Theorem oa4to6lem3 962

Description: Lemma for orthoarguesian law (4-variable to 6-variable proof). (Contributed by NM, 18-Dec-1998.)

Hypotheses

Ref	Expression
oa4to6lem.1	a⊥ ≤ b
oa4to6lem.2	c⊥ ≤ d
oa4to6lem.3	e⊥ ≤ f
oa4to6lem.4	g = (((a ∩ b) ∪ (c ∩ d)) ∪ (e ∩ f))

Assertion

Ref	Expression
oa4to6lem3	f ≤ (e →1 g)

Proof of Theorem oa4to6lem3

Step	Hyp	Ref	Expression
1		leor 159	. . . 4 f ≤ (e⊥ ∪ f)
2		comid 187	. . . . . . . . 9 e C e
3	2	comcom3 454	. . . . . . . 8 e⊥ C e
4		oa4to6lem.3	. . . . . . . . 9 e⊥ ≤ f
5	4	lecom 180	. . . . . . . 8 e⊥ C f
6	3, 5	fh3 471	. . . . . . 7 (e⊥ ∪ (e ∩ f)) = ((e⊥ ∪ e) ∩ (e⊥ ∪ f))
7		ancom 74	. . . . . . . 8 (1 ∩ (e⊥ ∪ f)) = ((e⊥ ∪ f) ∩ 1)
8		df-t 41	. . . . . . . . . 10 1 = (e ∪ e⊥ )
9		ax-a2 31	. . . . . . . . . 10 (e ∪ e⊥ ) = (e⊥ ∪ e)
10	8, 9	ax-r2 36	. . . . . . . . 9 1 = (e⊥ ∪ e)
11	10	ran 78	. . . . . . . 8 (1 ∩ (e⊥ ∪ f)) = ((e⊥ ∪ e) ∩ (e⊥ ∪ f))
12		an1 106	. . . . . . . 8 ((e⊥ ∪ f) ∩ 1) = (e⊥ ∪ f)
13	7, 11, 12	3tr2 64	. . . . . . 7 ((e⊥ ∪ e) ∩ (e⊥ ∪ f)) = (e⊥ ∪ f)
14	6, 13	ax-r2 36	. . . . . 6 (e⊥ ∪ (e ∩ f)) = (e⊥ ∪ f)
15	14	ax-r1 35	. . . . 5 (e⊥ ∪ f) = (e⊥ ∪ (e ∩ f))
16		anidm 111	. . . . . . . . 9 (e ∩ e) = e
17	16	ran 78	. . . . . . . 8 ((e ∩ e) ∩ f) = (e ∩ f)
18	17	ax-r1 35	. . . . . . 7 (e ∩ f) = ((e ∩ e) ∩ f)
19		anass 76	. . . . . . 7 ((e ∩ e) ∩ f) = (e ∩ (e ∩ f))
20	18, 19	ax-r2 36	. . . . . 6 (e ∩ f) = (e ∩ (e ∩ f))
21	20	lor 70	. . . . 5 (e⊥ ∪ (e ∩ f)) = (e⊥ ∪ (e ∩ (e ∩ f)))
22	15, 21	ax-r2 36	. . . 4 (e⊥ ∪ f) = (e⊥ ∪ (e ∩ (e ∩ f)))
23	1, 22	lbtr 139	. . 3 f ≤ (e⊥ ∪ (e ∩ (e ∩ f)))
24		leor 159	. . . . 5 (e ∩ f) ≤ (((a ∩ b) ∪ (c ∩ d)) ∪ (e ∩ f))
25	24	lelan 167	. . . 4 (e ∩ (e ∩ f)) ≤ (e ∩ (((a ∩ b) ∪ (c ∩ d)) ∪ (e ∩ f)))
26	25	lelor 166	. . 3 (e⊥ ∪ (e ∩ (e ∩ f))) ≤ (e⊥ ∪ (e ∩ (((a ∩ b) ∪ (c ∩ d)) ∪ (e ∩ f))))
27	23, 26	letr 137	. 2 f ≤ (e⊥ ∪ (e ∩ (((a ∩ b) ∪ (c ∩ d)) ∪ (e ∩ f))))
28		oa4to6lem.4	. . . . 5 g = (((a ∩ b) ∪ (c ∩ d)) ∪ (e ∩ f))
29	28	ud1lem0a 255	. . . 4 (e →1 g) = (e →1 (((a ∩ b) ∪ (c ∩ d)) ∪ (e ∩ f)))
30		df-i1 44	. . . 4 (e →1 (((a ∩ b) ∪ (c ∩ d)) ∪ (e ∩ f))) = (e⊥ ∪ (e ∩ (((a ∩ b) ∪ (c ∩ d)) ∪ (e ∩ f))))
31	29, 30	ax-r2 36	. . 3 (e →1 g) = (e⊥ ∪ (e ∩ (((a ∩ b) ∪ (c ∩ d)) ∪ (e ∩ f))))
32	31	ax-r1 35	. 2 (e⊥ ∪ (e ∩ (((a ∩ b) ∪ (c ∩ d)) ∪ (e ∩ f)))) = (e →1 g)
33	27, 32	lbtr 139	1 f ≤ (e →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:  oa4to6lem4  963