a ->1 c) = (a' v (a ^ c))
4932, 47, 48le3tr1 140 1 ((a == b) ^ (b ->1 c)) =< (a ->1 c)
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   == tb 5   v wo 6   ^ wa 7  0wf 9   ->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:  mlaoml  833
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