Theorem oa3-u2 972

Description: Derivation of a "universal" 3-OA. The hypothesis is a substitution instance of the proper 4-OA.

Hypothesis

Ref	Expression
oa3-u2.1	(((a_|_ ->1 c) ->1 c) ^ ((a_|_ ->1 c) v (c ^ ((((a_|_ ->1 c) ^ c) v (((a_|_ ->1 c) ->1 c) ^ (c ->1 c))) v ((((a_|_ ->1 c) ^ (b_|_ ->1 c)) v (((a_|_ ->1 c) ->1 c) ^ ((b_|_ ->1 c) ->1 c))) ^ ((c ^ (b_|_ ->1 c)) v ((c ->1 c) ^ ((b_|_ ->1 c) ->1 c)))))))) =< c

Assertion

Ref	Expression
oa3-u2	((a ->1 c) ^ ((a_|_ ->1 c) v (c ^ ((a ->1 c) v ((b ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v ((a_|_ ->1 c) ^ (b_|_ ->1 c)))))))) =< c

Proof of Theorem oa3-u2

Step	Hyp	Ref	Expression
1		oa3-u2lem 966	. . 3 ((a ->1 c) ^ ((a_|_ ->1 c) v (c ^ ((((a_|_ ->1 c) ^ c) v ((a ->1 c) ^ 1)) v ((((a_|_ ->1 c) ^ (b_|_ ->1 c)) v ((a ->1 c) ^ (b ->1 c))) ^ ((c ^ (b_|_ ->1 c)) v (1 ^ (b ->1 c)))))))) = ((a ->1 c) ^ ((a_|_ ->1 c) v (c ^ ((a ->1 c) v ((b ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v ((a_|_ ->1 c) ^ (b_|_ ->1 c))))))))
2	1	ax-r1 34	. 2 ((a ->1 c) ^ ((a_|_ ->1 c) v (c ^ ((a ->1 c) v ((b ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v ((a_|_ ->1 c) ^ (b_|_ ->1 c)))))))) = ((a ->1 c) ^ ((a_|_ ->1 c) v (c ^ ((((a_|_ ->1 c) ^ c) v ((a ->1 c) ^ 1)) v ((((a_|_ ->1 c) ^ (b_|_ ->1 c)) v ((a ->1 c) ^ (b ->1 c))) ^ ((c ^ (b_|_ ->1 c)) v (1 ^ (b ->1 c))))))))
3		u1lem9ab 761	. . 3 (a_|_ ->1 c)_|_ =< (a ->1 c)
4		le1 138	. . 3 c_|_ =< 1
5		u1lem9ab 761	. . 3 (b_|_ ->1 c)_|_ =< (b ->1 c)
6		or32 75	. . . . 5 ((((a_|_ ->1 c) ^ (a ->1 c)) v (c ^ 1)) v ((b_|_ ->1 c) ^ (b ->1 c))) = ((((a_|_ ->1 c) ^ (a ->1 c)) v ((b_|_ ->1 c) ^ (b ->1 c))) v (c ^ 1))
7		ancom 68	. . . . . . . 8 ((a_|_ ->1 c) ^ (a ->1 c)) = ((a ->1 c) ^ (a_|_ ->1 c))
8		u1lem8 758	. . . . . . . 8 ((a ->1 c) ^ (a_|_ ->1 c)) = ((a ^ c) v (a_|_ ^ c))
9	7, 8	ax-r2 35	. . . . . . 7 ((a_|_ ->1 c) ^ (a ->1 c)) = ((a ^ c) v (a_|_ ^ c))
10		ancom 68	. . . . . . . 8 ((b_|_ ->1 c) ^ (b ->1 c)) = ((b ->1 c) ^ (b_|_ ->1 c))
11		u1lem8 758	. . . . . . . 8 ((b ->1 c) ^ (b_|_ ->1 c)) = ((b ^ c) v (b_|_ ^ c))
12	10, 11	ax-r2 35	. . . . . . 7 ((b_|_ ->1 c) ^ (b ->1 c)) = ((b ^ c) v (b_|_ ^ c))
13	9, 12	2or 67	. . . . . 6 (((a_|_ ->1 c) ^ (a ->1 c)) v ((b_|_ ->1 c) ^ (b ->1 c))) = (((a ^ c) v (a_|_ ^ c)) v ((b ^ c) v (b_|_ ^ c)))
14		an1 98	. . . . . 6 (c ^ 1) = c
15	13, 14	2or 67	. . . . 5 ((((a_|_ ->1 c) ^ (a ->1 c)) v ((b_|_ ->1 c) ^ (b ->1 c))) v (c ^ 1)) = ((((a ^ c) v (a_|_ ^ c)) v ((b ^ c) v (b_|_ ^ c))) v c)
16		lear 153	. . . . . . . 8 (a ^ c) =< c
17		lear 153	. . . . . . . 8 (a_|_ ^ c) =< c
18	16, 17	lel2or 162	. . . . . . 7 ((a ^ c) v (a_|_ ^ c)) =< c
19		lear 153	. . . . . . . 8 (b ^ c) =< c
20		lear 153	. . . . . . . 8 (b_|_ ^ c) =< c
21	19, 20	lel2or 162	. . . . . . 7 ((b ^ c) v (b_|_ ^ c)) =< c
22	18, 21	lel2or 162	. . . . . 6 (((a ^ c) v (a_|_ ^ c)) v ((b ^ c) v (b_|_ ^ c))) =< c
23	22	df-le2 123	. . . . 5 ((((a ^ c) v (a_|_ ^ c)) v ((b ^ c) v (b_|_ ^ c))) v c) = c
24	6, 15, 23	3tr 62	. . . 4 ((((a_|_ ->1 c) ^ (a ->1 c)) v (c ^ 1)) v ((b_|_ ->1 c) ^ (b ->1 c))) = c
25	24	ax-r1 34	. . 3 c = ((((a_|_ ->1 c) ^ (a ->1 c)) v (c ^ 1)) v ((b_|_ ->1 c) ^ (b ->1 c)))
26		oa3-u2.1	. . 3 (((a_|_ ->1 c) ->1 c) ^ ((a_|_ ->1 c) v (c ^ ((((a_|_ ->1 c) ^ c) v (((a_|_ ->1 c) ->1 c) ^ (c ->1 c))) v ((((a_|_ ->1 c) ^ (b_|_ ->1 c)) v (((a_|_ ->1 c) ->1 c) ^ ((b_|_ ->1 c) ->1 c))) ^ ((c ^ (b_|_ ->1 c)) v ((c ->1 c) ^ ((b_|_ ->1 c) ->1 c)))))))) =< c
27	3, 4, 5, 25, 26	oa4to6dual 944	. 2 ((a ->1 c) ^ ((a_|_ ->1 c) v (c ^ ((((a_|_ ->1 c) ^ c) v ((a ->1 c) ^ 1)) v ((((a_|_ ->1 c) ^ (b_|_ ->1 c)) v ((a ->1 c) ^ (b ->1 c))) ^ ((c ^ (b_|_ ->1 c)) v (1 ^ (b ->1 c)))))))) =< c
28	2, 27	bltr 130	1 ((a ->1 c) ^ ((a_|_ ->1 c) v (c ^ ((a ->1 c) v ((b ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v ((a_|_ ->1 c) ^ (b_|_ ->1 c)))))))) =< c

Syntax hints:   =< 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  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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