Theorem u1lemanb 597

Description: Lemma for Sasaki implication study.

Assertion

Ref	Expression
u1lemanb	((a ->1 b) ^ b_|_) = (a_|_ ^ b_|_)

Proof of Theorem u1lemanb

Step	Hyp	Ref	Expression
1		df-i1 43	. . 3 (a ->1 b) = (a_|_ v (a ^ b))
2	1	ran 71	. 2 ((a ->1 b) ^ b_|_) = ((a_|_ v (a ^ b)) ^ b_|_)
3		ax-a2 30	. . . 4 (a_|_ v (a ^ b)) = ((a ^ b) v a_|_)
4	3	ran 71	. . 3 ((a_|_ v (a ^ b)) ^ b_|_) = (((a ^ b) v a_|_) ^ b_|_)
5		coman2 178	. . . . . 6 (a ^ b) C b
6	5	comcom2 175	. . . . 5 (a ^ b) C b_|_
7		coman1 177	. . . . . 6 (a ^ b) C a
8	7	comcom2 175	. . . . 5 (a ^ b) C a_|_
9	6, 8	fh2r 456	. . . 4 (((a ^ b) v a_|_) ^ b_|_) = (((a ^ b) ^ b_|_) v (a_|_ ^ b_|_))
10		ax-a2 30	. . . . 5 (((a ^ b) ^ b_|_) v (a_|_ ^ b_|_)) = ((a_|_ ^ b_|_) v ((a ^ b) ^ b_|_))
11		anass 69	. . . . . . . 8 ((a ^ b) ^ b_|_) = (a ^ (b ^ b_|_))
12		dff 93	. . . . . . . . . . 11 0 = (b ^ b_|_)
13	12	lan 70	. . . . . . . . . 10 (a ^ 0) = (a ^ (b ^ b_|_))
14	13	ax-r1 34	. . . . . . . . 9 (a ^ (b ^ b_|_)) = (a ^ 0)
15		an0 100	. . . . . . . . 9 (a ^ 0) = 0
16	14, 15	ax-r2 35	. . . . . . . 8 (a ^ (b ^ b_|_)) = 0
17	11, 16	ax-r2 35	. . . . . . 7 ((a ^ b) ^ b_|_) = 0
18	17	lor 66	. . . . . 6 ((a_|_ ^ b_|_) v ((a ^ b) ^ b_|_)) = ((a_|_ ^ b_|_) v 0)
19		or0 94	. . . . . 6 ((a_|_ ^ b_|_) v 0) = (a_|_ ^ b_|_)
20	18, 19	ax-r2 35	. . . . 5 ((a_|_ ^ b_|_) v ((a ^ b) ^ b_|_)) = (a_|_ ^ b_|_)
21	10, 20	ax-r2 35	. . . 4 (((a ^ b) ^ b_|_) v (a_|_ ^ b_|_)) = (a_|_ ^ b_|_)
22	9, 21	ax-r2 35	. . 3 (((a ^ b) v a_|_) ^ b_|_) = (a_|_ ^ b_|_)
23	4, 22	ax-r2 35	. 2 ((a_|_ v (a ^ b)) ^ b_|_) = (a_|_ ^ b_|_)
24	2, 23	ax-r2 35	1 ((a ->1 b) ^ b_|_) = (a_|_ ^ b_|_)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  0wf 10   ->1 wi1 13

This theorem is referenced by:  u1lemnob 652  u3lem14a 773  negantlem5 835

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-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

metamath.org