Theorem oa3to4lem1 925

Description: Lemma for orthoarguesian law (Godowski/Greechie 3-variable to 4-variable proof).

Hypotheses

Ref	Expression
oa3to4lem.1	a_|_ =< b
oa3to4lem.2	c_|_ =< d
oa3to4lem.3	g = ((a ^ b) v (c ^ d))

Assertion

Ref	Expression
oa3to4lem1	b =< (a ->1 g)

Proof of Theorem oa3to4lem1

Step	Hyp	Ref	Expression
1		leor 151	. . . 4 b =< (a_|_ v b)
2		comid 179	. . . . . . . . 9 a C a
3	2	comcom3 436	. . . . . . . 8 a_|_ C a
4		oa3to4lem.1	. . . . . . . . 9 a_|_ =< b
5	4	lecom 172	. . . . . . . 8 a_|_ C b
6	3, 5	fh3 453	. . . . . . 7 (a_|_ v (a ^ b)) = ((a_|_ v a) ^ (a_|_ v b))
7		ancom 68	. . . . . . . 8 (1 ^ (a_|_ v b)) = ((a_|_ v b) ^ 1)
8		df-t 40	. . . . . . . . . 10 1 = (a v a_|_)
9		ax-a2 30	. . . . . . . . . 10 (a v a_|_) = (a_|_ v a)
10	8, 9	ax-r2 35	. . . . . . . . 9 1 = (a_|_ v a)
11	10	ran 71	. . . . . . . 8 (1 ^ (a_|_ v b)) = ((a_|_ v a) ^ (a_|_ v b))
12		an1 98	. . . . . . . 8 ((a_|_ v b) ^ 1) = (a_|_ v b)
13	7, 11, 12	3tr2 61	. . . . . . 7 ((a_|_ v a) ^ (a_|_ v b)) = (a_|_ v b)
14	6, 13	ax-r2 35	. . . . . 6 (a_|_ v (a ^ b)) = (a_|_ v b)
15	14	ax-r1 34	. . . . 5 (a_|_ v b) = (a_|_ v (a ^ b))
16		anidm 103	. . . . . . . . 9 (a ^ a) = a
17	16	ran 71	. . . . . . . 8 ((a ^ a) ^ b) = (a ^ b)
18	17	ax-r1 34	. . . . . . 7 (a ^ b) = ((a ^ a) ^ b)
19		anass 69	. . . . . . 7 ((a ^ a) ^ b) = (a ^ (a ^ b))
20	18, 19	ax-r2 35	. . . . . 6 (a ^ b) = (a ^ (a ^ b))
21	20	lor 66	. . . . 5 (a_|_ v (a ^ b)) = (a_|_ v (a ^ (a ^ b)))
22	15, 21	ax-r2 35	. . . 4 (a_|_ v b) = (a_|_ v (a ^ (a ^ b)))
23	1, 22	lbtr 131	. . 3 b =< (a_|_ v (a ^ (a ^ b)))
24		leo 150	. . . . 5 (a ^ b) =< ((a ^ b) v (c ^ d))
25	24	lelan 159	. . . 4 (a ^ (a ^ b)) =< (a ^ ((a ^ b) v (c ^ d)))
26	25	lelor 158	. . 3 (a_|_ v (a ^ (a ^ b))) =< (a_|_ v (a ^ ((a ^ b) v (c ^ d))))
27	23, 26	letr 129	. 2 b =< (a_|_ v (a ^ ((a ^ b) v (c ^ d))))
28		oa3to4lem.3	. . . . 5 g = ((a ^ b) v (c ^ d))
29	28	ud1lem0a 247	. . . 4 (a ->1 g) = (a ->1 ((a ^ b) v (c ^ d)))
30		df-i1 43	. . . 4 (a ->1 ((a ^ b) v (c ^ d))) = (a_|_ v (a ^ ((a ^ b) v (c ^ d))))
31	29, 30	ax-r2 35	. . 3 (a ->1 g) = (a_|_ v (a ^ ((a ^ b) v (c ^ d))))
32	31	ax-r1 34	. 2 (a_|_ v (a ^ ((a ^ b) v (c ^ d)))) = (a ->1 g)
33	27, 32	lbtr 131	1 b =< (a ->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:  oa3to4lem3 927

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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