Theorem woml7 419

Description: Variant of weakly orthomodular law.

Assertion

Ref	Expression
woml7	(((a →2 b) ∩ (b →2 a))⊥ ∪ (a ≡ b)) = 1

Proof of Theorem woml7

Step	Hyp	Ref	Expression
1		df-i2 44	. . . . . . . 8 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
2		ax-a2 30	. . . . . . . 8 (b ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ b)
3	1, 2	ax-r2 35	. . . . . . 7 (a →2 b) = ((a⊥ ∩ b⊥ ) ∪ b)
4		df-i2 44	. . . . . . . 8 (b →2 a) = (a ∪ (b⊥ ∩ a⊥ ))
5		ax-a2 30	. . . . . . . 8 (a ∪ (b⊥ ∩ a⊥ )) = ((b⊥ ∩ a⊥ ) ∪ a)
6		ancom 68	. . . . . . . . 9 (b⊥ ∩ a⊥ ) = (a⊥ ∩ b⊥ )
7	6	ax-r5 37	. . . . . . . 8 ((b⊥ ∩ a⊥ ) ∪ a) = ((a⊥ ∩ b⊥ ) ∪ a)
8	4, 5, 7	3tr 62	. . . . . . 7 (b →2 a) = ((a⊥ ∩ b⊥ ) ∪ a)
9	3, 8	2an 72	. . . . . 6 ((a →2 b) ∩ (b →2 a)) = (((a⊥ ∩ b⊥ ) ∪ b) ∩ ((a⊥ ∩ b⊥ ) ∪ a))
10		ancom 68	. . . . . 6 (((a⊥ ∩ b⊥ ) ∪ b) ∩ ((a⊥ ∩ b⊥ ) ∪ a)) = (((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))
11	9, 10	ax-r2 35	. . . . 5 ((a →2 b) ∩ (b →2 a)) = (((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))
12	11	ax-r4 36	. . . 4 ((a →2 b) ∩ (b →2 a))⊥ = (((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))⊥
13		id 58	. . . 4 (((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))⊥ = (((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))⊥
14	12, 13	ax-r2 35	. . 3 ((a →2 b) ∩ (b →2 a))⊥ = (((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))⊥
15		dfb 86	. . 3 (a ≡ b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
16	14, 15	2or 67	. 2 (((a →2 b) ∩ (b →2 a))⊥ ∪ (a ≡ b)) = ((((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))
17		1b 109	. . 3 (1 ≡ ((((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))) = ((((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))
18	17	ax-r1 34	. 2 ((((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))) = (1 ≡ ((((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))))
19		df-t 40	. . . . 5 1 = (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ )
20		ax-a2 30	. . . . 5 (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ ) = (((a ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))
21	19, 20	ax-r2 35	. . . 4 1 = (((a ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))
22	21	bi1 110	. . 3 (1 ≡ (((a ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))) = 1
23		wa2 184	. . . . . 6 (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ≡ ((a⊥ ∩ b⊥ ) ∪ (a ∩ b))) = 1
24		wcoman1 395	. . . . . . . . 9 C ((a⊥ ∩ b⊥ ), a⊥ ) = 1
25	24	wcomcom3 398	. . . . . . . 8 C ((a⊥ ∩ b⊥ )⊥ , a⊥ ) = 1
26	25	wcomcom5 402	. . . . . . 7 C ((a⊥ ∩ b⊥ ), a) = 1
27		ancom 68	. . . . . . . . . . 11 (a⊥ ∩ b⊥ ) = (b⊥ ∩ a⊥ )
28	27	bi1 110	. . . . . . . . . 10 ((a⊥ ∩ b⊥ ) ≡ (b⊥ ∩ a⊥ )) = 1
29		wcoman1 395	. . . . . . . . . 10 C ((b⊥ ∩ a⊥ ), b⊥ ) = 1
30	28, 29	wbctr 392	. . . . . . . . 9 C ((a⊥ ∩ b⊥ ), b⊥ ) = 1
31	30	wcomcom3 398	. . . . . . . 8 C ((a⊥ ∩ b⊥ )⊥ , b⊥ ) = 1
32	31	wcomcom5 402	. . . . . . 7 C ((a⊥ ∩ b⊥ ), b) = 1
33	26, 32	wfh3 407	. . . . . 6 (((a⊥ ∩ b⊥ ) ∪ (a ∩ b)) ≡ (((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))) = 1
34	23, 33	wr2 353	. . . . 5 (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ≡ (((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))) = 1
35	34	wr4 191	. . . 4 (((a ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ ≡ (((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))⊥ ) = 1
36	35	wr5-2v 348	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))) ≡ ((((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))) = 1
37	22, 36	wr2 353	. 2 (1 ≡ ((((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))) = 1
38	16, 18, 37	3tr 62	1 (((a →2 b) ∩ (b →2 a))⊥ ∪ (a ≡ b)) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 9   →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-wom 343

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-i2 44  df-le 121  df-le1 122  df-le2 123  df-cmtr 126

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