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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^⊥ →₁c) →₁c) ∩ ((a^⊥ →₁c) ∪ (c ∩ ((((a^⊥ →₁c) ∩ c) ∪ (((a^⊥ →₁c) →₁c) ∩ (c →₁c))) ∪ ((((a^⊥ →₁c) ∩ (b^⊥ →₁c)) ∪ (((a^⊥ →₁c) →₁c) ∩ ((b^⊥ →₁c) →₁c))) ∩ ((c ∩ (b^⊥ →₁c)) ∪ ((c →₁c) ∩ ((b^⊥ →₁c) →₁c)))))))) ≤ c

**Assertion**
Ref	Expression
oa3-u2	((a →₁c) ∩ ((a^⊥ →₁c) ∪ (c ∩ ((a →₁c) ∪ ((b →₁c) ∩ (((a →₁c) ∩ (b →₁c)) ∪ ((a^⊥ →₁c) ∩ (b^⊥ →₁c)))))))) ≤ c

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

Colors of variables: term

Syntax hints: ≤ wle 2 ^⊥ wn 4 ∪ wo 6 ∩ wa 7 1wt 9 →₁wi1 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

metamath.org