Theorem oa4to6lem2 941

Description: Lemma for orthoarguesian law (4-variable to 6-variable proof).

Hypotheses

Ref	Expression
oa4to6lem.1	a_|_ =< b
oa4to6lem.2	c_|_ =< d
oa4to6lem.3	e_|_ =< f
oa4to6lem.4	g = (((a ^ b) v (c ^ d)) v (e ^ f))

Assertion

Ref	Expression
oa4to6lem2	d =< (c ->1 g)

Proof of Theorem oa4to6lem2

Step	Hyp	Ref	Expression
1		leor 151	. . . 4 d =< (c_|_ v d)
2		comid 179	. . . . . . . . 9 c C c
3	2	comcom3 436	. . . . . . . 8 c_|_ C c
4		oa4to6lem.2	. . . . . . . . 9 c_|_ =< d
5	4	lecom 172	. . . . . . . 8 c_|_ C d
6	3, 5	fh3 453	. . . . . . 7 (c_|_ v (c ^ d)) = ((c_|_ v c) ^ (c_|_ v d))
7		ancom 68	. . . . . . . 8 (1 ^ (c_|_ v d)) = ((c_|_ v d) ^ 1)
8		df-t 40	. . . . . . . . . 10 1 = (c v c_|_)
9		ax-a2 30	. . . . . . . . . 10 (c v c_|_) = (c_|_ v c)
10	8, 9	ax-r2 35	. . . . . . . . 9 1 = (c_|_ v c)
11	10	ran 71	. . . . . . . 8 (1 ^ (c_|_ v d)) = ((c_|_ v c) ^ (c_|_ v d))
12		an1 98	. . . . . . . 8 ((c_|_ v d) ^ 1) = (c_|_ v d)
13	7, 11, 12	3tr2 61	. . . . . . 7 ((c_|_ v c) ^ (c_|_ v d)) = (c_|_ v d)
14	6, 13	ax-r2 35	. . . . . 6 (c_|_ v (c ^ d)) = (c_|_ v d)
15	14	ax-r1 34	. . . . 5 (c_|_ v d) = (c_|_ v (c ^ d))
16		anidm 103	. . . . . . . . 9 (c ^ c) = c
17	16	ran 71	. . . . . . . 8 ((c ^ c) ^ d) = (c ^ d)
18	17	ax-r1 34	. . . . . . 7 (c ^ d) = ((c ^ c) ^ d)
19		anass 69	. . . . . . 7 ((c ^ c) ^ d) = (c ^ (c ^ d))
20	18, 19	ax-r2 35	. . . . . 6 (c ^ d) = (c ^ (c ^ d))
21	20	lor 66	. . . . 5 (c_|_ v (c ^ d)) = (c_|_ v (c ^ (c ^ d)))
22	15, 21	ax-r2 35	. . . 4 (c_|_ v d) = (c_|_ v (c ^ (c ^ d)))
23	1, 22	lbtr 131	. . 3 d =< (c_|_ v (c ^ (c ^ d)))
24		leor 151	. . . . . 6 (c ^ d) =< (((a ^ b) v (e ^ f)) v (c ^ d))
25		or32 75	. . . . . 6 (((a ^ b) v (e ^ f)) v (c ^ d)) = (((a ^ b) v (c ^ d)) v (e ^ f))
26	24, 25	lbtr 131	. . . . 5 (c ^ d) =< (((a ^ b) v (c ^ d)) v (e ^ f))
27	26	lelan 159	. . . 4 (c ^ (c ^ d)) =< (c ^ (((a ^ b) v (c ^ d)) v (e ^ f)))
28	27	lelor 158	. . 3 (c_|_ v (c ^ (c ^ d))) =< (c_|_ v (c ^ (((a ^ b) v (c ^ d)) v (e ^ f))))
29	23, 28	letr 129	. 2 d =< (c_|_ v (c ^ (((a ^ b) v (c ^ d)) v (e ^ f))))
30		oa4to6lem.4	. . . . 5 g = (((a ^ b) v (c ^ d)) v (e ^ f))
31	30	ud1lem0a 247	. . . 4 (c ->1 g) = (c ->1 (((a ^ b) v (c ^ d)) v (e ^ f)))
32		df-i1 43	. . . 4 (c ->1 (((a ^ b) v (c ^ d)) v (e ^ f))) = (c_|_ v (c ^ (((a ^ b) v (c ^ d)) v (e ^ f))))
33	31, 32	ax-r2 35	. . 3 (c ->1 g) = (c_|_ v (c ^ (((a ^ b) v (c ^ d)) v (e ^ f))))
34	33	ax-r1 34	. 2 (c_|_ v (c ^ (((a ^ b) v (c ^ d)) v (e ^ f)))) = (c ->1 g)
35	29, 34	lbtr 131	1 d =< (c ->1 g)

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

This theorem is referenced by:  oa4to6lem4 943

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-le1 122  df-le2 123  df-c1 124  df-c2 125

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