Theorem u5lemle2 701

Description: Relevance implication to l.e.

Hypothesis

Ref	Expression
u5lemle2.1	(a ->5 b) = 1

Assertion

Ref	Expression
u5lemle2	a =< b

Proof of Theorem u5lemle2

Step	Hyp	Ref	Expression
1		df-i5 47	. . . . . 6 (a ->5 b) = (((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_))
2	1	ax-r1 34	. . . . 5 (((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_)) = (a ->5 b)
3		u5lemle2.1	. . . . 5 (a ->5 b) = 1
4	2, 3	ax-r2 35	. . . 4 (((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_)) = 1
5	4	lan 70	. . 3 (a ^ (((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_))) = (a ^ 1)
6		comanr1 446	. . . . . 6 a C (a ^ b)
7		comanr1 446	. . . . . . 7 a_|_ C (a_|_ ^ b)
8	7	comcom6 441	. . . . . 6 a C (a_|_ ^ b)
9	6, 8	com2or 465	. . . . 5 a C ((a ^ b) v (a_|_ ^ b))
10		comanr1 446	. . . . . 6 a_|_ C (a_|_ ^ b_|_)
11	10	comcom6 441	. . . . 5 a C (a_|_ ^ b_|_)
12	9, 11	fh1 451	. . . 4 (a ^ (((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_))) = ((a ^ ((a ^ b) v (a_|_ ^ b))) v (a ^ (a_|_ ^ b_|_)))
13	6, 8	fh1 451	. . . . . . 7 (a ^ ((a ^ b) v (a_|_ ^ b))) = ((a ^ (a ^ b)) v (a ^ (a_|_ ^ b)))
14		anass 69	. . . . . . . . . . 11 ((a ^ a) ^ b) = (a ^ (a ^ b))
15	14	ax-r1 34	. . . . . . . . . 10 (a ^ (a ^ b)) = ((a ^ a) ^ b)
16		anidm 103	. . . . . . . . . . 11 (a ^ a) = a
17	16	ran 71	. . . . . . . . . 10 ((a ^ a) ^ b) = (a ^ b)
18	15, 17	ax-r2 35	. . . . . . . . 9 (a ^ (a ^ b)) = (a ^ b)
19		ancom 68	. . . . . . . . . 10 ((a ^ a_|_) ^ b) = (b ^ (a ^ a_|_))
20		anass 69	. . . . . . . . . 10 ((a ^ a_|_) ^ b) = (a ^ (a_|_ ^ b))
21		dff 93	. . . . . . . . . . . . 13 0 = (a ^ a_|_)
22	21	ax-r1 34	. . . . . . . . . . . 12 (a ^ a_|_) = 0
23	22	lan 70	. . . . . . . . . . 11 (b ^ (a ^ a_|_)) = (b ^ 0)
24		an0 100	. . . . . . . . . . 11 (b ^ 0) = 0
25	23, 24	ax-r2 35	. . . . . . . . . 10 (b ^ (a ^ a_|_)) = 0
26	19, 20, 25	3tr2 61	. . . . . . . . 9 (a ^ (a_|_ ^ b)) = 0
27	18, 26	2or 67	. . . . . . . 8 ((a ^ (a ^ b)) v (a ^ (a_|_ ^ b))) = ((a ^ b) v 0)
28		or0 94	. . . . . . . 8 ((a ^ b) v 0) = (a ^ b)
29	27, 28	ax-r2 35	. . . . . . 7 ((a ^ (a ^ b)) v (a ^ (a_|_ ^ b))) = (a ^ b)
30	13, 29	ax-r2 35	. . . . . 6 (a ^ ((a ^ b) v (a_|_ ^ b))) = (a ^ b)
31		ancom 68	. . . . . . 7 ((a ^ a_|_) ^ b_|_) = (b_|_ ^ (a ^ a_|_))
32		anass 69	. . . . . . 7 ((a ^ a_|_) ^ b_|_) = (a ^ (a_|_ ^ b_|_))
33	21	lan 70	. . . . . . . . 9 (b_|_ ^ 0) = (b_|_ ^ (a ^ a_|_))
34	33	ax-r1 34	. . . . . . . 8 (b_|_ ^ (a ^ a_|_)) = (b_|_ ^ 0)
35		an0 100	. . . . . . . 8 (b_|_ ^ 0) = 0
36	34, 35	ax-r2 35	. . . . . . 7 (b_|_ ^ (a ^ a_|_)) = 0
37	31, 32, 36	3tr2 61	. . . . . 6 (a ^ (a_|_ ^ b_|_)) = 0
38	30, 37	2or 67	. . . . 5 ((a ^ ((a ^ b) v (a_|_ ^ b))) v (a ^ (a_|_ ^ b_|_))) = ((a ^ b) v 0)
39	38, 28	ax-r2 35	. . . 4 ((a ^ ((a ^ b) v (a_|_ ^ b))) v (a ^ (a_|_ ^ b_|_))) = (a ^ b)
40	12, 39	ax-r2 35	. . 3 (a ^ (((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_))) = (a ^ b)
41		an1 98	. . 3 (a ^ 1) = a
42	5, 40, 41	3tr2 61	. 2 (a ^ b) = a
43	42	df2le1 127	1 a =< b

Syntax hints:   = wb 1   =< wle 2  _|_wn 4   v wo 6   ^ wa 7  1wt 9  0wf 10   ->5 wi5 17

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-i5 47  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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