Theorem u1lemc6 688

Description: Commutation theorem for Sasaki implication.

Assertion

Ref	Expression
u1lemc6	(a →1 b) C (a⊥ →1 b)

Proof of Theorem u1lemc6

Step	Hyp	Ref	Expression
1		lea 152	. . . . . 6 (a ∩ (a⊥ ∪ b⊥ )) ≤ a
2		ax-a1 29	. . . . . 6 a = a⊥ ⊥
3	1, 2	lbtr 131	. . . . 5 (a ∩ (a⊥ ∪ b⊥ )) ≤ a⊥ ⊥
4		leo 150	. . . . 5 a⊥ ⊥ ≤ (a⊥ ⊥ ∪ (a⊥ ∩ b))
5	3, 4	letr 129	. . . 4 (a ∩ (a⊥ ∪ b⊥ )) ≤ (a⊥ ⊥ ∪ (a⊥ ∩ b))
6		ud1lem0c 269	. . . 4 (a →1 b)⊥ = (a ∩ (a⊥ ∪ b⊥ ))
7		df-i1 43	. . . 4 (a⊥ →1 b) = (a⊥ ⊥ ∪ (a⊥ ∩ b))
8	5, 6, 7	le3tr1 132	. . 3 (a →1 b)⊥ ≤ (a⊥ →1 b)
9	8	lecom 172	. 2 (a →1 b)⊥ C (a⊥ →1 b)
10	9	comcom6 441	1 (a →1 b) C (a⊥ →1 b)

Syntax hints:   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 13

This theorem is referenced by:  negantlem2 831

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  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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