Theorem u4lem2 729

Description: Lemma for unified implication study.

Assertion

Ref	Expression
u4lem2	(((a →4 b) →4 a) →4 a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))

Proof of Theorem u4lem2

Step	Hyp	Ref	Expression
1		u4lemc1 665	. . . 4 a C ((a →4 b) →4 a)
2	1	comcom 435	. . 3 ((a →4 b) →4 a) C a
3	2	u4lemc4 686	. 2 (((a →4 b) →4 a) →4 a) = (((a →4 b) →4 a)⊥ ∪ a)
4		u4lem1n 724	. . . 4 ((a →4 b) →4 a)⊥ = ((((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
5	4	ax-r5 37	. . 3 (((a →4 b) →4 a)⊥ ∪ a) = (((((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ a)
6		ax-a3 31	. . . 4 (((((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ a) = ((((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a) ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a))
7		lear 153	. . . . . . 7 (((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a) ≤ a
8		leor 151	. . . . . . 7 a ≤ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a)
9	7, 8	letr 129	. . . . . 6 (((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a) ≤ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a)
10	9	df-le2 123	. . . . 5 ((((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a) ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a)) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a)
11		ax-a2 30	. . . . 5 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
12	10, 11	ax-r2 35	. . . 4 ((((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a) ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a)) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
13	6, 12	ax-r2 35	. . 3 (((((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
14	5, 13	ax-r2 35	. 2 (((a →4 b) →4 a)⊥ ∪ a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
15	3, 14	ax-r2 35	1 (((a →4 b) →4 a) →4 a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₄ wi4 16

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-i4 46  df-le1 122  df-le2 123  df-c1 124  df-c2 125

metamath.org