Theorem gomaex3h4 905

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

Hypotheses

Ref	Expression
gomaex3h4.11	r = ((p⊥ →1 q)⊥ ∩ (c ∪ d))
gomaex3h4.15	j = (c ∪ d)⊥
gomaex3h4.16	k = r

Assertion

Ref	Expression
gomaex3h4	j ≤ k⊥

Proof of Theorem gomaex3h4

Step	Hyp	Ref	Expression
1		gomaex3h4.11	. . . 4 r = ((p⊥ →1 q)⊥ ∩ (c ∪ d))
2		lear 161	. . . 4 ((p⊥ →1 q)⊥ ∩ (c ∪ d)) ≤ (c ∪ d)
3	1, 2	bltr 138	. . 3 r ≤ (c ∪ d)
4	3	lecon 154	. 2 (c ∪ d)⊥ ≤ r⊥
5		gomaex3h4.15	. 2 j = (c ∪ d)⊥
6		gomaex3h4.16	. . 3 k = r
7	6	ax-r4 37	. 2 k⊥ = r⊥
8	4, 5, 7	le3tr1 140	1 j ≤ k⊥

This proof depends on syntax axioms:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12

This proof depends on axioms:  ax-a1 30  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

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

This theorem is used by:  gomaex3lem5  918