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) v (a_|_ ^ b_|_))
3	2	lan 70	. . . 4 (a ^ (a == b)) = (a ^ ((a ^ b) v (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) v (a_|_ ^ b_|_))) = ((a ^ (a ^ b)) v (a ^ (a_|_ ^ b_|_)))
11		or0 94	. . . . . 6 ((a ^ b) v 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) v 0) = ((a ^ (a ^ b)) v (a ^ (a_|_ ^ b_|_)))
25		ancom 68	. . . . . 6 (a ^ b) = (b ^ a)
26	11, 24, 25	3tr2 61	. . . . 5 ((a ^ (a ^ b)) v (a ^ (a_|_ ^ b_|_))) = (b ^ a)
27	10, 26	ax-r2 35	. . . 4 (a ^ ((a ^ b) v (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   v 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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