Theorem negantlem9 841

Description: Negated antecedent identity.

Hypothesis

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

Assertion

Ref	Expression
negantlem9	(a →3 c) ≤ (b →3 c)

Proof of Theorem negantlem9

Step	Hyp	Ref	Expression
1		leao4 157	. . . . 5 (a⊥ ∩ c) ≤ (b⊥ ∪ c)
2		leor 151	. . . . . 6 (a⊥ ∩ c) ≤ (a ∪ (a⊥ ∩ c))
3		negant.1	. . . . . . . . 9 (a →1 c) = (b →1 c)
4	3	sac 817	. . . . . . . 8 (a⊥ →1 c) = (b⊥ →1 c)
5		df-i1 43	. . . . . . . . 9 (a⊥ →1 c) = (a⊥ ⊥ ∪ (a⊥ ∩ c))
6		ax-a1 29	. . . . . . . . . . 11 a = a⊥ ⊥
7	6	ax-r5 37	. . . . . . . . . 10 (a ∪ (a⊥ ∩ c)) = (a⊥ ⊥ ∪ (a⊥ ∩ c))
8	7	ax-r1 34	. . . . . . . . 9 (a⊥ ⊥ ∪ (a⊥ ∩ c)) = (a ∪ (a⊥ ∩ c))
9	5, 8	ax-r2 35	. . . . . . . 8 (a⊥ →1 c) = (a ∪ (a⊥ ∩ c))
10		df-i1 43	. . . . . . . . 9 (b⊥ →1 c) = (b⊥ ⊥ ∪ (b⊥ ∩ c))
11		ax-a1 29	. . . . . . . . . . 11 b = b⊥ ⊥
12	11	ax-r5 37	. . . . . . . . . 10 (b ∪ (b⊥ ∩ c)) = (b⊥ ⊥ ∪ (b⊥ ∩ c))
13	12	ax-r1 34	. . . . . . . . 9 (b⊥ ⊥ ∪ (b⊥ ∩ c)) = (b ∪ (b⊥ ∩ c))
14	10, 13	ax-r2 35	. . . . . . . 8 (b⊥ →1 c) = (b ∪ (b⊥ ∩ c))
15	4, 9, 14	3tr2 61	. . . . . . 7 (a ∪ (a⊥ ∩ c)) = (b ∪ (b⊥ ∩ c))
16		leo 150	. . . . . . . 8 b ≤ (b ∪ (b⊥ ∩ c⊥ ))
17	16	leror 144	. . . . . . 7 (b ∪ (b⊥ ∩ c)) ≤ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c))
18	15, 17	bltr 130	. . . . . 6 (a ∪ (a⊥ ∩ c)) ≤ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c))
19	2, 18	letr 129	. . . . 5 (a⊥ ∩ c) ≤ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c))
20	1, 19	ler2an 165	. . . 4 (a⊥ ∩ c) ≤ ((b⊥ ∪ c) ∩ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c)))
21		leao1 154	. . . . . 6 (a⊥ ∩ c⊥ ) ≤ (a⊥ ∪ c)
22	3	negantlem8 838	. . . . . 6 (a⊥ ∪ c) = (b⊥ ∪ c)
23	21, 22	lbtr 131	. . . . 5 (a⊥ ∩ c⊥ ) ≤ (b⊥ ∪ c)
24	3	negantlem5 835	. . . . . 6 (a⊥ ∩ c⊥ ) = (b⊥ ∩ c⊥ )
25		leor 151	. . . . . . 7 (b⊥ ∩ c⊥ ) ≤ (b ∪ (b⊥ ∩ c⊥ ))
26	25	ler 141	. . . . . 6 (b⊥ ∩ c⊥ ) ≤ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c))
27	24, 26	bltr 130	. . . . 5 (a⊥ ∩ c⊥ ) ≤ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c))
28	23, 27	ler2an 165	. . . 4 (a⊥ ∩ c⊥ ) ≤ ((b⊥ ∪ c) ∩ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c)))
29	20, 28	lel2or 162	. . 3 ((a⊥ ∩ c) ∪ (a⊥ ∩ c⊥ )) ≤ ((b⊥ ∪ c) ∩ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c)))
30		lear 153	. . . . 5 (a ∩ (a⊥ ∪ c)) ≤ (a⊥ ∪ c)
31	30, 22	lbtr 131	. . . 4 (a ∩ (a⊥ ∪ c)) ≤ (b⊥ ∪ c)
32		leo 150	. . . . . . 7 a ≤ (a ∪ (a⊥ ∩ c))
33	32, 15	lbtr 131	. . . . . 6 a ≤ (b ∪ (b⊥ ∩ c))
34	33, 17	letr 129	. . . . 5 a ≤ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c))
35	34	lel 143	. . . 4 (a ∩ (a⊥ ∪ c)) ≤ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c))
36	31, 35	ler2an 165	. . 3 (a ∩ (a⊥ ∪ c)) ≤ ((b⊥ ∪ c) ∩ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c)))
37	29, 36	lel2or 162	. 2 (((a⊥ ∩ c) ∪ (a⊥ ∩ c⊥ )) ∪ (a ∩ (a⊥ ∪ c))) ≤ ((b⊥ ∪ c) ∩ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c)))
38		df-i3 45	. 2 (a →3 c) = (((a⊥ ∩ c) ∪ (a⊥ ∩ c⊥ )) ∪ (a ∩ (a⊥ ∪ c)))
39		dfi3b 481	. 2 (b →3 c) = ((b⊥ ∪ c) ∩ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c)))
40	37, 38, 39	le3tr1 132	1 (a →3 c) ≤ (b →3 c)

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 13   →₃ wi3 15

This theorem is referenced by:  negant3 842

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-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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