Theorem 4oa 1018

Description: Variant of proper 4-OA.

Hypotheses

Ref	Expression
4oa.1	e = (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))))
4oa.2	f = (((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) ∪ e)

Assertion

Ref	Expression
4oa	((a →1 d) ∩ f) ≤ (b →1 d)

Proof of Theorem 4oa

Step	Hyp	Ref	Expression
1		4oa.2	. . 3 f = (((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) ∪ e)
2	1	lan 70	. 2 ((a →1 d) ∩ f) = ((a →1 d) ∩ (((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) ∪ e))
3		axoa4a 1016	. . . 4 (((b⊥ ⊥ →1 d) →1 d) ∩ ((b⊥ ⊥ →1 d) ∪ ((a⊥ ⊥ →1 d) ∩ ((((b⊥ ⊥ →1 d) ∩ (a⊥ ⊥ →1 d)) ∪ (((b⊥ ⊥ →1 d) →1 d) ∩ ((a⊥ ⊥ →1 d) →1 d))) ∪ ((((b⊥ ⊥ →1 d) ∩ (c⊥ ⊥ →1 d)) ∪ (((b⊥ ⊥ →1 d) →1 d) ∩ ((c⊥ ⊥ →1 d) →1 d))) ∩ (((a⊥ ⊥ →1 d) ∩ (c⊥ ⊥ →1 d)) ∪ (((a⊥ ⊥ →1 d) →1 d) ∩ ((c⊥ ⊥ →1 d) →1 d)))))))) ≤ ((((b⊥ ⊥ →1 d) ∩ d) ∪ ((a⊥ ⊥ →1 d) ∩ d)) ∪ ((c⊥ ⊥ →1 d) ∩ d))
4		id 58	. . . 4 (b⊥ ⊥ →1 d) = (b⊥ ⊥ →1 d)
5		id 58	. . . 4 (a⊥ ⊥ →1 d) = (a⊥ ⊥ →1 d)
6		id 58	. . . 4 (c⊥ ⊥ →1 d) = (c⊥ ⊥ →1 d)
7	3, 4, 5, 6	oa4to4u2 954	. . 3 ((b⊥ →1 d) ∩ ((b⊥ ⊥ →1 d) ∪ ((a⊥ ⊥ →1 d) ∩ ((((b⊥ →1 d) ∩ (a⊥ →1 d)) ∪ ((b⊥ ⊥ →1 d) ∩ (a⊥ ⊥ →1 d))) ∪ ((((b⊥ →1 d) ∩ (c⊥ →1 d)) ∪ ((b⊥ ⊥ →1 d) ∩ (c⊥ ⊥ →1 d))) ∩ (((a⊥ →1 d) ∩ (c⊥ →1 d)) ∪ ((a⊥ ⊥ →1 d) ∩ (c⊥ ⊥ →1 d)))))))) ≤ d
8		4oa.1	. . 3 e = (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))))
9	7, 8	oa4uto4g 955	. 2 ((a →1 d) ∩ (((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) ∪ e)) ≤ (b →1 d)
10	2, 9	bltr 130	1 ((a →1 d) ∩ f) ≤ (b →1 d)

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 13

This theorem is referenced by:  4oaiii 1019

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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