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) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ ))
2	1	ax-r1 34	. . . . 5 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) = (a →5 b)
3		u5lemle2.1	. . . . 5 (a →5 b) = 1
4	2, 3	ax-r2 35	. . . 4 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) = 1
5	4	lan 70	. . 3 (a ∩ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (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) ∪ (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) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ ))) = ((a ∩ ((a ∩ b) ∪ (a⊥ ∩ b))) ∪ (a ∩ (a⊥ ∩ b⊥ )))
13	6, 8	fh1 451	. . . . . . 7 (a ∩ ((a ∩ b) ∪ (a⊥ ∩ b))) = ((a ∩ (a ∩ b)) ∪ (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)) ∪ (a ∩ (a⊥ ∩ b))) = ((a ∩ b) ∪ 0)
28		or0 94	. . . . . . . 8 ((a ∩ b) ∪ 0) = (a ∩ b)
29	27, 28	ax-r2 35	. . . . . . 7 ((a ∩ (a ∩ b)) ∪ (a ∩ (a⊥ ∩ b))) = (a ∩ b)
30	13, 29	ax-r2 35	. . . . . 6 (a ∩ ((a ∩ b) ∪ (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) ∪ (a⊥ ∩ b))) ∪ (a ∩ (a⊥ ∩ b⊥ ))) = ((a ∩ b) ∪ 0)
39	38, 28	ax-r2 35	. . . 4 ((a ∩ ((a ∩ b) ∪ (a⊥ ∩ b))) ∪ (a ∩ (a⊥ ∩ b⊥ ))) = (a ∩ b)
40	12, 39	ax-r2 35	. . 3 (a ∩ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (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   ∪ 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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