Theorem negantlem3 850

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
negantlem3	(a⊥ ∩ c) ≤ (b⊥ →1 c)

Proof of Theorem negantlem3

Step	Hyp	Ref	Expression
1		leo 158	. . . 4 a⊥ ≤ (a⊥ ∪ (a ∩ c))
2		df-i1 44	. . . . . 6 (a →1 c) = (a⊥ ∪ (a ∩ c))
3	2	ax-r1 35	. . . . 5 (a⊥ ∪ (a ∩ c)) = (a →1 c)
4		negant.1	. . . . 5 (a →1 c) = (b →1 c)
5	3, 4	ax-r2 36	. . . 4 (a⊥ ∪ (a ∩ c)) = (b →1 c)
6	1, 5	lbtr 139	. . 3 a⊥ ≤ (b →1 c)
7	6	leran 153	. 2 (a⊥ ∩ c) ≤ ((b →1 c) ∩ c)
8		lea 160	. . . 4 (b ∩ c) ≤ b
9	8	leror 152	. . 3 ((b ∩ c) ∪ (b⊥ ∩ c)) ≤ (b ∪ (b⊥ ∩ c))
10		u1lemab 610	. . 3 ((b →1 c) ∩ c) = ((b ∩ c) ∪ (b⊥ ∩ c))
11		df-i1 44	. . . 4 (b⊥ →1 c) = (b⊥ ⊥ ∪ (b⊥ ∩ c))
12		ax-a1 30	. . . . . 6 b = b⊥ ⊥
13	12	ax-r5 38	. . . . 5 (b ∪ (b⊥ ∩ c)) = (b⊥ ⊥ ∪ (b⊥ ∩ c))
14	13	ax-r1 35	. . . 4 (b⊥ ⊥ ∪ (b⊥ ∩ c)) = (b ∪ (b⊥ ∩ c))
15	11, 14	ax-r2 36	. . 3 (b⊥ →1 c) = (b ∪ (b⊥ ∩ c))
16	9, 10, 15	le3tr1 140	. 2 ((b →1 c) ∩ c) ≤ (b⊥ →1 c)
17	7, 16	letr 137	1 (a⊥ ∩ c) ≤ (b⊥ →1 c)

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