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Theorem mhlemlem1 874
Description: Lemma for Lemma 7.1 of Kalmbach, p. 91.
Hypothesis
Ref Expression
mhlem.1 (a v b) =< (c v d)'
Assertion
Ref Expression
mhlemlem1 (((a v b) v c) ^ (a v (c v d))) = (a v c)
Proof of Theorem mhlemlem1
StepHypRef Expression
1 leo 158 . . . . 5 a =< (a v b)
21ler 149 . . . 4 a =< ((a v b) v c)
32lecom 180 . . 3 a C ((a v b) v c)
4 mhlem.1 . . . . . 6 (a v b) =< (c v d)'
51, 4letr 137 . . . . 5 a =< (c v d)'
65lecom 180 . . . 4 a C (c v d)'
76comcom7 460 . . 3 a C (c v d)
83, 7fh2 470 . 2 (((a v b) v c) ^ (a v (c v d))) = ((((a v b) v c) ^ a) v (((a v b) v c) ^ (c v d)))
9 ancom 74 . . . 4 (((a v b) v c) ^ a) = (a ^ ((a v b) v c))
10 ax-a3 32 . . . . 5 ((a v b) v c) = (a v (b v c))
1110lan 77 . . . 4 (a ^ ((a v b) v c)) = (a ^ (a v (b v c)))
12 anabs 121 . . . 4 (a ^ (a v (b v c))) = a
139, 11, 123tr 65 . . 3 (((a v b) v c) ^ a) = a
14 comor1 461 . . . . 5 (c v d) C c
154lecon3 157 . . . . . . 7 (c v d) =< (a v b)'
1615lecom 180 . . . . . 6 (c v d) C (a v b)'
1716comcom7 460 . . . . 5 (c v d) C (a v b)
1814, 17fh1rc 479 . . . 4 (((a v b) v c) ^ (c v d)) = (((a v b) ^ (c v d)) v (c ^ (c v d)))
194ortha 438 . . . . 5 ((a v b) ^ (c v d)) = 0
20 anabs 121 . . . . 5 (c ^ (c v d)) = c
2119, 202or 72 . . . 4 (((a v b) ^ (c v d)) v (c ^ (c v d))) = (0 v c)
22 or0r 103 . . . 4 (0 v c) = c
2318, 21, 223tr 65 . . 3 (((a v b) v c) ^ (c v d)) = c
2413, 232or 72 . 2 ((((a v b) v c) ^ a) v (((a v b) v c) ^ (c v d))) = (a v c)
258, 24ax-r2 36 1 (((a v b) v c) ^ (a v (c v d))) = (a v c)
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  '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:  mhlemlem2  875  mhlem  876
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