Theorem negantlem2 849

Description: Lemma for negated antecedent identity. (Contributed by NM, 6-Aug-2001.)

Hypothesis

Ref	Expression
negant.1	(a →1 c) = (b →1 c)

Assertion

Ref	Expression
negantlem2	a ≤ (b⊥ →1 c)

Proof of Theorem negantlem2

Step	Hyp	Ref	Expression
1		leo 158	. 2 a ≤ (a ∪ (b⊥ →1 c))
2		i1orni1 847	. . . . . 6 ((b →1 c) ∪ (b⊥ →1 c)) = 1
3	2	lan 77	. . . . 5 ((a ∪ (b⊥ →1 c)) ∩ ((b →1 c) ∪ (b⊥ →1 c))) = ((a ∪ (b⊥ →1 c)) ∩ 1)
4	3	ax-r1 35	. . . 4 ((a ∪ (b⊥ →1 c)) ∩ 1) = ((a ∪ (b⊥ →1 c)) ∩ ((b →1 c) ∪ (b⊥ →1 c)))
5		an1 106	. . . . 5 ((a ∪ (b⊥ →1 c)) ∩ 1) = (a ∪ (b⊥ →1 c))
6	5	ax-r1 35	. . . 4 (a ∪ (b⊥ →1 c)) = ((a ∪ (b⊥ →1 c)) ∩ 1)
7		u1lemc6 706	. . . . 5 (b →1 c) C (b⊥ →1 c)
8		negant.1	. . . . . . 7 (a →1 c) = (b →1 c)
9	8	negantlem1 848	. . . . . 6 a C (b →1 c)
10	9	comcom 453	. . . . 5 (b →1 c) C a
11	7, 10	fh4rc 482	. . . 4 ((a ∩ (b →1 c)) ∪ (b⊥ →1 c)) = ((a ∪ (b⊥ →1 c)) ∩ ((b →1 c) ∪ (b⊥ →1 c)))
12	4, 6, 11	3tr1 63	. . 3 (a ∪ (b⊥ →1 c)) = ((a ∩ (b →1 c)) ∪ (b⊥ →1 c))
13		ancom 74	. . . . . . . 8 (a ∩ (a →1 c)) = ((a →1 c) ∩ a)
14	8	lan 77	. . . . . . . 8 (a ∩ (a →1 c)) = (a ∩ (b →1 c))
15		u1lemaa 600	. . . . . . . 8 ((a →1 c) ∩ a) = (a ∩ c)
16	13, 14, 15	3tr2 64	. . . . . . 7 (a ∩ (b →1 c)) = (a ∩ c)
17		lear 161	. . . . . . 7 (a ∩ c) ≤ c
18	16, 17	bltr 138	. . . . . 6 (a ∩ (b →1 c)) ≤ c
19		lear 161	. . . . . 6 (a ∩ (b →1 c)) ≤ (b →1 c)
20	18, 19	ler2an 173	. . . . 5 (a ∩ (b →1 c)) ≤ (c ∩ (b →1 c))
21		lea 160	. . . . . . . 8 (b ∩ c) ≤ b
22		ax-a1 30	. . . . . . . 8 b = b⊥ ⊥
23	21, 22	lbtr 139	. . . . . . 7 (b ∩ c) ≤ b⊥ ⊥
24	23	leror 152	. . . . . 6 ((b ∩ c) ∪ (b⊥ ∩ c)) ≤ (b⊥ ⊥ ∪ (b⊥ ∩ c))
25		ancom 74	. . . . . . 7 (c ∩ (b →1 c)) = ((b →1 c) ∩ c)
26		u1lemab 610	. . . . . . 7 ((b →1 c) ∩ c) = ((b ∩ c) ∪ (b⊥ ∩ c))
27	25, 26	ax-r2 36	. . . . . 6 (c ∩ (b →1 c)) = ((b ∩ c) ∪ (b⊥ ∩ c))
28		df-i1 44	. . . . . 6 (b⊥ →1 c) = (b⊥ ⊥ ∪ (b⊥ ∩ c))
29	24, 27, 28	le3tr1 140	. . . . 5 (c ∩ (b →1 c)) ≤ (b⊥ →1 c)
30	20, 29	letr 137	. . . 4 (a ∩ (b →1 c)) ≤ (b⊥ →1 c)
31		leid 148	. . . 4 (b⊥ →1 c) ≤ (b⊥ →1 c)
32	30, 31	lel2or 170	. . 3 ((a ∩ (b →1 c)) ∪ (b⊥ →1 c)) ≤ (b⊥ →1 c)
33	12, 32	bltr 138	. 2 (a ∪ (b⊥ →1 c)) ≤ (b⊥ →1 c)
34	1, 33	letr 137	1 a ≤ (b⊥ →1 c)

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₁ wi1 12

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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  negantlem4  851