Theorem 4oagen1 1021

Description: "Generalized" 4-OA.

Hypotheses

Ref	Expression
4oa.1	e = (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
4oa.2	f = (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v e)
4oagen1.1	g =< f

Assertion

Ref	Expression
4oagen1	((a ->1 d) ^ (g v ((a ->1 d) ^ (b ->1 d)))) = ((a ->1 d) ^ (b ->1 d))

Proof of Theorem 4oagen1

Step	Hyp	Ref	Expression
1		4oagen1.1	. . . . . . 7 g =< f
2		4oa.2	. . . . . . . 8 f = (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v e)
3		or32 75	. . . . . . . 8 (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v e) = (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))
4	2, 3	ax-r2 35	. . . . . . 7 f = (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))
5	1, 4	lbtr 131	. . . . . 6 g =< (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))
6	5	leror 144	. . . . 5 (g v ((a ->1 d) ^ (b ->1 d))) =< ((((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d))) v ((a ->1 d) ^ (b ->1 d)))
7		ax-a3 31	. . . . . 6 ((((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d))) v ((a ->1 d) ^ (b ->1 d))) = (((a ^ b) v e) v (((a ->1 d) ^ (b ->1 d)) v ((a ->1 d) ^ (b ->1 d))))
8		oridm 102	. . . . . . . 8 (((a ->1 d) ^ (b ->1 d)) v ((a ->1 d) ^ (b ->1 d))) = ((a ->1 d) ^ (b ->1 d))
9	8	lor 66	. . . . . . 7 (((a ^ b) v e) v (((a ->1 d) ^ (b ->1 d)) v ((a ->1 d) ^ (b ->1 d)))) = (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))
10	4	ax-r1 34	. . . . . . 7 (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d))) = f
11	9, 10	ax-r2 35	. . . . . 6 (((a ^ b) v e) v (((a ->1 d) ^ (b ->1 d)) v ((a ->1 d) ^ (b ->1 d)))) = f
12	7, 11	ax-r2 35	. . . . 5 ((((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d))) v ((a ->1 d) ^ (b ->1 d))) = f
13	6, 12	lbtr 131	. . . 4 (g v ((a ->1 d) ^ (b ->1 d))) =< f
14	13	lelan 159	. . 3 ((a ->1 d) ^ (g v ((a ->1 d) ^ (b ->1 d)))) =< ((a ->1 d) ^ f)
15		4oa.1	. . . 4 e = (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
16	15, 2	4oath1 1020	. . 3 ((a ->1 d) ^ f) = ((a ->1 d) ^ (b ->1 d))
17	14, 16	lbtr 131	. 2 ((a ->1 d) ^ (g v ((a ->1 d) ^ (b ->1 d)))) =< ((a ->1 d) ^ (b ->1 d))
18		lea 152	. . 3 ((a ->1 d) ^ (b ->1 d)) =< (a ->1 d)
19		leor 151	. . 3 ((a ->1 d) ^ (b ->1 d)) =< (g v ((a ->1 d) ^ (b ->1 d)))
20	18, 19	ler2an 165	. 2 ((a ->1 d) ^ (b ->1 d)) =< ((a ->1 d) ^ (g v ((a ->1 d) ^ (b ->1 d))))
21	17, 20	lebi 137	1 ((a ->1 d) ^ (g v ((a ->1 d) ^ (b ->1 d)))) = ((a ->1 d) ^ (b ->1 d))

Syntax hints:   = wb 1   =< wle 2   v wo 6   ^ wa 7   ->1 wi1 13

This theorem is referenced by:  4oagen1b 1022

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  ax-4oa 1012

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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