Theorem oml4 469

Description: Orthomodular law.

Assertion

Ref	Expression
oml4	((a ≡ b) ∩ a) ≤ b

Proof of Theorem oml4

Step	Hyp	Ref	Expression
1		ancom 68	. . 3 ((a ≡ b) ∩ a) = (a ∩ (a ≡ b))
2		dfb 86	. . . . 5 (a ≡ b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
3	2	lan 70	. . . 4 (a ∩ (a ≡ b)) = (a ∩ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))
4		coman1 177	. . . . . . 7 (a ∩ b) C a
5	4	comcom 435	. . . . . 6 a C (a ∩ b)
6		coman1 177	. . . . . . . . 9 (a⊥ ∩ b⊥ ) C a⊥
7	6	comcom 435	. . . . . . . 8 a⊥ C (a⊥ ∩ b⊥ )
8	7	comcom2 175	. . . . . . 7 a⊥ C (a⊥ ∩ b⊥ )⊥
9	8	comcom5 440	. . . . . 6 a C (a⊥ ∩ b⊥ )
10	5, 9	fh1 451	. . . . 5 (a ∩ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))) = ((a ∩ (a ∩ b)) ∪ (a ∩ (a⊥ ∩ b⊥ )))
11		or0 94	. . . . . 6 ((a ∩ b) ∪ 0) = (a ∩ b)
12		anidm 103	. . . . . . . . . 10 (a ∩ a) = a
13	12	ran 71	. . . . . . . . 9 ((a ∩ a) ∩ b) = (a ∩ b)
14	13	ax-r1 34	. . . . . . . 8 (a ∩ b) = ((a ∩ a) ∩ b)
15		anass 69	. . . . . . . 8 ((a ∩ a) ∩ b) = (a ∩ (a ∩ b))
16	14, 15	ax-r2 35	. . . . . . 7 (a ∩ b) = (a ∩ (a ∩ b))
17		ancom 68	. . . . . . . . 9 (b⊥ ∩ 0) = (0 ∩ b⊥ )
18		an0 100	. . . . . . . . 9 (b⊥ ∩ 0) = 0
19		dff 93	. . . . . . . . . 10 0 = (a ∩ a⊥ )
20	19	ran 71	. . . . . . . . 9 (0 ∩ b⊥ ) = ((a ∩ a⊥ ) ∩ b⊥ )
21	17, 18, 20	3tr2 61	. . . . . . . 8 0 = ((a ∩ a⊥ ) ∩ b⊥ )
22		anass 69	. . . . . . . 8 ((a ∩ a⊥ ) ∩ b⊥ ) = (a ∩ (a⊥ ∩ b⊥ ))
23	21, 22	ax-r2 35	. . . . . . 7 0 = (a ∩ (a⊥ ∩ b⊥ ))
24	16, 23	2or 67	. . . . . 6 ((a ∩ b) ∪ 0) = ((a ∩ (a ∩ b)) ∪ (a ∩ (a⊥ ∩ b⊥ )))
25		ancom 68	. . . . . 6 (a ∩ b) = (b ∩ a)
26	11, 24, 25	3tr2 61	. . . . 5 ((a ∩ (a ∩ b)) ∪ (a ∩ (a⊥ ∩ b⊥ ))) = (b ∩ a)
27	10, 26	ax-r2 35	. . . 4 (a ∩ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))) = (b ∩ a)
28	3, 27	ax-r2 35	. . 3 (a ∩ (a ≡ b)) = (b ∩ a)
29	1, 28	ax-r2 35	. 2 ((a ≡ b) ∩ a) = (b ∩ a)
30		lea 152	. 2 (b ∩ a) ≤ b
31	29, 30	bltr 130	1 ((a ≡ b) ∩ a) ≤ b

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  0wf 10

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-le1 122  df-le2 123  df-c1 124  df-c2 125

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