Theorem womle2a 295

Description: An equivalent to the WOM law. (Contributed by NM, 24-Jan-1999.)

Hypothesis

Ref	Expression
womle2a.1	(a ∩ (a →2 b)) ≤ ((a →2 b)⊥ ∪ (a →1 b))

Assertion

Ref	Expression
womle2a	((a →2 b)⊥ ∪ (a →1 b)) = 1

Proof of Theorem womle2a

Step	Hyp	Ref	Expression
1		or4 84	. . 3 (((a →2 b)⊥ ∪ (a →2 b)⊥ ) ∪ ((a →1 b) ∪ a⊥ )) = (((a →2 b)⊥ ∪ (a →1 b)) ∪ ((a →2 b)⊥ ∪ a⊥ ))
2		oridm 110	. . . 4 ((a →2 b)⊥ ∪ (a →2 b)⊥ ) = (a →2 b)⊥
3		df-i1 44	. . . . . 6 (a →1 b) = (a⊥ ∪ (a ∩ b))
4	3	ax-r5 38	. . . . 5 ((a →1 b) ∪ a⊥ ) = ((a⊥ ∪ (a ∩ b)) ∪ a⊥ )
5		oridm 110	. . . . . . 7 (a⊥ ∪ a⊥ ) = a⊥
6	5	ax-r5 38	. . . . . 6 ((a⊥ ∪ a⊥ ) ∪ (a ∩ b)) = (a⊥ ∪ (a ∩ b))
7		or32 82	. . . . . 6 ((a⊥ ∪ (a ∩ b)) ∪ a⊥ ) = ((a⊥ ∪ a⊥ ) ∪ (a ∩ b))
8	6, 7, 3	3tr1 63	. . . . 5 ((a⊥ ∪ (a ∩ b)) ∪ a⊥ ) = (a →1 b)
9	4, 8	ax-r2 36	. . . 4 ((a →1 b) ∪ a⊥ ) = (a →1 b)
10	2, 9	2or 72	. . 3 (((a →2 b)⊥ ∪ (a →2 b)⊥ ) ∪ ((a →1 b) ∪ a⊥ )) = ((a →2 b)⊥ ∪ (a →1 b))
11		ax-a2 31	. . . . 5 ((a →2 b)⊥ ∪ a⊥ ) = (a⊥ ∪ (a →2 b)⊥ )
12		oran3 93	. . . . 5 (a⊥ ∪ (a →2 b)⊥ ) = (a ∩ (a →2 b))⊥
13	11, 12	ax-r2 36	. . . 4 ((a →2 b)⊥ ∪ a⊥ ) = (a ∩ (a →2 b))⊥
14	13	lor 70	. . 3 (((a →2 b)⊥ ∪ (a →1 b)) ∪ ((a →2 b)⊥ ∪ a⊥ )) = (((a →2 b)⊥ ∪ (a →1 b)) ∪ (a ∩ (a →2 b))⊥ )
15	1, 10, 14	3tr2 64	. 2 ((a →2 b)⊥ ∪ (a →1 b)) = (((a →2 b)⊥ ∪ (a →1 b)) ∪ (a ∩ (a →2 b))⊥ )
16		le1 146	. . 3 (((a →2 b)⊥ ∪ (a →1 b)) ∪ (a ∩ (a →2 b))⊥ ) ≤ 1
17		df-t 41	. . . 4 1 = ((a ∩ (a →2 b)) ∪ (a ∩ (a →2 b))⊥ )
18		womle2a.1	. . . . 5 (a ∩ (a →2 b)) ≤ ((a →2 b)⊥ ∪ (a →1 b))
19	18	leror 152	. . . 4 ((a ∩ (a →2 b)) ∪ (a ∩ (a →2 b))⊥ ) ≤ (((a →2 b)⊥ ∪ (a →1 b)) ∪ (a ∩ (a →2 b))⊥ )
20	17, 19	bltr 138	. . 3 1 ≤ (((a →2 b)⊥ ∪ (a →1 b)) ∪ (a ∩ (a →2 b))⊥ )
21	16, 20	lebi 145	. 2 (((a →2 b)⊥ ∪ (a →1 b)) ∪ (a ∩ (a →2 b))⊥ ) = 1
22	15, 21	ax-r2 36	1 ((a →2 b)⊥ ∪ (a →1 b)) = 1

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₁ wi1 12   →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

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131

This theorem is referenced by:  womle  298