Theorem oa4to6lem3 942

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
oa4to6lem3	f =< (e ->1 g)

Proof of Theorem oa4to6lem3

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