Theorem mlalem 832

Description: Lemma for Mladen's OML. (Contributed by NM, 4-Nov-1998.)

Assertion

Ref	Expression
mlalem	((a ≡ b) ∩ (b →1 c)) ≤ (a →1 c)

Proof of Theorem mlalem

Step	Hyp	Ref	Expression
1		comanr2 465	. . . . . 6 b C (a ∩ b)
2	1	comcom3 454	. . . . 5 b⊥ C (a ∩ b)
3		comanr1 464	. . . . . 6 b C (b ∩ c)
4	3	comcom3 454	. . . . 5 b⊥ C (b ∩ c)
5	2, 4	fh2 470	. . . 4 ((a ∩ b) ∩ (b⊥ ∪ (b ∩ c))) = (((a ∩ b) ∩ b⊥ ) ∪ ((a ∩ b) ∩ (b ∩ c)))
6		anass 76	. . . . . . 7 ((a ∩ b) ∩ b⊥ ) = (a ∩ (b ∩ b⊥ ))
7		dff 101	. . . . . . . . 9 0 = (b ∩ b⊥ )
8	7	ax-r1 35	. . . . . . . 8 (b ∩ b⊥ ) = 0
9	8	lan 77	. . . . . . 7 (a ∩ (b ∩ b⊥ )) = (a ∩ 0)
10		an0 108	. . . . . . 7 (a ∩ 0) = 0
11	6, 9, 10	3tr 65	. . . . . 6 ((a ∩ b) ∩ b⊥ ) = 0
12		le0 147	. . . . . 6 0 ≤ (a⊥ ∪ (a ∩ c))
13	11, 12	bltr 138	. . . . 5 ((a ∩ b) ∩ b⊥ ) ≤ (a⊥ ∪ (a ∩ c))
14		anass 76	. . . . . . 7 ((a ∩ b) ∩ (b ∩ c)) = (a ∩ (b ∩ (b ∩ c)))
15		an12 81	. . . . . . 7 (a ∩ (b ∩ (b ∩ c))) = (b ∩ (a ∩ (b ∩ c)))
16		anass 76	. . . . . . . . 9 ((b ∩ a) ∩ (b ∩ c)) = (b ∩ (a ∩ (b ∩ c)))
17	16	ax-r1 35	. . . . . . . 8 (b ∩ (a ∩ (b ∩ c))) = ((b ∩ a) ∩ (b ∩ c))
18		an4 86	. . . . . . . 8 ((b ∩ a) ∩ (b ∩ c)) = ((b ∩ b) ∩ (a ∩ c))
19	17, 18	ax-r2 36	. . . . . . 7 (b ∩ (a ∩ (b ∩ c))) = ((b ∩ b) ∩ (a ∩ c))
20	14, 15, 19	3tr 65	. . . . . 6 ((a ∩ b) ∩ (b ∩ c)) = ((b ∩ b) ∩ (a ∩ c))
21		lear 161	. . . . . . 7 ((b ∩ b) ∩ (a ∩ c)) ≤ (a ∩ c)
22		leor 159	. . . . . . 7 (a ∩ c) ≤ (a⊥ ∪ (a ∩ c))
23	21, 22	letr 137	. . . . . 6 ((b ∩ b) ∩ (a ∩ c)) ≤ (a⊥ ∪ (a ∩ c))
24	20, 23	bltr 138	. . . . 5 ((a ∩ b) ∩ (b ∩ c)) ≤ (a⊥ ∪ (a ∩ c))
25	13, 24	lel2or 170	. . . 4 (((a ∩ b) ∩ b⊥ ) ∪ ((a ∩ b) ∩ (b ∩ c))) ≤ (a⊥ ∪ (a ∩ c))
26	5, 25	bltr 138	. . 3 ((a ∩ b) ∩ (b⊥ ∪ (b ∩ c))) ≤ (a⊥ ∪ (a ∩ c))
27		anass 76	. . . 4 ((a⊥ ∩ b⊥ ) ∩ (b⊥ ∪ (b ∩ c))) = (a⊥ ∩ (b⊥ ∩ (b⊥ ∪ (b ∩ c))))
28		lea 160	. . . . 5 (a⊥ ∩ (b⊥ ∩ (b⊥ ∪ (b ∩ c)))) ≤ a⊥
29		leo 158	. . . . 5 a⊥ ≤ (a⊥ ∪ (a ∩ c))
30	28, 29	letr 137	. . . 4 (a⊥ ∩ (b⊥ ∩ (b⊥ ∪ (b ∩ c)))) ≤ (a⊥ ∪ (a ∩ c))
31	27, 30	bltr 138	. . 3 ((a⊥ ∩ b⊥ ) ∩ (b⊥ ∪ (b ∩ c))) ≤ (a⊥ ∪ (a ∩ c))
32	26, 31	lel2or 170	. 2 (((a ∩ b) ∩ (b⊥ ∪ (b ∩ c))) ∪ ((a⊥ ∩ b⊥ ) ∩ (b⊥ ∪ (b ∩ c)))) ≤ (a⊥ ∪ (a ∩ c))
33		dfb 94	. . . 4 (a ≡ b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
34		df-i1 44	. . . 4 (b →1 c) = (b⊥ ∪ (b ∩ c))
35	33, 34	2an 79	. . 3 ((a ≡ b) ∩ (b →1 c)) = (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ (b⊥ ∪ (b ∩ c)))
36		lear 161	. . . . . 6 (a⊥ ∩ b⊥ ) ≤ b⊥
37		leo 158	. . . . . 6 b⊥ ≤ (b⊥ ∪ (b ∩ c))
38	36, 37	letr 137	. . . . 5 (a⊥ ∩ b⊥ ) ≤ (b⊥ ∪ (b ∩ c))
39	38	lecom 180	. . . 4 (a⊥ ∩ b⊥ ) C (b⊥ ∪ (b ∩ c))
40		coman1 185	. . . . . . 7 (a⊥ ∩ b⊥ ) C a⊥
41		coman2 186	. . . . . . 7 (a⊥ ∩ b⊥ ) C b⊥
42	40, 41	com2or 483	. . . . . 6 (a⊥ ∩ b⊥ ) C (a⊥ ∪ b⊥ )
43		oran3 93	. . . . . 6 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
44	42, 43	cbtr 182	. . . . 5 (a⊥ ∩ b⊥ ) C (a ∩ b)⊥
45	44	comcom7 460	. . . 4 (a⊥ ∩ b⊥ ) C (a ∩ b)
46	39, 45	fh2rc 480	. . 3 (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ (b⊥ ∪ (b ∩ c))) = (((a ∩ b) ∩ (b⊥ ∪ (b ∩ c))) ∪ ((a⊥ ∩ b⊥ ) ∩ (b⊥ ∪ (b ∩ c))))
47	35, 46	ax-r2 36	. 2 ((a ≡ b) ∩ (b →1 c)) = (((a ∩ b) ∩ (b⊥ ∪ (b ∩ c))) ∪ ((a⊥ ∩ b⊥ ) ∩ (b⊥ ∪ (b ∩ c))))
48		df-i1 44	. 2 (a →1 c) = (a⊥ ∪ (a ∩ c))
49	32, 47, 48	le3tr1 140	1 ((a ≡ b) ∩ (b →1 c)) ≤ (a →1 c)

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  0wf 9   →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:  mlaoml  833