Theorem mhlem2 860

Description: Lemma for Marsden-Herman distributive law.

Hypotheses

Ref	Expression
mh.1	a C c
mh.2	a C d
mh.3	b C c
mh.4	b C d

Assertion

Ref	Expression
mhlem2	(((a v c) ^ (c_|_ v b_|_)) ^ ((b v d) ^ (a_|_ v d_|_))) = (((a ^ c_|_) ^ (b ^ d_|_)) v ((c ^ b_|_) ^ (d ^ a_|_)))

Proof of Theorem mhlem2

Step	Hyp	Ref	Expression
1		mh.1	. . . 4 a C c
2		mh.3	. . . . 5 b C c
3	2	comcom3 436	. . . 4 b_|_ C c
4	1, 3	mhlem1 859	. . 3 ((a v c) ^ (c_|_ v b_|_)) = ((a ^ c_|_) v (c ^ b_|_))
5		ax-a2 30	. . . . 5 (a_|_ v d_|_) = (d_|_ v a_|_)
6	5	lan 70	. . . 4 ((b v d) ^ (a_|_ v d_|_)) = ((b v d) ^ (d_|_ v a_|_))
7		mh.4	. . . . 5 b C d
8		mh.2	. . . . . 6 a C d
9	8	comcom3 436	. . . . 5 a_|_ C d
10	7, 9	mhlem1 859	. . . 4 ((b v d) ^ (d_|_ v a_|_)) = ((b ^ d_|_) v (d ^ a_|_))
11	6, 10	ax-r2 35	. . 3 ((b v d) ^ (a_|_ v d_|_)) = ((b ^ d_|_) v (d ^ a_|_))
12	4, 11	2an 72	. 2 (((a v c) ^ (c_|_ v b_|_)) ^ ((b v d) ^ (a_|_ v d_|_))) = (((a ^ c_|_) v (c ^ b_|_)) ^ ((b ^ d_|_) v (d ^ a_|_)))
13		leao2 155	. . . . . 6 (a ^ c_|_) =< (c_|_ v b)
14		leao3 156	. . . . . 6 (a ^ c_|_) =< (d_|_ v a)
15	13, 14	ler2an 165	. . . . 5 (a ^ c_|_) =< ((c_|_ v b) ^ (d_|_ v a))
16		leao3 156	. . . . . 6 (b ^ d_|_) =< (c_|_ v b)
17		leao2 155	. . . . . 6 (b ^ d_|_) =< (d_|_ v a)
18	16, 17	ler2an 165	. . . . 5 (b ^ d_|_) =< ((c_|_ v b) ^ (d_|_ v a))
19	15, 18	lel2or 162	. . . 4 ((a ^ c_|_) v (b ^ d_|_)) =< ((c_|_ v b) ^ (d_|_ v a))
20		oran2 84	. . . . . 6 (c_|_ v b) = (c ^ b_|_)_|_
21		oran2 84	. . . . . 6 (d_|_ v a) = (d ^ a_|_)_|_
22	20, 21	2an 72	. . . . 5 ((c_|_ v b) ^ (d_|_ v a)) = ((c ^ b_|_)_|_ ^ (d ^ a_|_)_|_)
23		anor3 82	. . . . 5 ((c ^ b_|_)_|_ ^ (d ^ a_|_)_|_) = ((c ^ b_|_) v (d ^ a_|_))_|_
24	22, 23	ax-r2 35	. . . 4 ((c_|_ v b) ^ (d_|_ v a)) = ((c ^ b_|_) v (d ^ a_|_))_|_
25	19, 24	lbtr 131	. . 3 ((a ^ c_|_) v (b ^ d_|_)) =< ((c ^ b_|_) v (d ^ a_|_))_|_
26	25	mhlem 858	. 2 (((a ^ c_|_) v (c ^ b_|_)) ^ ((b ^ d_|_) v (d ^ a_|_))) = (((a ^ c_|_) ^ (b ^ d_|_)) v ((c ^ b_|_) ^ (d ^ a_|_)))
27	12, 26	ax-r2 35	1 (((a v c) ^ (c_|_ v b_|_)) ^ ((b v d) ^ (a_|_ v d_|_))) = (((a ^ c_|_) ^ (b ^ d_|_)) v ((c ^ b_|_) ^ (d ^ a_|_)))

Syntax hints:   = wb 1   C wc 3  _|_wn 4   v wo 6   ^ wa 7

This theorem is referenced by:  mh 861

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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