Theorem 3vded21 817

Description: A 3-variable theorem. Experiment with weak deduction theorem. (Contributed by NM, 31-Oct-1998.)

Hypotheses

Ref	Expression
3vded21.1	c ≤ ((a →0 b) →0 (c →2 b))
3vded21.2	c ≤ (a →0 b)

Assertion

Ref	Expression
3vded21	c ≤ b

Proof of Theorem 3vded21

Step	Hyp	Ref	Expression
1		3vded21.2	. . . . . . 7 c ≤ (a →0 b)
2		df-i0 43	. . . . . . 7 (a →0 b) = (a⊥ ∪ b)
3	1, 2	lbtr 139	. . . . . 6 c ≤ (a⊥ ∪ b)
4		3vded21.1	. . . . . . 7 c ≤ ((a →0 b) →0 (c →2 b))
5		df-i0 43	. . . . . . . 8 ((a →0 b) →0 (c →2 b)) = ((a →0 b)⊥ ∪ (c →2 b))
6	2	ax-r4 37	. . . . . . . . 9 (a →0 b)⊥ = (a⊥ ∪ b)⊥
7		df-i2 45	. . . . . . . . . 10 (c →2 b) = (b ∪ (c⊥ ∩ b⊥ ))
8		anor3 90	. . . . . . . . . . 11 (c⊥ ∩ b⊥ ) = (c ∪ b)⊥
9	8	lor 70	. . . . . . . . . 10 (b ∪ (c⊥ ∩ b⊥ )) = (b ∪ (c ∪ b)⊥ )
10	7, 9	ax-r2 36	. . . . . . . . 9 (c →2 b) = (b ∪ (c ∪ b)⊥ )
11	6, 10	2or 72	. . . . . . . 8 ((a →0 b)⊥ ∪ (c →2 b)) = ((a⊥ ∪ b)⊥ ∪ (b ∪ (c ∪ b)⊥ ))
12		ax-a2 31	. . . . . . . 8 ((a⊥ ∪ b)⊥ ∪ (b ∪ (c ∪ b)⊥ )) = ((b ∪ (c ∪ b)⊥ ) ∪ (a⊥ ∪ b)⊥ )
13	5, 11, 12	3tr 65	. . . . . . 7 ((a →0 b) →0 (c →2 b)) = ((b ∪ (c ∪ b)⊥ ) ∪ (a⊥ ∪ b)⊥ )
14	4, 13	lbtr 139	. . . . . 6 c ≤ ((b ∪ (c ∪ b)⊥ ) ∪ (a⊥ ∪ b)⊥ )
15	3, 14	ler2an 173	. . . . 5 c ≤ ((a⊥ ∪ b) ∩ ((b ∪ (c ∪ b)⊥ ) ∪ (a⊥ ∪ b)⊥ ))
16		comor2 462	. . . . . . . 8 (a⊥ ∪ b) C b
17	3	leror 152	. . . . . . . . . . . 12 (c ∪ b) ≤ ((a⊥ ∪ b) ∪ b)
18		ax-a3 32	. . . . . . . . . . . . 13 ((a⊥ ∪ b) ∪ b) = (a⊥ ∪ (b ∪ b))
19		oridm 110	. . . . . . . . . . . . . 14 (b ∪ b) = b
20	19	lor 70	. . . . . . . . . . . . 13 (a⊥ ∪ (b ∪ b)) = (a⊥ ∪ b)
21	18, 20	ax-r2 36	. . . . . . . . . . . 12 ((a⊥ ∪ b) ∪ b) = (a⊥ ∪ b)
22	17, 21	lbtr 139	. . . . . . . . . . 11 (c ∪ b) ≤ (a⊥ ∪ b)
