Theorem oaidlem1 286

Description: Lemma for OA identity-like law.

Hypothesis

Ref	Expression
oaidlem1.1	(a ^ b) =< c

Assertion

Ref	Expression
oaidlem1	(a_|_ v (b ->1 c)) = 1

Proof of Theorem oaidlem1

Step	Hyp	Ref	Expression
1		df-i1 43	. . 3 (b ->1 c) = (b_|_ v (b ^ c))
2	1	lor 66	. 2 (a_|_ v (b ->1 c)) = (a_|_ v (b_|_ v (b ^ c)))
3		oran3 85	. . . 4 (a_|_ v b_|_) = (a ^ b)_|_
4	3	ax-r5 37	. . 3 ((a_|_ v b_|_) v (b ^ c)) = ((a ^ b)_|_ v (b ^ c))
5		ax-a3 31	. . 3 ((a_|_ v b_|_) v (b ^ c)) = (a_|_ v (b_|_ v (b ^ c)))
6		lear 153	. . . . 5 (a ^ b) =< b
7		oaidlem1.1	. . . . 5 (a ^ b) =< c
8	6, 7	ler2an 165	. . . 4 (a ^ b) =< (b ^ c)
9	8	sklem 222	. . 3 ((a ^ b)_|_ v (b ^ c)) = 1
10	4, 5, 9	3tr2 61	. 2 (a_|_ v (b_|_ v (b ^ c))) = 1
11	2, 10	ax-r2 35	1 (a_|_ v (b ->1 c)) = 1

Syntax hints:   = wb 1   =< wle 2  _|_wn 4   v wo 6   ^ wa 7  1wt 9   ->1 wi1 13

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

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i1 43  df-le1 122  df-le2 123

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