Theorem oa3to4lem5 949

Description: Lemma for orthoarguesian law (Godowski/Greechie 3-variable to 4-variable proof). (Contributed by NM, 19-Dec-1998.)

Hypothesis

Ref	Expression
oa3to4lem5.1	((a ∪ b) ∩ (c ∪ d)) ≤ (a ∪ (b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d)))))

Assertion

Ref	Expression
oa3to4lem5	((b ∪ a) ∩ (d ∪ c)) ≤ (a ∪ (b ∩ (d ∪ ((b ∪ d) ∩ (a ∪ c)))))

Proof of Theorem oa3to4lem5

Step	Hyp	Ref	Expression
1		oa3to4lem5.1	. 2 ((a ∪ b) ∩ (c ∪ d)) ≤ (a ∪ (b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d)))))
2		ax-a2 31	. . 3 (b ∪ a) = (a ∪ b)
3		ax-a2 31	. . 3 (d ∪ c) = (c ∪ d)
4	2, 3	2an 79	. 2 ((b ∪ a) ∩ (d ∪ c)) = ((a ∪ b) ∩ (c ∪ d))
5		ancom 74	. . . . 5 ((b ∪ d) ∩ (a ∪ c)) = ((a ∪ c) ∩ (b ∪ d))
6	5	lor 70	. . . 4 (d ∪ ((b ∪ d) ∩ (a ∪ c))) = (d ∪ ((a ∪ c) ∩ (b ∪ d)))
7	6	lan 77	. . 3 (b ∩ (d ∪ ((b ∪ d) ∩ (a ∪ c)))) = (b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d))))
8	7	lor 70	. 2 (a ∪ (b ∩ (d ∪ ((b ∪ d) ∩ (a ∪ c))))) = (a ∪ (b ∩ (d ∪ ((a ∪ c) ∩ (b ∪ d)))))
9	1, 4, 8	le3tr1 140	1 ((b ∪ a) ∩ (d ∪ c)) ≤ (a ∪ (b ∩ (d ∪ ((b ∪ d) ∩ (a ∪ c)))))

Syntax hints:   ≤ wle 2   ∪ wo 6   ∩ wa 7

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-le1 130  df-le2 131

This theorem is referenced by:  oa3to4  951