23	22	lecom 180	. . . . . . . . . 10 (c ∪ b) C (a⊥ ∪ b)
24	23	comcom 453	. . . . . . . . 9 (a⊥ ∪ b) C (c ∪ b)
25	24	comcom2 183	. . . . . . . 8 (a⊥ ∪ b) C (c ∪ b)⊥
26	16, 25	com2or 483	. . . . . . 7 (a⊥ ∪ b) C (b ∪ (c ∪ b)⊥ )
27		comid 187	. . . . . . . 8 (a⊥ ∪ b) C (a⊥ ∪ b)
28	27	comcom2 183	. . . . . . 7 (a⊥ ∪ b) C (a⊥ ∪ b)⊥
29	26, 28	fh1 469	. . . . . 6 ((a⊥ ∪ b) ∩ ((b ∪ (c ∪ b)⊥ ) ∪ (a⊥ ∪ b)⊥ )) = (((a⊥ ∪ b) ∩ (b ∪ (c ∪ b)⊥ )) ∪ ((a⊥ ∪ b) ∩ (a⊥ ∪ b)⊥ ))
30		or0 102	. . . . . . 7 ((((a⊥ ∪ b) ∩ b) ∪ ((a⊥ ∪ b) ∩ (c ∪ b)⊥ )) ∪ 0) = (((a⊥ ∪ b) ∩ b) ∪ ((a⊥ ∪ b) ∩ (c ∪ b)⊥ ))
31	16, 25	fh1 469	. . . . . . . . 9 ((a⊥ ∪ b) ∩ (b ∪ (c ∪ b)⊥ )) = (((a⊥ ∪ b) ∩ b) ∪ ((a⊥ ∪ b) ∩ (c ∪ b)⊥ ))
32	31	ax-r1 35	. . . . . . . 8 (((a⊥ ∪ b) ∩ b) ∪ ((a⊥ ∪ b) ∩ (c ∪ b)⊥ )) = ((a⊥ ∪ b) ∩ (b ∪ (c ∪ b)⊥ ))
33		dff 101	. . . . . . . 8 0 = ((a⊥ ∪ b) ∩ (a⊥ ∪ b)⊥ )
34	32, 33	2or 72	. . . . . . 7 ((((a⊥ ∪ b) ∩ b) ∪ ((a⊥ ∪ b) ∩ (c ∪ b)⊥ )) ∪ 0) = (((a⊥ ∪ b) ∩ (b ∪ (c ∪ b)⊥ )) ∪ ((a⊥ ∪ b) ∩ (a⊥ ∪ b)⊥ ))
35		ax-a2 31	. . . . . . . . . 10 (a⊥ ∪ b) = (b ∪ a⊥ )
36	35	ran 78	. . . . . . . . 9 ((a⊥ ∪ b) ∩ b) = ((b ∪ a⊥ ) ∩ b)
37		ancom 74	. . . . . . . . 9 ((b ∪ a⊥ ) ∩ b) = (b ∩ (b ∪ a⊥ ))
38		anabs 121	. . . . . . . . 9 (b ∩ (b ∪ a⊥ )) = b
39	36, 37, 38	3tr 65	. . . . . . . 8 ((a⊥ ∪ b) ∩ b) = b
40	39	ax-r5 38	. . . . . . 7 (((a⊥ ∪ b) ∩ b) ∪ ((a⊥ ∪ b) ∩ (c ∪ b)⊥ )) = (b ∪ ((a⊥ ∪ b) ∩ (c ∪ b)⊥ ))
41	30, 34, 40	3tr2 64	. . . . . 6 (((a⊥ ∪ b) ∩ (b ∪ (c ∪ b)⊥ )) ∪ ((a⊥ ∪ b) ∩ (a⊥ ∪ b)⊥ )) = (b ∪ ((a⊥ ∪ b) ∩ (c ∪ b)⊥ ))
42	29, 41	ax-r2 36	. . . . 5 ((a⊥ ∪ b) ∩ ((b ∪ (c ∪ b)⊥ ) ∪ (a⊥ ∪ b)⊥ )) = (b ∪ ((a⊥ ∪ b) ∩ (c ∪ b)⊥ ))
43	15, 42	lbtr 139	. . . 4 c ≤ (b ∪ ((a⊥ ∪ b) ∩ (c ∪ b)⊥ ))
