Theorem gon2n 898

Description: Lemma for converting n-variable to 2n-variable Godowski equations. (Contributed by NM, 19-Nov-1999.)

Hypotheses

Ref	Expression
govar.1	a ≤ b⊥
govar.2	b ≤ c⊥
gon2n.3	((c →2 a) ∩ d) ≤ (a →2 c)
gon2n.4	e ≤ d

Assertion

Ref	Expression
gon2n	((a ∪ b) ∩ e) ≤ (b ∪ c)

Proof of Theorem gon2n

Step	Hyp	Ref	Expression
1		lea 160	. . 3 ((a ∪ b) ∩ e) ≤ (a ∪ b)
2		govar.1	. . . . . 6 a ≤ b⊥
3		govar.2	. . . . . 6 b ≤ c⊥
4	2, 3	govar2 897	. . . . 5 (a ∪ b) ≤ (c →2 a)
5		gon2n.4	. . . . 5 e ≤ d
6	4, 5	le2an 169	. . . 4 ((a ∪ b) ∩ e) ≤ ((c →2 a) ∩ d)
7		gon2n.3	. . . 4 ((c →2 a) ∩ d) ≤ (a →2 c)
8	6, 7	letr 137	. . 3 ((a ∪ b) ∩ e) ≤ (a →2 c)
9	1, 8	ler2an 173	. 2 ((a ∪ b) ∩ e) ≤ ((a ∪ b) ∩ (a →2 c))
10	2, 3	govar 896	. 2 ((a ∪ b) ∩ (a →2 c)) ≤ (b ∪ c)
11	9, 10	letr 137	1 ((a ∪ b) ∩ e) ≤ (b ∪ c)

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 13

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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  go2n4  899  go2n6  901