Theorem u4lemle2 718

Description: Non-tollens implication to l.e. (Contributed by NM, 11-Jan-1998.)

Hypothesis

Ref	Expression
u4lemle2.1	(a →4 b) = 1

Assertion

Ref	Expression
u4lemle2	a ≤ b

Proof of Theorem u4lemle2

Step	Hyp	Ref	Expression
1		df-i4 47	. . . . . 6 (a →4 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
2	1	ax-r1 35	. . . . 5 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) = (a →4 b)
3		u4lemle2.1	. . . . 5 (a →4 b) = 1
4	2, 3	ax-r2 36	. . . 4 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) = 1
5	4	lan 77	. . 3 (a ∩ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))) = (a ∩ 1)
6		comanr1 464	. . . . . . 7 a C (a ∩ b)
7		comanr1 464	. . . . . . . 8 a⊥ C (a⊥ ∩ b)
8	7	comcom6 459	. . . . . . 7 a C (a⊥ ∩ b)
9	6, 8	com2or 483	. . . . . 6 a C ((a ∩ b) ∪ (a⊥ ∩ b))
10	9	comcom 453	. . . . 5 ((a ∩ b) ∪ (a⊥ ∩ b)) C a
11		comor1 461	. . . . . . . . . 10 (a⊥ ∪ b) C a⊥
12	11	comcom7 460	. . . . . . . . 9 (a⊥ ∪ b) C a
13		comor2 462	. . . . . . . . 9 (a⊥ ∪ b) C b
14	12, 13	com2an 484	. . . . . . . 8 (a⊥ ∪ b) C (a ∩ b)
15	11, 13	com2an 484	. . . . . . . 8 (a⊥ ∪ b) C (a⊥ ∩ b)
16	14, 15	com2or 483	. . . . . . 7 (a⊥ ∪ b) C ((a ∩ b) ∪ (a⊥ ∩ b))
17	16	comcom 453	. . . . . 6 ((a ∩ b) ∪ (a⊥ ∩ b)) C (a⊥ ∪ b)
18		comanr2 465	. . . . . . . . 9 b C (a ∩ b)
19	18	comcom3 454	. . . . . . . 8 b⊥ C (a ∩ b)
20		comanr2 465	. . . . . . . . 9 b C (a⊥ ∩ b)
21	20	comcom3 454	. . . . . . . 8 b⊥ C (a⊥ ∩ b)
22	19, 21	com2or 483	. . . . . . 7 b⊥ C ((a ∩ b) ∪ (a⊥ ∩ b))
23	22	comcom 453	. . . . . 6 ((a ∩ b) ∪ (a⊥ ∩ b)) C b⊥
24	17, 23	com2an 484	. . . . 5 ((a ∩ b) ∪ (a⊥ ∩ b)) C ((a⊥ ∪ b) ∩ b⊥ )
25	10, 24	fh2 470	. . . 4 (a ∩ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))) = ((a ∩ ((a ∩ b) ∪ (a⊥ ∩ b))) ∪ (a ∩ ((a⊥ ∪ b) ∩ b⊥ )))
26	6, 8	fh1 469	. . . . . . 7 (a ∩ ((a ∩ b) ∪ (a⊥ ∩ b))) = ((a ∩ (a ∩ b)) ∪ (a ∩ (a⊥ ∩ b)))
27		anidm 111	. . . . . . . . . . . . 13 (a ∩ a) = a
28	27	ran 78	. . . . . . . . . . . 12 ((a ∩ a) ∩ b) = (a ∩ b)
29	28	ax-r1 35	. . . . . . . . . . 11 (a ∩ b) = ((a ∩ a) ∩ b)
30		anass 76	. . . . . . . . . . 11 ((a ∩ a) ∩ b) = (a ∩ (a ∩ b))
31	29, 30	ax-r2 36	. . . . . . . . . 10 (a ∩ b) = (a ∩ (a ∩ b))
32		dff 101	. . . . . . . . . . . . 13 0 = (a ∩ a⊥ )
33	32	lan 77	. . . . . . . . . . . 12 (b ∩ 0) = (b ∩ (a ∩ a⊥ ))
34		an0 108	. . . . . . . . . . . 12 (b ∩ 0) = 0
35		ancom 74	. . . . . . . . . . . 12 (b ∩ (a ∩ a⊥ )) = ((a ∩ a⊥ ) ∩ b)
36	33, 34, 35	3tr2 64	. . . . . . . . . . 11 0 = ((a ∩ a⊥ ) ∩ b)
37		anass 76	. . . . . . . . . . 11 ((a ∩ a⊥ ) ∩ b) = (a ∩ (a⊥ ∩ b))
38	36, 37	ax-r2 36	. . . . . . . . . 10 0 = (a ∩ (a⊥ ∩ b))
39	31, 38	2or 72	. . . . . . . . 9 ((a ∩ b) ∪ 0) = ((a ∩ (a ∩ b)) ∪ (a ∩ (a⊥ ∩ b)))
40	39	ax-r1 35	. . . . . . . 8 ((a ∩ (a ∩ b)) ∪ (a ∩ (a⊥ ∩ b))) = ((a ∩ b) ∪ 0)
41		or0 102	. . . . . . . 8 ((a ∩ b) ∪ 0) = (a ∩ b)
42	40, 41	ax-r2 36	. . . . . . 7 ((a ∩ (a ∩ b)) ∪ (a ∩ (a⊥ ∩ b))) = (a ∩ b)
43	26, 42	ax-r2 36	. . . . . 6 (a ∩ ((a ∩ b) ∪ (a⊥ ∩ b))) = (a ∩ b)
44		anor1 88	. . . . . . . 8 (a ∩ b⊥ ) = (a⊥ ∪ b)⊥
45	44	lan 77	. . . . . . 7 ((a⊥ ∪ b) ∩ (a ∩ b⊥ )) = ((a⊥ ∪ b) ∩ (a⊥ ∪ b)⊥ )
46		an12 81	. . . . . . 7 (a ∩ ((a⊥ ∪ b) ∩ b⊥ )) = ((a⊥ ∪ b) ∩ (a ∩ b⊥ ))
47		dff 101	. . . . . . 7 0 = ((a⊥ ∪ b) ∩ (a⊥ ∪ b)⊥ )
48	45, 46, 47	3tr1 63	. . . . . 6 (a ∩ ((a⊥ ∪ b) ∩ b⊥ )) = 0
49	43, 48	2or 72	. . . . 5 ((a ∩ ((a ∩ b) ∪ (a⊥ ∩ b))) ∪ (a ∩ ((a⊥ ∪ b) ∩ b⊥ ))) = ((a ∩ b) ∪ 0)
50	49, 41	ax-r2 36	. . . 4 ((a ∩ ((a ∩ b) ∪ (a⊥ ∩ b))) ∪ (a ∩ ((a⊥ ∪ b) ∩ b⊥ ))) = (a ∩ b)
51	25, 50	ax-r2 36	. . 3 (a ∩ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))) = (a ∩ b)
52		an1 106	. . 3 (a ∩ 1) = a
53	5, 51, 52	3tr2 64	. 2 (a ∩ b) = a
54	53	df2le1 135	1 a ≤ b

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   →₄ wi4 15

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-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by: (None)