Theorem gomaex3lem7 900

Description: Lemma for Godowski 6-var -> Mayet Example 3.

Hypotheses

Ref	Expression
gomaex3lem5.1	a =< b_|_
gomaex3lem5.2	b =< c_|_
gomaex3lem5.3	c =< d_|_
gomaex3lem5.5	e =< f_|_
gomaex3lem5.6	f =< a_|_
gomaex3lem5.8	(((i ->2 g) ^ (g ->2 y)) ^ (((y ->2 w) ^ (w ->2 n)) ^ ((n ->2 k) ^ (k ->2 i)))) =< (g ->2 i)
gomaex3lem5.9	p = ((a v b) ->1 (d v e)_|_)_|_
gomaex3lem5.10	q = ((e v f) ->1 (b v c)_|_)_|_
gomaex3lem5.11	r = ((p_|_ ->1 q)_|_ ^ (c v d))
gomaex3lem5.12	g = a
gomaex3lem5.13	h = b
gomaex3lem5.14	i = c
gomaex3lem5.15	j = (c v d)_|_
gomaex3lem5.16	k = r
gomaex3lem5.17	m = (p_|_ ->1 q)
gomaex3lem5.18	n = (p_|_ ->1 q)_|_
gomaex3lem5.19	u = (p_|_ ^ q)
gomaex3lem5.20	w = q_|_
gomaex3lem5.21	x = q
gomaex3lem5.22	y = (e v f)_|_
gomaex3lem5.23	z = f

Assertion

Ref	Expression
gomaex3lem7	(((a v b) ^ d_|_) ^ (((r v (p_|_ ->1 q)) ^ p_|_) ^ e_|_)) =< (b v c)

Proof of Theorem gomaex3lem7

Step	Hyp	Ref	Expression
1		gomaex3lem5.3	. . . . . 6 c =< d_|_
2	1	gomaex3lem1 894	. . . . 5 (c v (c v d)_|_) = d_|_
3	2	lan 70	. . . 4 ((a v b) ^ (c v (c v d)_|_)) = ((a v b) ^ d_|_)
4		gomaex3lem3 896	. . . . . 6 ((p_|_ ->1 q)_|_ v (p_|_ ^ q)) = p_|_
5	4	lan 70	. . . . 5 ((r v (p_|_ ->1 q)) ^ ((p_|_ ->1 q)_|_ v (p_|_ ^ q))) = ((r v (p_|_ ->1 q)) ^ p_|_)
6		ancom 68	. . . . . 6 ((q_|_ v q) ^ ((e v f)_|_ v f)) = (((e v f)_|_ v f) ^ (q_|_ v q))
7		gomaex3lem5.5	. . . . . . . 8 e =< f_|_
8	7	gomaex3lem2 895	. . . . . . 7 ((e v f)_|_ v f) = e_|_
9		ax-a2 30	. . . . . . . 8 (q_|_ v q) = (q v q_|_)
10		df-t 40	. . . . . . . . 9 1 = (q v q_|_)
11	10	ax-r1 34	. . . . . . . 8 (q v q_|_) = 1
12	9, 11	ax-r2 35	. . . . . . 7 (q_|_ v q) = 1
13	8, 12	2an 72	. . . . . 6 (((e v f)_|_ v f) ^ (q_|_ v q)) = (e_|_ ^ 1)
14		an1 98	. . . . . 6 (e_|_ ^ 1) = e_|_
15	6, 13, 14	3tr 62	. . . . 5 ((q_|_ v q) ^ ((e v f)_|_ v f)) = e_|_
16	5, 15	2an 72	. . . 4 (((r v (p_|_ ->1 q)) ^ ((p_|_ ->1 q)_|_ v (p_|_ ^ q))) ^ ((q_|_ v q) ^ ((e v f)_|_ v f))) = (((r v (p_|_ ->1 q)) ^ p_|_) ^ e_|_)
17	3, 16	2an 72	. . 3 (((a v b) ^ (c v (c v d)_|_)) ^ (((r v (p_|_ ->1 q)) ^ ((p_|_ ->1 q)_|_ v (p_|_ ^ q))) ^ ((q_|_ v q) ^ ((e v f)_|_ v f)))) = (((a v b) ^ d_|_) ^ (((r v (p_|_ ->1 q)) ^ p_|_) ^ e_|_))
18	17	ax-r1 34	. 2 (((a v b) ^ d_|_) ^ (((r v (p_|_ ->1 q)) ^ p_|_) ^ e_|_)) = (((a v b) ^ (c v (c v d)_|_)) ^ (((r v (p_|_ ->1 q)) ^ ((p_|_ ->1 q)_|_ v (p_|_ ^ q))) ^ ((q_|_ v q) ^ ((e v f)_|_ v f))))
19		gomaex3lem5.1	. . 3 a =< b_|_
20		gomaex3lem5.2	. . 3 b =< c_|_
21		gomaex3lem5.6	. . 3 f =< a_|_
22		gomaex3lem5.8	. . 3 (((i ->2 g) ^ (g ->2 y)) ^ (((y ->2 w) ^ (w ->2 n)) ^ ((n ->2 k) ^ (k ->2 i)))) =< (g ->2 i)
23		gomaex3lem5.9	. . 3 p = ((a v b) ->1 (d v e)_|_)_|_
24		gomaex3lem5.10	. . 3 q = ((e v f) ->1 (b v c)_|_)_|_
25		gomaex3lem5.11	. . 3 r = ((p_|_ ->1 q)_|_ ^ (c v d))
26		gomaex3lem5.12	. . 3 g = a
27		gomaex3lem5.13	. . 3 h = b
28		gomaex3lem5.14	. . 3 i = c
29		gomaex3lem5.15	. . 3 j = (c v d)_|_
30		gomaex3lem5.16	. . 3 k = r
31		gomaex3lem5.17	. . 3 m = (p_|_ ->1 q)
32		gomaex3lem5.18	. . 3 n = (p_|_ ->1 q)_|_
33		gomaex3lem5.19	. . 3 u = (p_|_ ^ q)
34		gomaex3lem5.20	. . 3 w = q_|_
35		gomaex3lem5.21	. . 3 x = q
36		gomaex3lem5.22	. . 3 y = (e v f)_|_
37		gomaex3lem5.23	. . 3 z = f
38	19, 20, 1, 7, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37	gomaex3lem6 899	. 2 (((a v b) ^ (c v (c v d)_|_)) ^ (((r v (p_|_ ->1 q)) ^ ((p_|_ ->1 q)_|_ v (p_|_ ^ q))) ^ ((q_|_ v q) ^ ((e v f)_|_ v f)))) =< (b v c)
39	18, 38	bltr 130	1 (((a v b) ^ d_|_) ^ (((r v (p_|_ ->1 q)) ^ p_|_) ^ e_|_)) =< (b v c)

Syntax hints:   = wb 1   =< wle 2  _|_wn 4   v wo 6   ^ wa 7  1wt 9   ->1 wi1 13   ->2 wi2 14

This theorem is referenced by:  gomaex3lem8 901

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-i1 43  df-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

metamath.org