Theorem u3lemc4 685

Description: Lemma for Kalmbach implication study.

Hypothesis

Ref	Expression
ulemc3.1	a C b

Assertion

Ref	Expression
u3lemc4	(a ->3 b) = (a_|_ v b)

Proof of Theorem u3lemc4

Step	Hyp	Ref	Expression
1		df-i3 45	. 2 (a ->3 b) = (((a_|_ ^ b) v (a_|_ ^ b_|_)) v (a ^ (a_|_ v b)))
2		ulemc3.1	. . . . . . . 8 a C b
3	2	comcom3 436	. . . . . . 7 a_|_ C b
4	2	comcom4 437	. . . . . . 7 a_|_ C b_|_
5	3, 4	fh1 451	. . . . . 6 (a_|_ ^ (b v b_|_)) = ((a_|_ ^ b) v (a_|_ ^ b_|_))
6	5	ax-r1 34	. . . . 5 ((a_|_ ^ b) v (a_|_ ^ b_|_)) = (a_|_ ^ (b v b_|_))
7		df-t 40	. . . . . . . 8 1 = (b v b_|_)
8	7	ax-r1 34	. . . . . . 7 (b v b_|_) = 1
9	8	lan 70	. . . . . 6 (a_|_ ^ (b v b_|_)) = (a_|_ ^ 1)
10		an1 98	. . . . . 6 (a_|_ ^ 1) = a_|_
11	9, 10	ax-r2 35	. . . . 5 (a_|_ ^ (b v b_|_)) = a_|_
12	6, 11	ax-r2 35	. . . 4 ((a_|_ ^ b) v (a_|_ ^ b_|_)) = a_|_
13		comid 179	. . . . . . 7 a C a
14	13	comcom2 175	. . . . . 6 a C a_|_
15	14, 2	fh1 451	. . . . 5 (a ^ (a_|_ v b)) = ((a ^ a_|_) v (a ^ b))
16		ax-a2 30	. . . . . 6 ((a ^ a_|_) v (a ^ b)) = ((a ^ b) v (a ^ a_|_))
17		dff 93	. . . . . . . . 9 0 = (a ^ a_|_)
18	17	ax-r1 34	. . . . . . . 8 (a ^ a_|_) = 0
19	18	lor 66	. . . . . . 7 ((a ^ b) v (a ^ a_|_)) = ((a ^ b) v 0)
20		or0 94	. . . . . . 7 ((a ^ b) v 0) = (a ^ b)
21	19, 20	ax-r2 35	. . . . . 6 ((a ^ b) v (a ^ a_|_)) = (a ^ b)
22	16, 21	ax-r2 35	. . . . 5 ((a ^ a_|_) v (a ^ b)) = (a ^ b)
23	15, 22	ax-r2 35	. . . 4 (a ^ (a_|_ v b)) = (a ^ b)
24	12, 23	2or 67	. . 3 (((a_|_ ^ b) v (a_|_ ^ b_|_)) v (a ^ (a_|_ v b))) = (a_|_ v (a ^ b))
25	14, 2	fh4 454	. . . 4 (a_|_ v (a ^ b)) = ((a_|_ v a) ^ (a_|_ v b))
26		ancom 68	. . . . 5 ((a_|_ v a) ^ (a_|_ v b)) = ((a_|_ v b) ^ (a_|_ v a))
27		ax-a2 30	. . . . . . . 8 (a_|_ v a) = (a v a_|_)
28		df-t 40	. . . . . . . . 9 1 = (a v a_|_)
29	28	ax-r1 34	. . . . . . . 8 (a v a_|_) = 1
30	27, 29	ax-r2 35	. . . . . . 7 (a_|_ v a) = 1
31	30	lan 70	. . . . . 6 ((a_|_ v b) ^ (a_|_ v a)) = ((a_|_ v b) ^ 1)
32		an1 98	. . . . . 6 ((a_|_ v b) ^ 1) = (a_|_ v b)
33	31, 32	ax-r2 35	. . . . 5 ((a_|_ v b) ^ (a_|_ v a)) = (a_|_ v b)
34	26, 33	ax-r2 35	. . . 4 ((a_|_ v a) ^ (a_|_ v b)) = (a_|_ v b)
35	25, 34	ax-r2 35	. . 3 (a_|_ v (a ^ b)) = (a_|_ v b)
36	24, 35	ax-r2 35	. 2 (((a_|_ ^ b) v (a_|_ ^ b_|_)) v (a ^ (a_|_ v b))) = (a_|_ v b)
37	1, 36	ax-r2 35	1 (a ->3 b) = (a_|_ v b)

Syntax hints:   = wb 1   C wc 3  _|_wn 4   v wo 6   ^ wa 7  1wt 9  0wf 10   ->3 wi3 15

This theorem is referenced by:  u3lemle1 694  u3lem1 718  u3lem2 728  u3lem5 745  u3lem6 749  u3lem7 756  u3lem8 765  u3lem9 766

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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