Theorem gomaex3lem5 898

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
gomaex3lem5	(((g v h) ^ (i v j)) ^ (((k v m) ^ (n v u)) ^ ((w v x) ^ (y v z)))) =< (h v i)

Proof of Theorem gomaex3lem5

Step	Hyp	Ref	Expression
1		gomaex3lem5.1	. . 3 a =< b_|_
2		gomaex3lem5.12	. . 3 g = a
3		gomaex3lem5.13	. . 3 h = b
4	1, 2, 3	gomaex3h1 882	. 2 g =< h_|_
5		gomaex3lem5.2	. . 3 b =< c_|_
6		gomaex3lem5.14	. . 3 i = c
7	5, 3, 6	gomaex3h2 883	. 2 h =< i_|_
8		gomaex3lem5.15	. . 3 j = (c v d)_|_
9	6, 8	gomaex3h3 884	. 2 i =< j_|_
10		gomaex3lem5.11	. . 3 r = ((p_|_ ->1 q)_|_ ^ (c v d))
11		gomaex3lem5.16	. . 3 k = r
12	10, 8, 11	gomaex3h4 885	. 2 j =< k_|_
13		gomaex3lem5.17	. . 3 m = (p_|_ ->1 q)
14	10, 11, 13	gomaex3h5 886	. 2 k =< m_|_
15		gomaex3lem5.18	. . 3 n = (p_|_ ->1 q)_|_
16	13, 15	gomaex3h6 887	. 2 m =< n_|_
17		gomaex3lem5.19	. . 3 u = (p_|_ ^ q)
18	15, 17	gomaex3h7 888	. 2 n =< u_|_
19		gomaex3lem5.20	. . 3 w = q_|_
20	17, 19	gomaex3h8 889	. 2 u =< w_|_
21		gomaex3lem5.21	. . 3 x = q
22	19, 21	gomaex3h9 890	. 2 w =< x_|_
23		gomaex3lem5.10	. . 3 q = ((e v f) ->1 (b v c)_|_)_|_
24		gomaex3lem5.22	. . 3 y = (e v f)_|_
25	23, 21, 24	gomaex3h10 891	. 2 x =< y_|_
26		gomaex3lem5.23	. . 3 z = f
27	24, 26	gomaex3h11 892	. 2 y =< z_|_
28		gomaex3lem5.6	. . 3 f =< a_|_
29	28, 2, 26	gomaex3h12 893	. 2 z =< g_|_
30		gomaex3lem5.8	. 2 (((i ->2 g) ^ (g ->2 y)) ^ (((y ->2 w) ^ (w ->2 n)) ^ ((n ->2 k) ^ (k ->2 i)))) =< (g ->2 i)
31	4, 7, 9, 12, 14, 16, 18, 20, 22, 25, 27, 29, 30	go2n6 881	1 (((g v h) ^ (i v j)) ^ (((k v m) ^ (n v u)) ^ ((w v x) ^ (y v z)))) =< (h v i)

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

This theorem is referenced by:  gomaex3lem6 899

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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