Theorem u5lemaa 604

Description: Lemma for relevance implication study.

Assertion

Ref	Expression
u5lemaa	((a ->5 b) ^ a) = (a ^ b)

Proof of Theorem u5lemaa

Step	Hyp	Ref	Expression
1		df-i5 48	. . 3 (a ->5 b) = (((a ^ b) v (a' ^ b)) v (a' ^ b'))
2	1	ran 78	. 2 ((a ->5 b) ^ a) = ((((a ^ b) v (a' ^ b)) v (a' ^ b')) ^ a)
3		comanr1 464	. . . . 5 a C (a ^ b)
4		comanr1 464	. . . . . 6 a' C (a' ^ b)
5	4	comcom6 459	. . . . 5 a C (a' ^ b)
6	3, 5	com2or 483	. . . 4 a C ((a ^ b) v (a' ^ b))
7		comanr1 464	. . . . 5 a' C (a' ^ b')
8	7	comcom6 459	. . . 4 a C (a' ^ b')
9	6, 8	fh1r 473	. . 3 ((((a ^ b) v (a' ^ b)) v (a' ^ b')) ^ a) = ((((a ^ b) v (a' ^ b)) ^ a) v ((a' ^ b') ^ a))
10	3, 5	fh1r 473	. . . . . 6 (((a ^ b) v (a' ^ b)) ^ a) = (((a ^ b) ^ a) v ((a' ^ b) ^ a))
11		an32 83	. . . . . . . . 9 ((a ^ b) ^ a) = ((a ^ a) ^ b)
12		anidm 111	. . . . . . . . . 10 (a ^ a) = a
13	12	ran 78	. . . . . . . . 9 ((a ^ a) ^ b) = (a ^ b)
14	11, 13	ax-r2 36	. . . . . . . 8 ((a ^ b) ^ a) = (a ^ b)
15		an32 83	. . . . . . . . 9 ((a' ^ b) ^ a) = ((a' ^ a) ^ b)
16		ancom 74	. . . . . . . . . 10 ((a' ^ a) ^ b) = (b ^ (a' ^ a))
17		ancom 74	. . . . . . . . . . . . . 14 (a ^ a') = (a' ^ a)
18	17	ax-r1 35	. . . . . . . . . . . . 13 (a' ^ a) = (a ^ a')
19		dff 101	. . . . . . . . . . . . . 14 0 = (a ^ a')
20	19	ax-r1 35	. . . . . . . . . . . . 13 (a ^ a') = 0
21	18, 20	ax-r2 36	. . . . . . . . . . . 12 (a' ^ a) = 0
22	21	lan 77	. . . . . . . . . . 11 (b ^ (a' ^ a)) = (b ^ 0)
23		an0 108	. . . . . . . . . . 11 (b ^ 0) = 0
24	22, 23	ax-r2 36	. . . . . . . . . 10 (b ^ (a' ^ a)) = 0
25	16, 24	ax-r2 36	. . . . . . . . 9 ((a' ^ a) ^ b) = 0
26	15, 25	ax-r2 36	. . . . . . . 8 ((a' ^ b) ^ a) = 0
27	14, 26	2or 72	. . . . . . 7 (((a ^ b) ^ a) v ((a' ^ b) ^ a)) = ((a ^ b) v 0)
28		or0 102	. . . . . . 7 ((a ^ b) v 0) = (a ^ b)
29	27, 28	ax-r2 36	. . . . . 6 (((a ^ b) ^ a) v ((a' ^ b) ^ a)) = (a ^ b)
30	10, 29	ax-r2 36	. . . . 5 (((a ^ b) v (a' ^ b)) ^ a) = (a ^ b)
31		ancom 74	. . . . 5 ((a' ^ b') ^ a) = (a ^ (a' ^ b'))
32	30, 31	2or 72	. . . 4 ((((a ^ b) v (a' ^ b)) ^ a) v ((a' ^ b') ^ a)) = ((a ^ b) v (a ^ (a' ^ b')))
33	3, 8	fh4 472	. . . . 5 ((a ^ b) v (a ^ (a' ^ b'))) = (((a ^ b) v a) ^ ((a ^ b) v (a' ^ b')))
34		ax-a2 31	. . . . . . . 8 ((a ^ b) v a) = (a v (a ^ b))
35		a5b 120	. . . . . . . 8 (a v (a ^ b)) = a
36	34, 35	ax-r2 36	. . . . . . 7 ((a ^ b) v a) = a
37	36	ran 78	. . . . . 6 (((a ^ b) v a) ^ ((a ^ b) v (a' ^ b'))) = (a ^ ((a ^ b) v (a' ^ b')))
38	3, 8	fh1 469	. . . . . . 7 (a ^ ((a ^ b) v (a' ^ b'))) = ((a ^ (a ^ b)) v (a ^ (a' ^ b')))
39		anass 76	. . . . . . . . . . 11 ((a ^ a) ^ b) = (a ^ (a ^ b))
40	39	ax-r1 35	. . . . . . . . . 10 (a ^ (a ^ b)) = ((a ^ a) ^ b)
41	40, 13	ax-r2 36	. . . . . . . . 9 (a ^ (a ^ b)) = (a ^ b)
42		anass 76	. . . . . . . . . . 11 ((a ^ a') ^ b') = (a ^ (a' ^ b'))
43	42	ax-r1 35	. . . . . . . . . 10 (a ^ (a' ^ b')) = ((a ^ a') ^ b')
44		ancom 74	. . . . . . . . . . 11 ((a ^ a') ^ b') = (b' ^ (a ^ a'))
45	19	lan 77	. . . . . . . . . . . . 13 (b' ^ 0) = (b' ^ (a ^ a'))
46	45	ax-r1 35	. . . . . . . . . . . 12 (b' ^ (a ^ a')) = (b' ^ 0)
47		an0 108	. . . . . . . . . . . 12 (b' ^ 0) = 0
48	46, 47	ax-r2 36	. . . . . . . . . . 11 (b' ^ (a ^ a')) = 0
49	44, 48	ax-r2 36	. . . . . . . . . 10 ((a ^ a') ^ b') = 0
50	43, 49	ax-r2 36	. . . . . . . . 9 (a ^ (a' ^ b')) = 0
51	41, 50	2or 72	. . . . . . . 8 ((a ^ (a ^ b)) v (a ^ (a' ^ b'))) = ((a ^ b) v 0)
52	51, 28	ax-r2 36	. . . . . . 7 ((a ^ (a ^ b)) v (a ^ (a' ^ b'))) = (a ^ b)
53	38, 52	ax-r2 36	. . . . . 6 (a ^ ((a ^ b) v (a' ^ b'))) = (a ^ b)
54	37, 53	ax-r2 36	. . . . 5 (((a ^ b) v a) ^ ((a ^ b) v (a' ^ b'))) = (a ^ b)
55	33, 54	ax-r2 36	. . . 4 ((a ^ b) v (a ^ (a' ^ b'))) = (a ^ b)
56	32, 55	ax-r2 36	. . 3 ((((a ^ b) v (a' ^ b)) ^ a) v ((a' ^ b') ^ a)) = (a ^ b)
57	9, 56	ax-r2 36	. 2 ((((a ^ b) v (a' ^ b)) v (a' ^ b')) ^ a) = (a ^ b)
58	2, 57	ax-r2 36	1 ((a ->5 b) ^ a) = (a ^ b)

Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  0wf 9   ->5 wi5 16

This theorem is referenced by:  u5lemnona 669  u5lembi 725

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i5 48  df-le1 130  df-le2 131  df-c1 132  df-c2 133

Copyright terms: Public domain