Theorem oa3to4lem4 948

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))
oa3to4lem.oa3	(a ∩ ((a →1 g) ∪ ((c →1 g) ∩ ((a ∩ c) ∪ ((a →1 g) ∩ (c →1 g)))))) ≤ ((a ∩ g) ∪ (c ∩ g))

Assertion

Ref	Expression
oa3to4lem4	(a ∩ (b ∪ (d ∩ ((a ∩ c) ∪ (b ∩ d))))) ≤ g

Proof of Theorem oa3to4lem4

Step	Hyp	Ref	Expression
1		oa3to4lem.1	. . 3 a⊥ ≤ b
2		oa3to4lem.2	. . 3 c⊥ ≤ d
3		oa3to4lem.3	. . 3 g = ((a ∩ b) ∪ (c ∩ d))
4	1, 2, 3	oa3to4lem3 947	. 2 (a ∩ (b ∪ (d ∩ ((a ∩ c) ∪ (b ∩ d))))) ≤ (a ∩ ((a →1 g) ∪ ((c →1 g) ∩ ((a ∩ c) ∪ ((a →1 g) ∩ (c →1 g))))))
5		oa3to4lem.oa3	. . 3 (a ∩ ((a →1 g) ∪ ((c →1 g) ∩ ((a ∩ c) ∪ ((a →1 g) ∩ (c →1 g)))))) ≤ ((a ∩ g) ∪ (c ∩ g))
6		lear 161	. . . 4 (a ∩ g) ≤ g
7		lear 161	. . . 4 (c ∩ g) ≤ g
8	6, 7	lel2or 170	. . 3 ((a ∩ g) ∪ (c ∩ g)) ≤ g
9	5, 8	letr 137	. 2 (a ∩ ((a →1 g) ∪ ((c →1 g) ∩ ((a ∩ c) ∪ ((a →1 g) ∩ (c →1 g)))))) ≤ g
10	4, 9	letr 137	1 (a ∩ (b ∪ (d ∩ ((a ∩ c) ∪ (b ∩ d))))) ≤ 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:  oa3to4lem6  950