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Theorem vneulem6 1134
Description: Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96
Hypothesis
Ref Expression
vneulem6.1 ((a v b) ^ (c v d)) = 0
Assertion
Ref Expression
vneulem6 (((a v b) v d) ^ ((b v c) v d)) = ((c ^ a) v (b v d))

Proof of Theorem vneulem6
StepHypRef Expression
1 orcom 73 . . . . . . 7 (a v b) = (b v a)
21ror 71 . . . . . 6 ((a v b) v d) = ((b v a) v d)
3 or32 82 . . . . . 6 ((b v a) v d) = ((b v d) v a)
42, 3tr 62 . . . . 5 ((a v b) v d) = ((b v d) v a)
5 or32 82 . . . . 5 ((b v c) v d) = ((b v d) v c)
64, 52an 79 . . . 4 (((a v b) v d) ^ ((b v c) v d)) = (((b v d) v a) ^ ((b v d) v c))
7 vneulem5 1133 . . . 4 (((b v d) v a) ^ ((b v d) v c)) = ((b v d) v (((b v d) v a) ^ c))
86, 7ax-r2 36 . . 3 (((a v b) v d) ^ ((b v c) v d)) = ((b v d) v (((b v d) v a) ^ c))
9 leor 159 . . . 4 (b v d) =< ((c ^ a) v (b v d))
10 or32 82 . . . . . . 7 ((b v d) v a) = ((b v a) v d)
1110ran 78 . . . . . 6 (((b v d) v a) ^ c) = (((b v a) v d) ^ c)
12 ax-a2 31 . . . . . . . . 9 (b v a) = (a v b)
13 ax-a2 31 . . . . . . . . 9 (d v c) = (c v d)
1412, 132an 79 . . . . . . . 8 ((b v a) ^ (d v c)) = ((a v b) ^ (c v d))
15 vneulem6.1 . . . . . . . 8 ((a v b) ^ (c v d)) = 0
1614, 15tr 62 . . . . . . 7 ((b v a) ^ (d v c)) = 0
1716vneulem4 1132 . . . . . 6 (((b v a) v d) ^ c) = (d ^ c)
1811, 17tr 62 . . . . 5 (((b v d) v a) ^ c) = (d ^ c)
19 leao3 164 . . . . . 6 (d ^ c) =< (b v d)
2019lerr 150 . . . . 5 (d ^ c) =< ((c ^ a) v (b v d))
2118, 20bltr 138 . . . 4 (((b v d) v a) ^ c) =< ((c ^ a) v (b v d))
229, 21lel2or 170 . . 3 ((b v d) v (((b v d) v a) ^ c)) =< ((c ^ a) v (b v d))
238, 22bltr 138 . 2 (((a v b) v d) ^ ((b v c) v d)) =< ((c ^ a) v (b v d))
24 leao2 163 . . . . 5 (c ^ a) =< (a v b)
2524ler 149 . . . 4 (c ^ a) =< ((a v b) v d)
26 leor 159 . . . . 5 b =< (a v b)
2726leror 152 . . . 4 (b v d) =< ((a v b) v d)
2825, 27lel2or 170 . . 3 ((c ^ a) v (b v d)) =< ((a v b) v d)
29 leao3 164 . . . . 5 (c ^ a) =< (b v c)
3029ler 149 . . . 4 (c ^ a) =< ((b v c) v d)
31 leo 158 . . . . 5 b =< (b v c)
3231leror 152 . . . 4 (b v d) =< ((b v c) v d)
3330, 32lel2or 170 . . 3 ((c ^ a) v (b v d)) =< ((b v c) v d)
3428, 33ler2an 173 . 2 ((c ^ a) v (b v d)) =< (((a v b) v d) ^ ((b v c) v d))
3523, 34lebi 145 1 (((a v b) v d) ^ ((b v c) v d)) = ((c ^ a) v (b v d))
Colors of variables: term
Syntax hints:   = wb 1   v wo 6   ^ wa 7  0wf 9
This theorem is referenced by:  vneulem8  1136
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-ml 1120
This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131
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