Theorem govar 896

Description: Lemma for converting n-variable Godowski equations to 2n-variable equations. (Contributed by NM, 19-Nov-1999.)

Hypotheses

Ref	Expression
govar.1	a ≤ b⊥
govar.2	b ≤ c⊥

Assertion

Ref	Expression
govar	((a ∪ b) ∩ (a →2 c)) ≤ (b ∪ c)

Proof of Theorem govar

Step	Hyp	Ref	Expression
1		df-i2 45	. . . 4 (a →2 c) = (c ∪ (a⊥ ∩ c⊥ ))
2	1	lan 77	. . 3 ((a ∪ b) ∩ (a →2 c)) = ((a ∪ b) ∩ (c ∪ (a⊥ ∩ c⊥ )))
3		ax-a2 31	. . . . 5 (a ∪ b) = (b ∪ a)
4	3	ran 78	. . . 4 ((a ∪ b) ∩ (c ∪ (a⊥ ∩ c⊥ ))) = ((b ∪ a) ∩ (c ∪ (a⊥ ∩ c⊥ )))
5		govar.2	. . . . . . . 8 b ≤ c⊥
6	5	lecom 180	. . . . . . 7 b C c⊥
7	6	comcom7 460	. . . . . 6 b C c
8		govar.1	. . . . . . . . . . 11 a ≤ b⊥
9	8	lecom 180	. . . . . . . . . 10 a C b⊥
10	9	comcom7 460	. . . . . . . . 9 a C b
11	10	comcom 453	. . . . . . . 8 b C a
12	11	comcom2 183	. . . . . . 7 b C a⊥
13	12, 6	com2an 484	. . . . . 6 b C (a⊥ ∩ c⊥ )
14	7, 13	com2or 483	. . . . 5 b C (c ∪ (a⊥ ∩ c⊥ ))
15	14, 11	fh2r 474	. . . 4 ((b ∪ a) ∩ (c ∪ (a⊥ ∩ c⊥ ))) = ((b ∩ (c ∪ (a⊥ ∩ c⊥ ))) ∪ (a ∩ (c ∪ (a⊥ ∩ c⊥ ))))
16	4, 15	ax-r2 36	. . 3 ((a ∪ b) ∩ (c ∪ (a⊥ ∩ c⊥ ))) = ((b ∩ (c ∪ (a⊥ ∩ c⊥ ))) ∪ (a ∩ (c ∪ (a⊥ ∩ c⊥ ))))
17		coman1 185	. . . . . . 7 (a⊥ ∩ c⊥ ) C a⊥
18	17	comcom7 460	. . . . . 6 (a⊥ ∩ c⊥ ) C a
19		coman2 186	. . . . . . 7 (a⊥ ∩ c⊥ ) C c⊥
20	19	comcom7 460	. . . . . 6 (a⊥ ∩ c⊥ ) C c
21	18, 20	fh2c 477	. . . . 5 (a ∩ (c ∪ (a⊥ ∩ c⊥ ))) = ((a ∩ c) ∪ (a ∩ (a⊥ ∩ c⊥ )))
22		dff 101	. . . . . . . . 9 0 = (a ∩ a⊥ )
23	22	ran 78	. . . . . . . 8 (0 ∩ c⊥ ) = ((a ∩ a⊥ ) ∩ c⊥ )
24	23	ax-r1 35	. . . . . . 7 ((a ∩ a⊥ ) ∩ c⊥ ) = (0 ∩ c⊥ )
25		anass 76	. . . . . . 7 ((a ∩ a⊥ ) ∩ c⊥ ) = (a ∩ (a⊥ ∩ c⊥ ))
26		an0r 109	. . . . . . 7 (0 ∩ c⊥ ) = 0
27	24, 25, 26	3tr2 64	. . . . . 6 (a ∩ (a⊥ ∩ c⊥ )) = 0
28	27	lor 70	. . . . 5 ((a ∩ c) ∪ (a ∩ (a⊥ ∩ c⊥ ))) = ((a ∩ c) ∪ 0)
29		or0 102	. . . . 5 ((a ∩ c) ∪ 0) = (a ∩ c)
30	21, 28, 29	3tr 65	. . . 4 (a ∩ (c ∪ (a⊥ ∩ c⊥ ))) = (a ∩ c)
31	30	lor 70	. . 3 ((b ∩ (c ∪ (a⊥ ∩ c⊥ ))) ∪ (a ∩ (c ∪ (a⊥ ∩ c⊥ )))) = ((b ∩ (c ∪ (a⊥ ∩ c⊥ ))) ∪ (a ∩ c))
32	2, 16, 31	3tr 65	. 2 ((a ∪ b) ∩ (a →2 c)) = ((b ∩ (c ∪ (a⊥ ∩ c⊥ ))) ∪ (a ∩ c))
33		lea 160	. . 3 (b ∩ (c ∪ (a⊥ ∩ c⊥ ))) ≤ b
34		lear 161	. . 3 (a ∩ c) ≤ c
35	33, 34	le2or 168	. 2 ((b ∩ (c ∪ (a⊥ ∩ c⊥ ))) ∪ (a ∩ c)) ≤ (b ∪ c)
36	32, 35	bltr 138	1 ((a ∪ b) ∩ (a →2 c)) ≤ (b ∪ c)

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →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:  gon2n  898