44	43	leran 153	. . 3 (c ∩ (c ∪ b)) ≤ ((b ∪ ((a⊥ ∪ b) ∩ (c ∪ b)⊥ )) ∩ (c ∪ b))
45		anabs 121	. . 3 (c ∩ (c ∪ b)) = c
46		comor2 462	. . . . 5 (c ∪ b) C b
47		comid 187	. . . . . . 7 (c ∪ b) C (c ∪ b)
48	47	comcom2 183	. . . . . 6 (c ∪ b) C (c ∪ b)⊥
49	23, 48	com2an 484	. . . . 5 (c ∪ b) C ((a⊥ ∪ b) ∩ (c ∪ b)⊥ )
50	46, 49	fh1r 473	. . . 4 ((b ∪ ((a⊥ ∪ b) ∩ (c ∪ b)⊥ )) ∩ (c ∪ b)) = ((b ∩ (c ∪ b)) ∪ (((a⊥ ∪ b) ∩ (c ∪ b)⊥ ) ∩ (c ∪ b)))
51		ax-a2 31	. . . . . . 7 (c ∪ b) = (b ∪ c)
52	51	lan 77	. . . . . 6 (b ∩ (c ∪ b)) = (b ∩ (b ∪ c))
53		anabs 121	. . . . . 6 (b ∩ (b ∪ c)) = b
54	52, 53	ax-r2 36	. . . . 5 (b ∩ (c ∪ b)) = b
55		an32 83	. . . . . 6 (((a⊥ ∪ b) ∩ (c ∪ b)⊥ ) ∩ (c ∪ b)) = (((a⊥ ∪ b) ∩ (c ∪ b)) ∩ (c ∪ b)⊥ )
56		anass 76	. . . . . 6 (((a⊥ ∪ b) ∩ (c ∪ b)) ∩ (c ∪ b)⊥ ) = ((a⊥ ∪ b) ∩ ((c ∪ b) ∩ (c ∪ b)⊥ ))
57		dff 101	. . . . . . . . 9 0 = ((c ∪ b) ∩ (c ∪ b)⊥ )
58	57	lan 77	. . . . . . . 8 ((a⊥ ∪ b) ∩ 0) = ((a⊥ ∪ b) ∩ ((c ∪ b) ∩ (c ∪ b)⊥ ))
59	58	ax-r1 35	. . . . . . 7 ((a⊥ ∪ b) ∩ ((c ∪ b) ∩ (c ∪ b)⊥ )) = ((a⊥ ∪ b) ∩ 0)
60		an0 108	. . . . . . 7 ((a⊥ ∪ b) ∩ 0) = 0
61	59, 60	ax-r2 36	. . . . . 6 ((a⊥ ∪ b) ∩ ((c ∪ b) ∩ (c ∪ b)⊥ )) = 0
62	55, 56, 61	3tr 65	. . . . 5 (((a⊥ ∪ b) ∩ (c ∪ b)⊥ ) ∩ (c ∪ b)) = 0
63	54, 62	2or 72	. . . 4 ((b ∩ (c ∪ b)) ∪ (((a⊥ ∪ b) ∩ (c ∪ b)⊥ ) ∩ (c ∪ b))) = (b ∪ 0)
64	50, 63	ax-r2 36	. . 3 ((b ∪ ((a⊥ ∪ b) ∩ (c ∪ b)⊥ )) ∩ (c ∪ b)) = (b ∪ 0)
65	44, 45, 64	le3tr2 141	. 2 c ≤ (b ∪ 0)
66		or0 102	. 2 (b ∪ 0) = b
67	65, 66	lbtr 139	1 c ≤ b

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →₀ wi0 11   →₂ wi2 13

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-i0 43  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  3vded22  818