Theorem 3vroa 813

Description: OA-like inference rule (requires OM only).

Hypothesis

Ref	Expression
3vroa.1	((a →2 b) ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))) = 1

Assertion

Ref	Expression
3vroa	(a →2 c) = 1

Proof of Theorem 3vroa

Step	Hyp	Ref	Expression
1		df-i2 44	. 2 (a →2 c) = (c ∪ (a⊥ ∩ c⊥ ))
2		or12 73	. . 3 (c ∪ ((a⊥ ∩ c⊥ ) ∪ (a⊥ ∩ c⊥ ))) = ((a⊥ ∩ c⊥ ) ∪ (c ∪ (a⊥ ∩ c⊥ )))
3		oridm 102	. . . 4 ((a⊥ ∩ c⊥ ) ∪ (a⊥ ∩ c⊥ )) = (a⊥ ∩ c⊥ )
4	3	lor 66	. . 3 (c ∪ ((a⊥ ∩ c⊥ ) ∪ (a⊥ ∩ c⊥ ))) = (c ∪ (a⊥ ∩ c⊥ ))
5		le1 138	. . . . . . . . . 10 (a →2 b) ≤ 1
6		3vroa.1	. . . . . . . . . . . 12 ((a →2 b) ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))) = 1
7	6	ax-r1 34	. . . . . . . . . . 11 1 = ((a →2 b) ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))))
8		lea 152	. . . . . . . . . . 11 ((a →2 b) ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))) ≤ (a →2 b)
9	7, 8	bltr 130	. . . . . . . . . 10 1 ≤ (a →2 b)
10	5, 9	lebi 137	. . . . . . . . 9 (a →2 b) = 1
11	10	ran 71	. . . . . . . 8 ((a →2 b) ∩ (a →2 c)) = (1 ∩ (a →2 c))
12		ancom 68	. . . . . . . 8 (1 ∩ (a →2 c)) = ((a →2 c) ∩ 1)
13	11, 12	ax-r2 35	. . . . . . 7 ((a →2 b) ∩ (a →2 c)) = ((a →2 c) ∩ 1)
14		an1 98	. . . . . . 7 ((a →2 c) ∩ 1) = (a →2 c)
15	13, 14, 1	3tr 62	. . . . . 6 ((a →2 b) ∩ (a →2 c)) = (c ∪ (a⊥ ∩ c⊥ ))
16	15	lor 66	. . . . 5 ((a⊥ ∩ c⊥ ) ∪ ((a →2 b) ∩ (a →2 c))) = ((a⊥ ∩ c⊥ ) ∪ (c ∪ (a⊥ ∩ c⊥ )))
17	16	ax-r1 34	. . . 4 ((a⊥ ∩ c⊥ ) ∪ (c ∪ (a⊥ ∩ c⊥ ))) = ((a⊥ ∩ c⊥ ) ∪ ((a →2 b) ∩ (a →2 c)))
18		le1 138	. . . . 5 ((a⊥ ∩ c⊥ ) ∪ ((a →2 b) ∩ (a →2 c))) ≤ 1
19		lear 153	. . . . . . . 8 ((a →2 b) ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))) ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
20		df-i0 42	. . . . . . . . 9 ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))) = ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))
21		anor3 82	. . . . . . . . . . 11 (b⊥ ∩ c⊥ ) = (b ∪ c)⊥
22	21	ax-r5 37	. . . . . . . . . 10 ((b⊥ ∩ c⊥ ) ∪ ((a →2 b) ∩ (a →2 c))) = ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))
23	22	ax-r1 34	. . . . . . . . 9 ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c))) = ((b⊥ ∩ c⊥ ) ∪ ((a →2 b) ∩ (a →2 c)))
24	20, 23	ax-r2 35	. . . . . . . 8 ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))) = ((b⊥ ∩ c⊥ ) ∪ ((a →2 b) ∩ (a →2 c)))
25	19, 6, 24	le3tr2 133	. . . . . . 7 1 ≤ ((b⊥ ∩ c⊥ ) ∪ ((a →2 b) ∩ (a →2 c)))
26		le1 138	. . . . . . 7 ((b⊥ ∩ c⊥ ) ∪ ((a →2 b) ∩ (a →2 c))) ≤ 1
27	25, 26	lebi 137	. . . . . 6 1 = ((b⊥ ∩ c⊥ ) ∪ ((a →2 b) ∩ (a →2 c)))
28	10	u2lemle2 698	. . . . . . . . 9 a ≤ b
29	28	lecon 146	. . . . . . . 8 b⊥ ≤ a⊥
30	29	leran 145	. . . . . . 7 (b⊥ ∩ c⊥ ) ≤ (a⊥ ∩ c⊥ )
31	30	leror 144	. . . . . 6 ((b⊥ ∩ c⊥ ) ∪ ((a →2 b) ∩ (a →2 c))) ≤ ((a⊥ ∩ c⊥ ) ∪ ((a →2 b) ∩ (a →2 c)))
32	27, 31	bltr 130	. . . . 5 1 ≤ ((a⊥ ∩ c⊥ ) ∪ ((a →2 b) ∩ (a →2 c)))
33	18, 32	lebi 137	. . . 4 ((a⊥ ∩ c⊥ ) ∪ ((a →2 b) ∩ (a →2 c))) = 1
34	17, 33	ax-r2 35	. . 3 ((a⊥ ∩ c⊥ ) ∪ (c ∪ (a⊥ ∩ c⊥ ))) = 1
35	2, 4, 34	3tr2 61	. 2 (c ∪ (a⊥ ∩ c⊥ )) = 1
36	1, 35	ax-r2 35	1 (a →2 c) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9   →₀ wi0 12   →₂ wi2 14

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i0 42  df-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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