Theorem negantlem3 832

Description: Lemma for negated antecedent identity.

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 150	. . . 4 a⊥ ≤ (a⊥ ∪ (a ∩ c))
2		df-i1 43	. . . . . 6 (a →1 c) = (a⊥ ∪ (a ∩ c))
3	2	ax-r1 34	. . . . 5 (a⊥ ∪ (a ∩ c)) = (a →1 c)
4		negant.1	. . . . 5 (a →1 c) = (b →1 c)
5	3, 4	ax-r2 35	. . . 4 (a⊥ ∪ (a ∩ c)) = (b →1 c)
6	1, 5	lbtr 131	. . 3 a⊥ ≤ (b →1 c)
7	6	leran 145	. 2 (a⊥ ∩ c) ≤ ((b →1 c) ∩ c)
8		lea 152	. . . 4 (b ∩ c) ≤ b
9	8	leror 144	. . 3 ((b ∩ c) ∪ (b⊥ ∩ c)) ≤ (b ∪ (b⊥ ∩ c))
10		u1lemab 592	. . 3 ((b →1 c) ∩ c) = ((b ∩ c) ∪ (b⊥ ∩ c))
11		df-i1 43	. . . 4 (b⊥ →1 c) = (b⊥ ⊥ ∪ (b⊥ ∩ c))
12		ax-a1 29	. . . . . 6 b = b⊥ ⊥
13	12	ax-r5 37	. . . . 5 (b ∪ (b⊥ ∩ c)) = (b⊥ ⊥ ∪ (b⊥ ∩ c))
14	13	ax-r1 34	. . . 4 (b⊥ ⊥ ∪ (b⊥ ∩ c)) = (b ∪ (b⊥ ∩ c))
15	11, 14	ax-r2 35	. . 3 (b⊥ →1 c) = (b ∪ (b⊥ ∩ c))
16	9, 10, 15	le3tr1 132	. 2 ((b →1 c) ∩ c) ≤ (b⊥ →1 c)
17	7, 16	letr 129	1 (a⊥ ∩ c) ≤ (b⊥ →1 c)

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