Theorem negantlem2 831

Description: Lemma for negated antecedent identity.

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 150	. 2 a =< (a v (b_|_ ->1 c))
2		i1orni1 829	. . . . . 6 ((b ->1 c) v (b_|_ ->1 c)) = 1
3	2	lan 70	. . . . 5 ((a v (b_|_ ->1 c)) ^ ((b ->1 c) v (b_|_ ->1 c))) = ((a v (b_|_ ->1 c)) ^ 1)
4	3	ax-r1 34	. . . 4 ((a v (b_|_ ->1 c)) ^ 1) = ((a v (b_|_ ->1 c)) ^ ((b ->1 c) v (b_|_ ->1 c)))
5		an1 98	. . . . 5 ((a v (b_|_ ->1 c)) ^ 1) = (a v (b_|_ ->1 c))
6	5	ax-r1 34	. . . 4 (a v (b_|_ ->1 c)) = ((a v (b_|_ ->1 c)) ^ 1)
7		u1lemc6 688	. . . . 5 (b ->1 c) C (b_|_ ->1 c)
8		negant.1	. . . . . . 7 (a ->1 c) = (b ->1 c)
9	8	negantlem1 830	. . . . . 6 a C (b ->1 c)
10	9	comcom 435	. . . . 5 (b ->1 c) C a
11	7, 10	fh4rc 464	. . . 4 ((a ^ (b ->1 c)) v (b_|_ ->1 c)) = ((a v (b_|_ ->1 c)) ^ ((b ->1 c) v (b_|_ ->1 c)))
12	4, 6, 11	3tr1 60	. . 3 (a v (b_|_ ->1 c)) = ((a ^ (b ->1 c)) v (b_|_ ->1 c))
13		ancom 68	. . . . . . . 8 (a ^ (a ->1 c)) = ((a ->1 c) ^ a)
14	8	lan 70	. . . . . . . 8 (a ^ (a ->1 c)) = (a ^ (b ->1 c))
15		u1lemaa 582	. . . . . . . 8 ((a ->1 c) ^ a) = (a ^ c)
16	13, 14, 15	3tr2 61	. . . . . . 7 (a ^ (b ->1 c)) = (a ^ c)
17		lear 153	. . . . . . 7 (a ^ c) =< c
18	16, 17	bltr 130	. . . . . 6 (a ^ (b ->1 c)) =< c
19		lear 153	. . . . . 6 (a ^ (b ->1 c)) =< (b ->1 c)
20	18, 19	ler2an 165	. . . . 5 (a ^ (b ->1 c)) =< (c ^ (b ->1 c))
21		lea 152	. . . . . . . 8 (b ^ c) =< b
22		ax-a1 29	. . . . . . . 8 b = b_|__|_
23	21, 22	lbtr 131	. . . . . . 7 (b ^ c) =< b_|__|_
24	23	leror 144	. . . . . 6 ((b ^ c) v (b_|_ ^ c)) =< (b_|__|_ v (b_|_ ^ c))
25		ancom 68	. . . . . . 7 (c ^ (b ->1 c)) = ((b ->1 c) ^ c)
26		u1lemab 592	. . . . . . 7 ((b ->1 c) ^ c) = ((b ^ c) v (b_|_ ^ c))
27	25, 26	ax-r2 35	. . . . . 6 (c ^ (b ->1 c)) = ((b ^ c) v (b_|_ ^ c))
28		df-i1 43	. . . . . 6 (b_|_ ->1 c) = (b_|__|_ v (b_|_ ^ c))
29	24, 27, 28	le3tr1 132	. . . . 5 (c ^ (b ->1 c)) =< (b_|_ ->1 c)
30	20, 29	letr 129	. . . 4 (a ^ (b ->1 c)) =< (b_|_ ->1 c)
31		leid 140	. . . 4 (b_|_ ->1 c) =< (b_|_ ->1 c)
32	30, 31	lel2or 162	. . 3 ((a ^ (b ->1 c)) v (b_|_ ->1 c)) =< (b_|_ ->1 c)
33	12, 32	bltr 130	. 2 (a v (b_|_ ->1 c)) =< (b_|_ ->1 c)
34	1, 33	letr 129	1 a =< (b_|_ ->1 c)

Syntax hints:   = wb 1   =< wle 2  _|_wn 4   v wo 6   ^ wa 7  1wt 9   ->1 wi1 13

This theorem is referenced by:  negantlem4 833

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-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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