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Theorem marsdenlem4 883
Description: Lemma for Marsden-Herman distributive law.
Hypotheses
Ref Expression
marsden.1 a C b
marsden.2 b C c
marsden.3 c C d
marsden.4 d C a
Assertion
Ref Expression
marsdenlem4 (((a' ^ b) v (a ^ d')) ^ (b' ^ d)) = 0
Proof of Theorem marsdenlem4
StepHypRef Expression
1 leao3 164 . . . . . 6 (b' ^ d) =< (a v b')
2 oran1 91 . . . . . 6 (a v b') = (a' ^ b)'
31, 2lbtr 139 . . . . 5 (b' ^ d) =< (a' ^ b)'
43lecom 180 . . . 4 (b' ^ d) C (a' ^ b)'
54comcom7 460 . . 3 (b' ^ d) C (a' ^ b)
6 leao4 165 . . . . . 6 (b' ^ d) =< (a' v d)
7 oran2 92 . . . . . 6 (a' v d) = (a ^ d')'
86, 7lbtr 139 . . . . 5 (b' ^ d) =< (a ^ d')'
98lecom 180 . . . 4 (b' ^ d) C (a ^ d')'
109comcom7 460 . . 3 (b' ^ d) C (a ^ d')
115, 10fh1r 473 . 2 (((a' ^ b) v (a ^ d')) ^ (b' ^ d)) = (((a' ^ b) ^ (b' ^ d)) v ((a ^ d') ^ (b' ^ d)))
12 ancom 74 . . . . 5 (b' ^ d) = (d ^ b')
1312lan 77 . . . 4 ((a' ^ b) ^ (b' ^ d)) = ((a' ^ b) ^ (d ^ b'))
14 an4 86 . . . 4 ((a' ^ b) ^ (d ^ b')) = ((a' ^ d) ^ (b ^ b'))
15 dff 101 . . . . . . 7 0 = (b ^ b')
1615lan 77 . . . . . 6 ((a' ^ d) ^ 0) = ((a' ^ d) ^ (b ^ b'))
1716ax-r1 35 . . . . 5 ((a' ^ d) ^ (b ^ b')) = ((a' ^ d) ^ 0)
18 an0 108 . . . . 5 ((a' ^ d) ^ 0) = 0
1917, 18ax-r2 36 . . . 4 ((a' ^ d) ^ (b ^ b')) = 0
2013, 14, 193tr 65 . . 3 ((a' ^ b) ^ (b' ^ d)) = 0
21 an4 86 . . . 4 ((a ^ d') ^ (b' ^ d)) = ((a ^ b') ^ (d' ^ d))
22 ancom 74 . . . . . 6 (d' ^ d) = (d ^ d')
23 dff 101 . . . . . . 7 0 = (d ^ d')
2423ax-r1 35 . . . . . 6 (d ^ d') = 0
2522, 24ax-r2 36 . . . . 5 (d' ^ d) = 0
2625lan 77 . . . 4 ((a ^ b') ^ (d' ^ d)) = ((a ^ b') ^ 0)
27 an0 108 . . . 4 ((a ^ b') ^ 0) = 0
2821, 26, 273tr 65 . . 3 ((a ^ d') ^ (b' ^ d)) = 0
2920, 282or 72 . 2 (((a' ^ b) ^ (b' ^ d)) v ((a ^ d') ^ (b' ^ d))) = (0 v 0)
30 or0 102 . 2 (0 v 0) = 0
3111, 29, 303tr 65 1 (((a' ^ b) v (a ^ d')) ^ (b' ^ d)) = 0
Colors of variables: term
Syntax hints:   = wb 1   C wc 3  'wn 4   v wo 6   ^ wa 7  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: (None)
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