Theorem negantlem9 859

Description: Negated antecedent identity. (Contributed by NM, 6-Aug-2001.)

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 165	. . . . 5 (a⊥ ∩ c) ≤ (b⊥ ∪ c)
2		leor 159	. . . . . 6 (a⊥ ∩ c) ≤ (a ∪ (a⊥ ∩ c))
3		negant.1	. . . . . . . . 9 (a →1 c) = (b →1 c)
4	3	sac 835	. . . . . . . 8 (a⊥ →1 c) = (b⊥ →1 c)
5		df-i1 44	. . . . . . . . 9 (a⊥ →1 c) = (a⊥ ⊥ ∪ (a⊥ ∩ c))
6		ax-a1 30	. . . . . . . . . . 11 a = a⊥ ⊥
7	6	ax-r5 38	. . . . . . . . . 10 (a ∪ (a⊥ ∩ c)) = (a⊥ ⊥ ∪ (a⊥ ∩ c))
8	7	ax-r1 35	. . . . . . . . 9 (a⊥ ⊥ ∪ (a⊥ ∩ c)) = (a ∪ (a⊥ ∩ c))
9	5, 8	ax-r2 36	. . . . . . . 8 (a⊥ →1 c) = (a ∪ (a⊥ ∩ c))
10		df-i1 44	. . . . . . . . 9 (b⊥ →1 c) = (b⊥ ⊥ ∪ (b⊥ ∩ c))
11		ax-a1 30	. . . . . . . . . . 11 b = b⊥ ⊥
12	11	ax-r5 38	. . . . . . . . . 10 (b ∪ (b⊥ ∩ c)) = (b⊥ ⊥ ∪ (b⊥ ∩ c))
13	12	ax-r1 35	. . . . . . . . 9 (b⊥ ⊥ ∪ (b⊥ ∩ c)) = (b ∪ (b⊥ ∩ c))
14	10, 13	ax-r2 36	. . . . . . . 8 (b⊥ →1 c) = (b ∪ (b⊥ ∩ c))
15	4, 9, 14	3tr2 64	. . . . . . 7 (a ∪ (a⊥ ∩ c)) = (b ∪ (b⊥ ∩ c))
16		leo 158	. . . . . . . 8 b ≤ (b ∪ (b⊥ ∩ c⊥ ))
17	16	leror 152	. . . . . . 7 (b ∪ (b⊥ ∩ c)) ≤ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c))
18	15, 17	bltr 138	. . . . . 6 (a ∪ (a⊥ ∩ c)) ≤ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c))
19	2, 18	letr 137	. . . . 5 (a⊥ ∩ c) ≤ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c))
20	1, 19	ler2an 173	. . . 4 (a⊥ ∩ c) ≤ ((b⊥ ∪ c) ∩ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c)))
21		leao1 162	. . . . . 6 (a⊥ ∩ c⊥ ) ≤ (a⊥ ∪ c)
22	3	negantlem8 856	. . . . . 6 (a⊥ ∪ c) = (b⊥ ∪ c)
23	21, 22	lbtr 139	. . . . 5 (a⊥ ∩ c⊥ ) ≤ (b⊥ ∪ c)
24	3	negantlem5 853	. . . . . 6 (a⊥ ∩ c⊥ ) = (b⊥ ∩ c⊥ )
25		leor 159	. . . . . . 7 (b⊥ ∩ c⊥ ) ≤ (b ∪ (b⊥ ∩ c⊥ ))
26	25	ler 149	. . . . . 6 (b⊥ ∩ c⊥ ) ≤ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c))
27	24, 26	bltr 138	. . . . 5 (a⊥ ∩ c⊥ ) ≤ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c))
28	23, 27	ler2an 173	. . . 4 (a⊥ ∩ c⊥ ) ≤ ((b⊥ ∪ c) ∩ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c)))
29	20, 28	lel2or 170	. . 3 ((a⊥ ∩ c) ∪ (a⊥ ∩ c⊥ )) ≤ ((b⊥ ∪ c) ∩ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c)))
30		lear 161	. . . . 5 (a ∩ (a⊥ ∪ c)) ≤ (a⊥ ∪ c)
31	30, 22	lbtr 139	. . . 4 (a ∩ (a⊥ ∪ c)) ≤ (b⊥ ∪ c)
32		leo 158	. . . . . . 7 a ≤ (a ∪ (a⊥ ∩ c))
33	32, 15	lbtr 139	. . . . . 6 a ≤ (b ∪ (b⊥ ∩ c))
34	33, 17	letr 137	. . . . 5 a ≤ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c))
35	34	lel 151	. . . 4 (a ∩ (a⊥ ∪ c)) ≤ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c))
36	31, 35	ler2an 173	. . 3 (a ∩ (a⊥ ∪ c)) ≤ ((b⊥ ∪ c) ∩ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c)))
37	29, 36	lel2or 170	. 2 (((a⊥ ∩ c) ∪ (a⊥ ∩ c⊥ )) ∪ (a ∩ (a⊥ ∪ c))) ≤ ((b⊥ ∪ c) ∩ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c)))
38		df-i3 46	. 2 (a →3 c) = (((a⊥ ∩ c) ∪ (a⊥ ∩ c⊥ )) ∪ (a ∩ (a⊥ ∪ c)))
39		dfi3b 499	. 2 (b →3 c) = ((b⊥ ∪ c) ∩ ((b ∪ (b⊥ ∩ c⊥ )) ∪ (b⊥ ∩ c)))
40	37, 38, 39	le3tr1 140	1 (a →3 c) ≤ (b →3 c)

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12   →₃ wi3 14

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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  negant3  860