Theorem i3lem1 486

Description: Lemma for Kalmbach implication.

Hypothesis

Ref	Expression
i3lem.1	(a ->3 b) = 1

Assertion

Ref	Expression
i3lem1	((a_|_ ^ b) v (a_|_ ^ b_|_)) = a_|_

Proof of Theorem i3lem1

Step	Hyp	Ref	Expression
1		coman1 177	. . . . . . 7 (a_|_ ^ b_|_) C a_|_
2	1	comcom 435	. . . . . 6 a_|_ C (a_|_ ^ b_|_)
3		comorr 176	. . . . . . 7 a_|_ C (a_|_ v b)
4		comorr 176	. . . . . . . 8 a C (a v (a_|_ ^ b))
5	4	comcom3 436	. . . . . . 7 a_|_ C (a v (a_|_ ^ b))
6	3, 5	com2an 466	. . . . . 6 a_|_ C ((a_|_ v b) ^ (a v (a_|_ ^ b)))
7	2, 6	fh1 451	. . . . 5 (a_|_ ^ ((a_|_ ^ b_|_) v ((a_|_ v b) ^ (a v (a_|_ ^ b))))) = ((a_|_ ^ (a_|_ ^ b_|_)) v (a_|_ ^ ((a_|_ v b) ^ (a v (a_|_ ^ b)))))
8		anass 69	. . . . . . . . 9 ((a_|_ ^ a_|_) ^ b_|_) = (a_|_ ^ (a_|_ ^ b_|_))
9	8	ax-r1 34	. . . . . . . 8 (a_|_ ^ (a_|_ ^ b_|_)) = ((a_|_ ^ a_|_) ^ b_|_)
10		anidm 103	. . . . . . . . 9 (a_|_ ^ a_|_) = a_|_
11	10	ran 71	. . . . . . . 8 ((a_|_ ^ a_|_) ^ b_|_) = (a_|_ ^ b_|_)
12	9, 11	ax-r2 35	. . . . . . 7 (a_|_ ^ (a_|_ ^ b_|_)) = (a_|_ ^ b_|_)
13		anass 69	. . . . . . . . 9 ((a_|_ ^ (a_|_ v b)) ^ (a v (a_|_ ^ b))) = (a_|_ ^ ((a_|_ v b) ^ (a v (a_|_ ^ b))))
14	13	ax-r1 34	. . . . . . . 8 (a_|_ ^ ((a_|_ v b) ^ (a v (a_|_ ^ b)))) = ((a_|_ ^ (a_|_ v b)) ^ (a v (a_|_ ^ b)))
15		a5c 113	. . . . . . . . . 10 (a_|_ ^ (a_|_ v b)) = a_|_
16	15	ran 71	. . . . . . . . 9 ((a_|_ ^ (a_|_ v b)) ^ (a v (a_|_ ^ b))) = (a_|_ ^ (a v (a_|_ ^ b)))
17		omlan 430	. . . . . . . . 9 (a_|_ ^ (a v (a_|_ ^ b))) = (a_|_ ^ b)
18	16, 17	ax-r2 35	. . . . . . . 8 ((a_|_ ^ (a_|_ v b)) ^ (a v (a_|_ ^ b))) = (a_|_ ^ b)
19	14, 18	ax-r2 35	. . . . . . 7 (a_|_ ^ ((a_|_ v b) ^ (a v (a_|_ ^ b)))) = (a_|_ ^ b)
20	12, 19	2or 67	. . . . . 6 ((a_|_ ^ (a_|_ ^ b_|_)) v (a_|_ ^ ((a_|_ v b) ^ (a v (a_|_ ^ b))))) = ((a_|_ ^ b_|_) v (a_|_ ^ b))
21		ax-a2 30	. . . . . 6 ((a_|_ ^ b_|_) v (a_|_ ^ b)) = ((a_|_ ^ b) v (a_|_ ^ b_|_))
22	20, 21	ax-r2 35	. . . . 5 ((a_|_ ^ (a_|_ ^ b_|_)) v (a_|_ ^ ((a_|_ v b) ^ (a v (a_|_ ^ b))))) = ((a_|_ ^ b) v (a_|_ ^ b_|_))
23	7, 22	ax-r2 35	. . . 4 (a_|_ ^ ((a_|_ ^ b_|_) v ((a_|_ v b) ^ (a v (a_|_ ^ b))))) = ((a_|_ ^ b) v (a_|_ ^ b_|_))
24	23	ax-r1 34	. . 3 ((a_|_ ^ b) v (a_|_ ^ b_|_)) = (a_|_ ^ ((a_|_ ^ b_|_) v ((a_|_ v b) ^ (a v (a_|_ ^ b)))))
25		df2i3 480	. . . . . 6 (a ->3 b) = ((a_|_ ^ b_|_) v ((a_|_ v b) ^ (a v (a_|_ ^ b))))
26	25	ax-r1 34	. . . . 5 ((a_|_ ^ b_|_) v ((a_|_ v b) ^ (a v (a_|_ ^ b)))) = (a ->3 b)
27		i3lem.1	. . . . 5 (a ->3 b) = 1
28	26, 27	ax-r2 35	. . . 4 ((a_|_ ^ b_|_) v ((a_|_ v b) ^ (a v (a_|_ ^ b)))) = 1
29	28	lan 70	. . 3 (a_|_ ^ ((a_|_ ^ b_|_) v ((a_|_ v b) ^ (a v (a_|_ ^ b))))) = (a_|_ ^ 1)
30	24, 29	ax-r2 35	. 2 ((a_|_ ^ b) v (a_|_ ^ b_|_)) = (a_|_ ^ 1)
31		an1 98	. 2 (a_|_ ^ 1) = a_|_
32	30, 31	ax-r2 35	1 ((a_|_ ^ b) v (a_|_ ^ b_|_)) = a_|_

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  1wt 9   ->3 wi3 15

This theorem is referenced by:  i3lem2 487  i3lem3 488  i3lem4 489

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-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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