Theorem oa3to4lem1 945

Description: Lemma for orthoarguesian law (Godowski/Greechie 3-variable to 4-variable proof). (Contributed by NM, 19-Dec-1998.)

Hypotheses

Ref	Expression
oa3to4lem.1	a⊥ ≤ b
oa3to4lem.2	c⊥ ≤ d
oa3to4lem.3	g = ((a ∩ b) ∪ (c ∩ d))

Assertion

Ref	Expression
oa3to4lem1	b ≤ (a →1 g)

Proof of Theorem oa3to4lem1

Step	Hyp	Ref	Expression
1		leor 159	. . . 4 b ≤ (a⊥ ∪ b)
2		comid 187	. . . . . . . . 9 a C a
3	2	comcom3 454	. . . . . . . 8 a⊥ C a
4		oa3to4lem.1	. . . . . . . . 9 a⊥ ≤ b
5	4	lecom 180	. . . . . . . 8 a⊥ C b
6	3, 5	fh3 471	. . . . . . 7 (a⊥ ∪ (a ∩ b)) = ((a⊥ ∪ a) ∩ (a⊥ ∪ b))
7		ancom 74	. . . . . . . 8 (1 ∩ (a⊥ ∪ b)) = ((a⊥ ∪ b) ∩ 1)
8		df-t 41	. . . . . . . . . 10 1 = (a ∪ a⊥ )
9		ax-a2 31	. . . . . . . . . 10 (a ∪ a⊥ ) = (a⊥ ∪ a)
10	8, 9	ax-r2 36	. . . . . . . . 9 1 = (a⊥ ∪ a)
11	10	ran 78	. . . . . . . 8 (1 ∩ (a⊥ ∪ b)) = ((a⊥ ∪ a) ∩ (a⊥ ∪ b))
12		an1 106	. . . . . . . 8 ((a⊥ ∪ b) ∩ 1) = (a⊥ ∪ b)
13	7, 11, 12	3tr2 64	. . . . . . 7 ((a⊥ ∪ a) ∩ (a⊥ ∪ b)) = (a⊥ ∪ b)
14	6, 13	ax-r2 36	. . . . . 6 (a⊥ ∪ (a ∩ b)) = (a⊥ ∪ b)
15	14	ax-r1 35	. . . . 5 (a⊥ ∪ b) = (a⊥ ∪ (a ∩ b))
16		anidm 111	. . . . . . . . 9 (a ∩ a) = a
17	16	ran 78	. . . . . . . 8 ((a ∩ a) ∩ b) = (a ∩ b)
18	17	ax-r1 35	. . . . . . 7 (a ∩ b) = ((a ∩ a) ∩ b)
19		anass 76	. . . . . . 7 ((a ∩ a) ∩ b) = (a ∩ (a ∩ b))
20	18, 19	ax-r2 36	. . . . . 6 (a ∩ b) = (a ∩ (a ∩ b))
21	20	lor 70	. . . . 5 (a⊥ ∪ (a ∩ b)) = (a⊥ ∪ (a ∩ (a ∩ b)))
22	15, 21	ax-r2 36	. . . 4 (a⊥ ∪ b) = (a⊥ ∪ (a ∩ (a ∩ b)))
23	1, 22	lbtr 139	. . 3 b ≤ (a⊥ ∪ (a ∩ (a ∩ b)))
24		leo 158	. . . . 5 (a ∩ b) ≤ ((a ∩ b) ∪ (c ∩ d))
25	24	lelan 167	. . . 4 (a ∩ (a ∩ b)) ≤ (a ∩ ((a ∩ b) ∪ (c ∩ d)))
26	25	lelor 166	. . 3 (a⊥ ∪ (a ∩ (a ∩ b))) ≤ (a⊥ ∪ (a ∩ ((a ∩ b) ∪ (c ∩ d))))
27	23, 26	letr 137	. 2 b ≤ (a⊥ ∪ (a ∩ ((a ∩ b) ∪ (c ∩ d))))
28		oa3to4lem.3	. . . . 5 g = ((a ∩ b) ∪ (c ∩ d))
29	28	ud1lem0a 255	. . . 4 (a →1 g) = (a →1 ((a ∩ b) ∪ (c ∩ d)))
30		df-i1 44	. . . 4 (a →1 ((a ∩ b) ∪ (c ∩ d))) = (a⊥ ∪ (a ∩ ((a ∩ b) ∪ (c ∩ d))))
31	29, 30	ax-r2 36	. . 3 (a →1 g) = (a⊥ ∪ (a ∩ ((a ∩ b) ∪ (c ∩ d))))
32	31	ax-r1 35	. 2 (a⊥ ∪ (a ∩ ((a ∩ b) ∪ (c ∩ d)))) = (a →1 g)
33	27, 32	lbtr 139	1 b ≤ (a →1 g)

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₁ wi1 12

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  oa3to4lem3  947