Theorem gomaex3lem9 902

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
gomaex3lem9	(((a v b) ^ (d v e)_|_) ^ (r v (p_|_ ->1 q))) =< (b v c)

Proof of Theorem gomaex3lem9

Step	Hyp	Ref	Expression
1		ancom 68	. . 3 (((a v b) ^ (d v e)_|_) ^ (r v (p_|_ ->1 q))) = ((r v (p_|_ ->1 q)) ^ ((a v b) ^ (d v e)_|_))
2		gomaex3lem5.9	. . . . . . 7 p = ((a v b) ->1 (d v e)_|_)_|_
3	2	gomaex3lem4 897	. . . . . 6 ((a v b) ^ (d v e)_|_) =< p_|_
4	3	df2le2 128	. . . . 5 (((a v b) ^ (d v e)_|_) ^ p_|_) = ((a v b) ^ (d v e)_|_)
5	4	ax-r1 34	. . . 4 ((a v b) ^ (d v e)_|_) = (((a v b) ^ (d v e)_|_) ^ p_|_)
6	5	lan 70	. . 3 ((r v (p_|_ ->1 q)) ^ ((a v b) ^ (d v e)_|_)) = ((r v (p_|_ ->1 q)) ^ (((a v b) ^ (d v e)_|_) ^ p_|_))
7		an12 74	. . 3 ((r v (p_|_ ->1 q)) ^ (((a v b) ^ (d v e)_|_) ^ p_|_)) = (((a v b) ^ (d v e)_|_) ^ ((r v (p_|_ ->1 q)) ^ p_|_))
8	1, 6, 7	3tr 62	. 2 (((a v b) ^ (d v e)_|_) ^ (r v (p_|_ ->1 q))) = (((a v b) ^ (d v e)_|_) ^ ((r v (p_|_ ->1 q)) ^ p_|_))
9		gomaex3lem5.1	. . 3 a =< b_|_
10		gomaex3lem5.2	. . 3 b =< c_|_
11		gomaex3lem5.3	. . 3 c =< d_|_
12		gomaex3lem5.5	. . 3 e =< f_|_
13		gomaex3lem5.6	. . 3 f =< a_|_
14		gomaex3lem5.8	. . 3 (((i ->2 g) ^ (g ->2 y)) ^ (((y ->2 w) ^ (w ->2 n)) ^ ((n ->2 k) ^ (k ->2 i)))) =< (g ->2 i)
15		gomaex3lem5.10	. . 3 q = ((e v f) ->1 (b v c)_|_)_|_
16		gomaex3lem5.11	. . 3 r = ((p_|_ ->1 q)_|_ ^ (c v d))
17		gomaex3lem5.12	. . 3 g = a
18		gomaex3lem5.13	. . 3 h = b
19		gomaex3lem5.14	. . 3 i = c
20		gomaex3lem5.15	. . 3 j = (c v d)_|_
21		gomaex3lem5.16	. . 3 k = r
22		gomaex3lem5.17	. . 3 m = (p_|_ ->1 q)
23		gomaex3lem5.18	. . 3 n = (p_|_ ->1 q)_|_
24		gomaex3lem5.19	. . 3 u = (p_|_ ^ q)
25		gomaex3lem5.20	. . 3 w = q_|_
26		gomaex3lem5.21	. . 3 x = q
27		gomaex3lem5.22	. . 3 y = (e v f)_|_
28		gomaex3lem5.23	. . 3 z = f
29	9, 10, 11, 12, 13, 14, 2, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28	gomaex3lem8 901	. 2 (((a v b) ^ (d v e)_|_) ^ ((r v (p_|_ ->1 q)) ^ p_|_)) =< (b v c)
30	8, 29	bltr 130	1 (((a v b) ^ (d v e)_|_) ^ (r v (p_|_ ->1 q))) =< (b v c)

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

This theorem is referenced by:  gomaex3lem10 903

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

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