Theorem gomaex3h7 908

Description: Hypothesis for Godowski 6-var -> Mayet Example 3. (Contributed by NM, 29-Nov-1999.)

Hypotheses

Ref	Expression
gomaex3h7.18	n = (p⊥ →1 q)⊥
gomaex3h7.19	u = (p⊥ ∩ q)

Assertion

Ref	Expression
gomaex3h7	n ≤ u⊥

Proof of Theorem gomaex3h7

Step	Hyp	Ref	Expression
1		leor 159	. . . 4 (p⊥ ∩ q) ≤ (p⊥ ⊥ ∪ (p⊥ ∩ q))
2		df-i1 44	. . . . 5 (p⊥ →1 q) = (p⊥ ⊥ ∪ (p⊥ ∩ q))
3	2	ax-r1 35	. . . 4 (p⊥ ⊥ ∪ (p⊥ ∩ q)) = (p⊥ →1 q)
4	1, 3	lbtr 139	. . 3 (p⊥ ∩ q) ≤ (p⊥ →1 q)
5	4	lecon 154	. 2 (p⊥ →1 q)⊥ ≤ (p⊥ ∩ q)⊥
6		gomaex3h7.18	. 2 n = (p⊥ →1 q)⊥
7		gomaex3h7.19	. . 3 u = (p⊥ ∩ q)
8	7	ax-r4 37	. 2 u⊥ = (p⊥ ∩ q)⊥
9	5, 6, 8	le3tr1 140	1 n ≤ u⊥

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-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-i1 44  df-le1 130  df-le2 131

This theorem is referenced by:  gomaex3lem5  918