Theorem go2n4 899

Description: 8-variable Godowski equation derived from 4-variable one. The last hypothesis is the 4-variable Godowski equation. (Contributed by NM, 19-Nov-1999.)

Hypotheses

Ref	Expression
go2n4.1	a ≤ b⊥
go2n4.2	b ≤ c⊥
go2n4.3	c ≤ d⊥
go2n4.4	d ≤ e⊥
go2n4.5	e ≤ f⊥
go2n4.6	f ≤ g⊥
go2n4.7	g ≤ h⊥
go2n4.8	h ≤ a⊥
go2n4.9	(((c →2 a) ∩ (a →2 g)) ∩ ((g →2 e) ∩ (e →2 c))) ≤ (a →2 c)

Assertion

Ref	Expression
go2n4	(((a ∪ b) ∩ (c ∪ d)) ∩ ((e ∪ f) ∩ (g ∪ h))) ≤ (b ∪ c)

Proof of Theorem go2n4

Step	Hyp	Ref	Expression
1		anass 76	. . 3 (((a ∪ b) ∩ (c ∪ d)) ∩ ((e ∪ f) ∩ (g ∪ h))) = ((a ∪ b) ∩ ((c ∪ d) ∩ ((e ∪ f) ∩ (g ∪ h))))
2		ancom 74	. . . 4 ((c ∪ d) ∩ ((e ∪ f) ∩ (g ∪ h))) = (((e ∪ f) ∩ (g ∪ h)) ∩ (c ∪ d))
3	2	lan 77	. . 3 ((a ∪ b) ∩ ((c ∪ d) ∩ ((e ∪ f) ∩ (g ∪ h)))) = ((a ∪ b) ∩ (((e ∪ f) ∩ (g ∪ h)) ∩ (c ∪ d)))
4	1, 3	ax-r2 36	. 2 (((a ∪ b) ∩ (c ∪ d)) ∩ ((e ∪ f) ∩ (g ∪ h))) = ((a ∪ b) ∩ (((e ∪ f) ∩ (g ∪ h)) ∩ (c ∪ d)))
5		go2n4.1	. . 3 a ≤ b⊥
6		go2n4.2	. . 3 b ≤ c⊥
7		anass 76	. . . . . 6 (((c →2 a) ∩ (a →2 g)) ∩ ((g →2 e) ∩ (e →2 c))) = ((c →2 a) ∩ ((a →2 g) ∩ ((g →2 e) ∩ (e →2 c))))
8		ancom 74	. . . . . . . 8 ((a →2 g) ∩ ((g →2 e) ∩ (e →2 c))) = (((g →2 e) ∩ (e →2 c)) ∩ (a →2 g))
9		an32 83	. . . . . . . 8 (((g →2 e) ∩ (e →2 c)) ∩ (a →2 g)) = (((g →2 e) ∩ (a →2 g)) ∩ (e →2 c))
10	8, 9	ax-r2 36	. . . . . . 7 ((a →2 g) ∩ ((g →2 e) ∩ (e →2 c))) = (((g →2 e) ∩ (a →2 g)) ∩ (e →2 c))
11	10	lan 77	. . . . . 6 ((c →2 a) ∩ ((a →2 g) ∩ ((g →2 e) ∩ (e →2 c)))) = ((c →2 a) ∩ (((g →2 e) ∩ (a →2 g)) ∩ (e →2 c)))
12	7, 11	ax-r2 36	. . . . 5 (((c →2 a) ∩ (a →2 g)) ∩ ((g →2 e) ∩ (e →2 c))) = ((c →2 a) ∩ (((g →2 e) ∩ (a →2 g)) ∩ (e →2 c)))
13	12	ax-r1 35	. . . 4 ((c →2 a) ∩ (((g →2 e) ∩ (a →2 g)) ∩ (e →2 c))) = (((c →2 a) ∩ (a →2 g)) ∩ ((g →2 e) ∩ (e →2 c)))
14		go2n4.9	. . . 4 (((c →2 a) ∩ (a →2 g)) ∩ ((g →2 e) ∩ (e →2 c))) ≤ (a →2 c)
15	13, 14	bltr 138	. . 3 ((c →2 a) ∩ (((g →2 e) ∩ (a →2 g)) ∩ (e →2 c))) ≤ (a →2 c)
16		go2n4.5	. . . . . 6 e ≤ f⊥
17		go2n4.6	. . . . . 6 f ≤ g⊥
18	16, 17	govar2 897	. . . . 5 (e ∪ f) ≤ (g →2 e)
19		go2n4.7	. . . . . 6 g ≤ h⊥
20		go2n4.8	. . . . . 6 h ≤ a⊥
21	19, 20	govar2 897	. . . . 5 (g ∪ h) ≤ (a →2 g)
22	18, 21	le2an 169	. . . 4 ((e ∪ f) ∩ (g ∪ h)) ≤ ((g →2 e) ∩ (a →2 g))
23		go2n4.3	. . . . 5 c ≤ d⊥
24		go2n4.4	. . . . 5 d ≤ e⊥
25	23, 24	govar2 897	. . . 4 (c ∪ d) ≤ (e →2 c)
26	22, 25	le2an 169	. . 3 (((e ∪ f) ∩ (g ∪ h)) ∩ (c ∪ d)) ≤ (((g →2 e) ∩ (a →2 g)) ∩ (e →2 c))
27	5, 6, 15, 26	gon2n 898	. 2 ((a ∪ b) ∩ (((e ∪ f) ∩ (g ∪ h)) ∩ (c ∪ d))) ≤ (b ∪ c)
28	4, 27	bltr 138	1 (((a ∪ b) ∩ (c ∪ d)) ∩ ((e ∪ f) ∩ (g ∪ h))) ≤ (b ∪ c)

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 13

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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  gomaex4  900