Theorem d4oa 976

Description: Variant of proper 4-OA proved from OA distributive law.

Hypotheses

Ref	Expression
d4oa.2	e = ((a ^ b) v ((a ->1 d) ^ (b ->1 d)))
d4oa.1	f = (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))

Assertion

Ref	Expression
d4oa	((a ->1 d) ^ (e v f)) =< (b ->1 d)

Proof of Theorem d4oa

Step	Hyp	Ref	Expression
1		ax-a2 30	. . . 4 (e v f) = (f v e)
2	1	lan 70	. . 3 ((a ->1 d) ^ (e v f)) = ((a ->1 d) ^ (f v e))
3		id 58	. . . 4 (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))) = (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
4		d4oa.2	. . . . 5 e = ((a ^ b) v ((a ->1 d) ^ (b ->1 d)))
5		d4oa.1	. . . . 5 f = (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
6	4, 5	2or 67	. . . 4 (e v f) = (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))))
7		leid 140	. . . 4 (a ->1 d) =< (a ->1 d)
8		leor 151	. . . 4 f =< (e v f)
9		leo 150	. . . 4 e =< (e v f)
10		leor 151	. . . . 5 ((a ->1 d) ^ (b ->1 d)) =< ((a ^ b) v ((a ->1 d) ^ (b ->1 d)))
11	4	ax-r1 34	. . . . 5 ((a ^ b) v ((a ->1 d) ^ (b ->1 d))) = e
12	10, 11	lbtr 131	. . . 4 ((a ->1 d) ^ (b ->1 d)) =< e
13	3, 6, 7, 8, 9, 12	ax-oadist 974	. . 3 ((a ->1 d) ^ (f v e)) = (((a ->1 d) ^ f) v ((a ->1 d) ^ e))
14	2, 13	ax-r2 35	. 2 ((a ->1 d) ^ (e v f)) = (((a ->1 d) ^ f) v ((a ->1 d) ^ e))
15	5	lan 70	. . . . . 6 ((a ->1 d) ^ f) = ((a ->1 d) ^ (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))))
16		anass 69	. . . . . . 7 (((a ->1 d) ^ ((a ^ c) v ((a ->1 d) ^ (c ->1 d)))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))) = ((a ->1 d) ^ (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))))
17	16	ax-r1 34	. . . . . 6 ((a ->1 d) ^ (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))) = (((a ->1 d) ^ ((a ^ c) v ((a ->1 d) ^ (c ->1 d)))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
18	15, 17	ax-r2 35	. . . . 5 ((a ->1 d) ^ f) = (((a ->1 d) ^ ((a ^ c) v ((a ->1 d) ^ (c ->1 d)))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
19		id 58	. . . . . . 7 ((a ^ c) v ((a ->1 d) ^ (c ->1 d))) = ((a ^ c) v ((a ->1 d) ^ (c ->1 d)))
20	19	d3oa 975	. . . . . 6 ((a ->1 d) ^ ((a ^ c) v ((a ->1 d) ^ (c ->1 d)))) =< (c ->1 d)
21	20	leran 145	. . . . 5 (((a ->1 d) ^ ((a ^ c) v ((a ->1 d) ^ (c ->1 d)))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))) =< ((c ->1 d) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
22	18, 21	bltr 130	. . . 4 ((a ->1 d) ^ f) =< ((c ->1 d) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
23		ancom 68	. . . . . 6 (b ^ c) = (c ^ b)
24		ancom 68	. . . . . 6 ((b ->1 d) ^ (c ->1 d)) = ((c ->1 d) ^ (b ->1 d))
25	23, 24	2or 67	. . . . 5 ((b ^ c) v ((b ->1 d) ^ (c ->1 d))) = ((c ^ b) v ((c ->1 d) ^ (b ->1 d)))
26	25	d3oa 975	. . . 4 ((c ->1 d) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))) =< (b ->1 d)
27	22, 26	letr 129	. . 3 ((a ->1 d) ^ f) =< (b ->1 d)
28	4	d3oa 975	. . 3 ((a ->1 d) ^ e) =< (b ->1 d)
29	27, 28	lel2or 162	. 2 (((a ->1 d) ^ f) v ((a ->1 d) ^ e)) =< (b ->1 d)
30	14, 29	bltr 130	1 ((a ->1 d) ^ (e v f)) =< (b ->1 d)

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

This theorem is referenced by:  d6oa 977

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-oadist 974

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i0 42  df-i1 43  df-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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