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Theorem marsdenlem3 882
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
marsdenlem3 (((b' ^ c) v (c' ^ d)) ^ (b ^ d')) = 0
Proof of Theorem marsdenlem3
StepHypRef Expression
1 lea 160 . . . . . . . 8 (b ^ d') =< b
21lecon 154 . . . . . . 7 b' =< (b ^ d')'
32lel 151 . . . . . 6 (b' ^ c) =< (b ^ d')'
43lecom 180 . . . . 5 (b' ^ c) C (b ^ d')'
54comcom7 460 . . . 4 (b' ^ c) C (b ^ d')
65comcom 453 . . 3 (b ^ d') C (b' ^ c)
7 lear 161 . . . . . . . 8 (c' ^ d) =< d
87lerr 150 . . . . . . 7 (c' ^ d) =< (b' v d)
9 oran2 92 . . . . . . 7 (b' v d) = (b ^ d')'
108, 9lbtr 139 . . . . . 6 (c' ^ d) =< (b ^ d')'
1110lecom 180 . . . . 5 (c' ^ d) C (b ^ d')'
1211comcom7 460 . . . 4 (c' ^ d) C (b ^ d')
1312comcom 453 . . 3 (b ^ d') C (c' ^ d)
146, 13fh1r 473 . 2 (((b' ^ c) v (c' ^ d)) ^ (b ^ d')) = (((b' ^ c) ^ (b ^ d')) v ((c' ^ d) ^ (b ^ d')))
15 an4 86 . . . 4 ((b' ^ c) ^ (b ^ d')) = ((b' ^ b) ^ (c ^ d'))
16 ancom 74 . . . . . 6 (b' ^ b) = (b ^ b')
17 dff 101 . . . . . . 7 0 = (b ^ b')
1817ax-r1 35 . . . . . 6 (b ^ b') = 0
1916, 18ax-r2 36 . . . . 5 (b' ^ b) = 0
2019ran 78 . . . 4 ((b' ^ b) ^ (c ^ d')) = (0 ^ (c ^ d'))
21 an0r 109 . . . 4 (0 ^ (c ^ d')) = 0
2215, 20, 213tr 65 . . 3 ((b' ^ c) ^ (b ^ d')) = 0
23 an4 86 . . . 4 ((c' ^ d) ^ (b ^ d')) = ((c' ^ b) ^ (d ^ d'))
24 dff 101 . . . . . 6 0 = (d ^ d')
2524ax-r1 35 . . . . 5 (d ^ d') = 0
2625lan 77 . . . 4 ((c' ^ b) ^ (d ^ d')) = ((c' ^ b) ^ 0)
27 an0 108 . . . 4 ((c' ^ b) ^ 0) = 0
2823, 26, 273tr 65 . . 3 ((c' ^ d) ^ (b ^ d')) = 0
2922, 282or 72 . 2 (((b' ^ c) ^ (b ^ d')) v ((c' ^ d) ^ (b ^ d'))) = (0 v 0)
30 or0 102 . 2 (0 v 0) = 0
3114, 29, 303tr 65 1 (((b' ^ c) v (c' ^ 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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