Theorem gomaex3lem4 917

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

Hypothesis

Ref	Expression
gomaex3lem4.9	p = ((a ∪ b) →1 (d ∪ e)⊥ )⊥

Assertion

Ref	Expression
gomaex3lem4	((a ∪ b) ∩ (d ∪ e)⊥ ) ≤ p⊥

Proof of Theorem gomaex3lem4

Step	Hyp	Ref	Expression
1		leor 159	. 2 ((a ∪ b) ∩ (d ∪ e)⊥ ) ≤ ((a ∪ b)⊥ ∪ ((a ∪ b) ∩ (d ∪ e)⊥ ))
2		ax-a1 30	. . 3 ((a ∪ b) →1 (d ∪ e)⊥ ) = ((a ∪ b) →1 (d ∪ e)⊥ )⊥ ⊥
3		df-i1 44	. . . 4 ((a ∪ b) →1 (d ∪ e)⊥ ) = ((a ∪ b)⊥ ∪ ((a ∪ b) ∩ (d ∪ e)⊥ ))
4	3	ax-r1 35	. . 3 ((a ∪ b)⊥ ∪ ((a ∪ b) ∩ (d ∪ e)⊥ )) = ((a ∪ b) →1 (d ∪ e)⊥ )
5		gomaex3lem4.9	. . . 4 p = ((a ∪ b) →1 (d ∪ e)⊥ )⊥
6	5	ax-r4 37	. . 3 p⊥ = ((a ∪ b) →1 (d ∪ e)⊥ )⊥ ⊥
7	2, 4, 6	3tr1 63	. 2 ((a ∪ b)⊥ ∪ ((a ∪ b) ∩ (d ∪ e)⊥ )) = p⊥
8	1, 7	lbtr 139	1 ((a ∪ b) ∩ (d ∪ e)⊥ ) ≤ p⊥

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:  gomaex3lem9  922