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Theorem mhlem1 877
Description: Lemma for Marsden-Herman distributive law.
Hypotheses
Ref Expression
mhlem1.1 a C b
mhlem1.2 c C b
Assertion
Ref Expression
mhlem1 ((a v b) ^ (b' v c)) = ((a ^ b') v (b ^ c))
Proof of Theorem mhlem1
StepHypRef Expression
1 df-t 41 . . . . 5 1 = (b v b')
21lan 77 . . . 4 ((a v b) ^ 1) = ((a v b) ^ (b v b'))
3 an1 106 . . . 4 ((a v b) ^ 1) = (a v b)
4 comor2 462 . . . . . 6 (a v b) C b
54comcom2 183 . . . . . 6 (a v b) C b'
64, 5fh1 469 . . . . 5 ((a v b) ^ (b v b')) = (((a v b) ^ b) v ((a v b) ^ b'))
7 ax-a2 31 . . . . 5 (((a v b) ^ b) v ((a v b) ^ b')) = (((a v b) ^ b') v ((a v b) ^ b))
8 mhlem1.1 . . . . . . . . . 10 a C b
98comcom2 183 . . . . . . . . 9 a C b'
109comcom 453 . . . . . . . 8 b' C a
11 comid 187 . . . . . . . . 9 b C b
1211comcom3 454 . . . . . . . 8 b' C b
1310, 12fh1r 473 . . . . . . 7 ((a v b) ^ b') = ((a ^ b') v (b ^ b'))
14 dff 101 . . . . . . . . 9 0 = (b ^ b')
1514lor 70 . . . . . . . 8 ((a ^ b') v 0) = ((a ^ b') v (b ^ b'))
1615ax-r1 35 . . . . . . 7 ((a ^ b') v (b ^ b')) = ((a ^ b') v 0)
17 or0 102 . . . . . . 7 ((a ^ b') v 0) = (a ^ b')
1813, 16, 173tr 65 . . . . . 6 ((a v b) ^ b') = (a ^ b')
19 ancom 74 . . . . . . 7 ((a v b) ^ b) = (b ^ (a v b))
20 ax-a2 31 . . . . . . . 8 (a v b) = (b v a)
2120lan 77 . . . . . . 7 (b ^ (a v b)) = (b ^ (b v a))
22 anabs 121 . . . . . . 7 (b ^ (b v a)) = b
2319, 21, 223tr 65 . . . . . 6 ((a v b) ^ b) = b
2418, 232or 72 . . . . 5 (((a v b) ^ b') v ((a v b) ^ b)) = ((a ^ b') v b)
256, 7, 243tr 65 . . . 4 ((a v b) ^ (b v b')) = ((a ^ b') v b)
262, 3, 253tr2 64 . . 3 (a v b) = ((a ^ b') v b)
2726ran 78 . 2 ((a v b) ^ (b' v c)) = (((a ^ b') v b) ^ (b' v c))
28 comorr 184 . . . . 5 b' C (b' v c)
2928comcom6 459 . . . 4 b C (b' v c)
30 comanr2 465 . . . . 5 b' C (a ^ b')
3130comcom6 459 . . . 4 b C (a ^ b')
3229, 31fh2rc 480 . . 3 (((a ^ b') v b) ^ (b' v c)) = (((a ^ b') ^ (b' v c)) v (b ^ (b' v c)))
33 leao2 163 . . . . 5 (a ^ b') =< (b' v c)
3433df2le2 136 . . . 4 ((a ^ b') ^ (b' v c)) = (a ^ b')
3534ax-r5 38 . . 3 (((a ^ b') ^ (b' v c)) v (b ^ (b' v c))) = ((a ^ b') v (b ^ (b' v c)))
3632, 35ax-r2 36 . 2 (((a ^ b') v b) ^ (b' v c)) = ((a ^ b') v (b ^ (b' v c)))
3711comcom2 183 . . . . 5 b C b'
38 mhlem1.2 . . . . . 6 c C b
3938comcom 453 . . . . 5 b C c
4037, 39fh1 469 . . . 4 (b ^ (b' v c)) = ((b ^ b') v (b ^ c))
4114ax-r5 38 . . . . 5 (0 v (b ^ c)) = ((b ^ b') v (b ^ c))
4241ax-r1 35 . . . 4 ((b ^ b') v (b ^ c)) = (0 v (b ^ c))
43 or0r 103 . . . 4 (0 v (b ^ c)) = (b ^ c)
4440, 42, 433tr 65 . . 3 (b ^ (b' v c)) = (b ^ c)
4544lor 70 . 2 ((a ^ b') v (b ^ (b' v c))) = ((a ^ b') v (b ^ c))
4627, 36, 453tr 65 1 ((a v b) ^ (b' v c)) = ((a ^ b') v (b ^ c))
Colors of variables: term
Syntax hints:   = wb 1   C wc 3  'wn 4   v wo 6   ^ wa 7  1wt 8  0wf 9
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-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  mhlem2  878
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