Theorem nom14 303

Description: Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper.

Assertion

Ref	Expression
nom14	(a →4 (a ∩ b)) = (a →1 b)

Proof of Theorem nom14

Step	Hyp	Ref	Expression
1		ax-a2 30	. . . . 5 ((a ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b))) = ((a⊥ ∩ (a ∩ b)) ∪ (a ∩ (a ∩ b)))
2		anass 69	. . . . . . . 8 ((a ∩ a) ∩ b) = (a ∩ (a ∩ b))
3	2	ax-r1 34	. . . . . . 7 (a ∩ (a ∩ b)) = ((a ∩ a) ∩ b)
4		anidm 103	. . . . . . . 8 (a ∩ a) = a
5	4	ran 71	. . . . . . 7 ((a ∩ a) ∩ b) = (a ∩ b)
6	3, 5	ax-r2 35	. . . . . 6 (a ∩ (a ∩ b)) = (a ∩ b)
7	6	lor 66	. . . . 5 ((a⊥ ∩ (a ∩ b)) ