Theorem mhlemlem1 856

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

Step	Hyp	Ref	Expression
1		leo 150	. . . . 5 a =< (a v b)
2	1	ler 141	. . . 4 a =< ((a v b) v c)
3	2	lecom 172	. . 3 a C ((a v b) v c)
4		mhlem.1	. . . . . 6 (a v b) =< (c v d)_|_
5	1, 4	letr 129	. . . . 5 a =< (c v d)_|_
6	5	lecom 172	. . . 4 a C (c v d)_|_
7	6	comcom7 442	. . 3 a C (c v d)
8	3, 7	fh2 452	. 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 68	. . . 4 (((a v b) v c) ^ a) = (a ^ ((a v b) v c))
10		ax-a3 31	. . . . 5 ((a v b) v c) = (a v (b v c))
11	10	lan 70	. . . 4 (a ^ ((a v b) v c)) = (a ^ (a v (b v c)))
12		a5c 113	. . . 4 (a ^ (a v (b v c))) = a
13	9, 11, 12	3tr 62	. . 3 (((a v b) v c) ^ a) = a
14		comor1 443	. . . . 5 (c v d) C c
15	4	lecon3 149	. . . . . . 7 (c v d) =< (a v b)_|_
16	15	lecom 172	. . . . . 6 (c v d) C (a v b)_|_
17	16	comcom7 442	. . . . 5 (c v d) C (a v b)
18	14, 17	fh1rc 461	. . . 4 (((a v b) v c) ^ (c v d)) = (((a v b) ^ (c v d)) v (c ^ (c v d)))
19	4	ortha 420	. . . . 5 ((a v b) ^ (c v d)) = 0
20		a5c 113	. . . . 5 (c ^ (c v d)) = c
21	19, 20	2or 67	. . . 4 (((a v b) ^ (c v d)) v (c ^ (c v d))) = (0 v c)
22		or0r 95	. . . 4 (0 v c) = c
23	18, 21, 22	3tr 62	. . 3 (((a v b) v c) ^ (c v d)) = c
24	13, 23	2or 67	. 2 ((((a v b) v c) ^ a) v (((a v b) v c) ^ (c v d))) = (a v c)
25	8, 24	ax-r2 35	1 (((a v b) v c) ^ (a v (c v d))) = (a v c)

Syntax hints:   = wb 1   =< wle 2  _|_wn 4   v wo 6   ^ wa 7  0wf 10

This theorem is referenced by:  mhlemlem2 857  mhlem 858

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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