Theorem oa4to6lem4 963

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
oa4to6lem4	(b ∩ (a ∪ (c ∩ (((a ∩ c) ∪ (b ∩ d)) ∪ (((a ∩ e) ∪ (b ∩ f)) ∩ ((c ∩ e) ∪ (d ∩ f))))))) ≤ ((a →1 g) ∩ (a ∪ (c ∩ (((a ∩ c) ∪ ((a →1 g) ∩ (c →1 g))) ∪ (((a ∩ e) ∪ ((a →1 g) ∩ (e →1 g))) ∩ ((c ∩ e) ∪ ((c →1 g) ∩ (e →1 g))))))))

Proof of Theorem oa4to6lem4

Step	Hyp	Ref	Expression
1		oa4to6lem.1	. . 3 a⊥ ≤ b
2		oa4to6lem.2	. . 3 c⊥ ≤ d
3		oa4to6lem.3	. . 3 e⊥ ≤ f
4		oa4to6lem.4	. . 3 g = (((a ∩ b) ∪ (c ∩ d)) ∪ (e ∩ f))
5	1, 2, 3, 4	oa4to6lem1 960	. 2 b ≤ (a →1 g)
6	1, 2, 3, 4	oa4to6lem2 961	. . . . . . 7 d ≤ (c →1 g)
7	5, 6	le2an 169	. . . . . 6 (b ∩ d) ≤ ((a →1 g) ∩ (c →1 g))
8	7	lelor 166	. . . . 5 ((a ∩ c) ∪ (b ∩ d)) ≤ ((a ∩ c) ∪ ((a →1 g) ∩ (c →1 g)))
9	1, 2, 3, 4	oa4to6lem3 962	. . . . . . . 8 f ≤ (e →1 g)
10	5, 9	le2an 169	. . . . . . 7 (b ∩ f) ≤ ((a →1 g) ∩ (e →1 g))
11	10	lelor 166	. . . . . 6 ((a ∩ e) ∪ (b ∩ f)) ≤ ((a ∩ e) ∪ ((a →1 g) ∩ (e →1 g)))
12	6, 9	le2an 169	. . . . . . 7 (d ∩ f) ≤ ((c →1 g) ∩ (e →1 g))
13	12	lelor 166	. . . . . 6 ((c ∩ e) ∪ (d ∩ f)) ≤ ((c ∩ e) ∪ ((c →1 g) ∩ (e →1 g)))
14	11, 13	le2an 169	. . . . 5 (((a ∩ e) ∪ (b ∩ f)) ∩ ((c ∩ e) ∪ (d ∩ f))) ≤ (((a ∩ e) ∪ ((a →1 g) ∩ (e →1 g))) ∩ ((c ∩ e) ∪ ((c →1 g) ∩ (e →1 g))))
15	8, 14	le2or 168	. . . 4 (((a ∩ c) ∪ (b ∩ d)) ∪ (((a ∩ e) ∪ (b ∩ f)) ∩ ((c ∩ e) ∪ (d ∩ f)))) ≤ (((a ∩ c) ∪ ((a →1 g) ∩ (c →1 g))) ∪ (((a ∩ e) ∪ ((a →1 g) ∩ (e →1 g))) ∩ ((c ∩ e) ∪ ((c →1 g) ∩ (e →1 g)))))
16	15	lelan 167	. . 3 (c ∩ (((a ∩ c) ∪ (b ∩ d)) ∪ (((a ∩ e) ∪ (b ∩ f)) ∩ ((c ∩ e) ∪ (d ∩ f))))) ≤ (c ∩ (((a ∩ c) ∪ ((a →1 g) ∩ (c →1 g))) ∪ (((a ∩ e) ∪ ((a →1 g) ∩ (e →1 g))) ∩ ((c ∩ e) ∪ ((c →1 g) ∩ (e →1 g))))))
17	16	lelor 166	. 2 (a ∪ (c ∩ (((a ∩ c) ∪ (b ∩ d)) ∪ (((a ∩ e) ∪ (b ∩ f)) ∩ ((c ∩ e) ∪ (d ∩ f)))))) ≤ (a ∪ (c ∩ (((a ∩ c) ∪ ((a →1 g) ∩ (c →1 g))) ∪ (((a ∩ e) ∪ ((a →1 g) ∩ (e →1 g))) ∩ ((c ∩ e) ∪ ((c →1 g) ∩ (e →1 g)))))))
18	5, 17	le2an 169	1 (b ∩ (a ∪ (c ∩ (((a ∩ c) ∪ (b ∩ d)) ∪ (((a ∩ e) ∪ (b ∩ f)) ∩ ((c ∩ e) ∪ (d ∩ f))))))) ≤ ((a →1 g) ∩ (a ∪ (c ∩ (((a ∩ c) ∪ ((a →1 g) ∩ (c →1 g))) ∪ (((a ∩ e) ∪ ((a →1 g) ∩ (e →1 g))) ∩ ((c ∩ e) ∪ ((c →1 g) ∩ (e →1 g))))))))

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →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:  oa4to6dual  964