Theorem oa3to4lem3 947

Description: Lemma for orthoarguesian law (Godowski/Greechie 3-variable to 4-variable proof).

Hypotheses

Ref	Expression
oa3to4lem.1	a⊥ ≤ b
oa3to4lem.2	c⊥ ≤ d
oa3to4lem.3	g = ((a ∩ b) ∪ (c ∩ d))

Assertion

Ref	Expression
oa3to4lem3	(a ∩ (b ∪ (d ∩ ((a ∩ c) ∪ (b ∩ d))))) ≤ (a ∩ ((a →1 g) ∪ ((c →1 g) ∩ ((a ∩ c) ∪ ((a →1 g) ∩ (c →1 g))))))

Proof of Theorem oa3to4lem3

Step	Hyp	Ref	Expression
1		oa3to4lem.1	. . . 4 a⊥ ≤ b
2		oa3to4lem.2	. . . 4 c⊥ ≤ d
3		oa3to4lem.3	. . . 4 g = ((a ∩ b) ∪ (c ∩ d))
4	1, 2, 3	oa3to4lem1 945	. . 3 b ≤ (a →1 g)
5	1, 2, 3	oa3to4lem2 946	. . . 4 d ≤ (c →1 g)
6	4, 5	le2an 169	. . . . 5 (b ∩ d) ≤ ((a →1 g) ∩ (c →1 g))
7	6	lelor 166	. . . 4 ((a ∩ c) ∪ (b ∩ d)) ≤ ((a ∩ c) ∪ ((a →1 g) ∩ (c →1 g)))
8	5, 7	le2an 169	. . 3 (d ∩ ((a ∩ c) ∪ (b ∩ d))) ≤ ((c →1 g) ∩ ((a ∩ c) ∪ ((a →1 g) ∩ (c →1 g))))
9	4, 8	le2or 168	. 2 (b ∪ (d ∩ ((a ∩ c) ∪ (b ∩ d)))) ≤ ((a →1 g) ∪ ((c →1 g) ∩ ((a ∩ c) ∪ ((a →1 g) ∩ (c →1 g)))))
10	9	lelan 167	1 (a ∩ (b ∪ (d ∩ ((a ∩ c) ∪ (b ∩ d))))) ≤ (a ∩ ((a →1 g) ∪ ((c →1 g) ∩ ((a ∩ c) ∪ ((a →1 g) ∩ (c →1 g))))))

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  oa3to4lem4  948