Theorem nom11 308

Description: Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper. (Contributed by NM, 7-Feb-1999.)

Assertion

Ref	Expression
nom11	(a →1 (a ∩ b)) = (a →1 b)

Proof of Theorem nom11

Step	Hyp	Ref	Expression
1		anass 76	. . . . 5 ((a ∩ a) ∩ b) = (a ∩ (a ∩ b))
2	1	ax-r1 35	. . . 4 (a ∩ (a ∩ b)) = ((a ∩ a) ∩ b)
3		anidm 111	. . . . 5 (a ∩ a) = a
4	3	ran 78	. . . 4 ((a ∩ a) ∩ b) = (a ∩ b)
5	2, 4	ax-r2 36	. . 3 (a ∩ (a ∩ b)) = (a ∩ b)
6	5	lor 70	. 2 (a⊥ ∪ (a ∩ (a ∩ b))) = (a⊥ ∪ (a ∩ b))
7		df-i1 44	. 2 (a →1 (a ∩ b)) = (a⊥ ∪ (a ∩ (a ∩ b)))
8		df-i1 44	. 2 (a →1 b) = (a⊥ ∪ (a ∩ b))
9	6, 7, 8	3tr1 63	1 (a →1 (a ∩ b)) = (a →1 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44

This theorem is referenced by:  nom42  327  lem3.3.7i1e3  1062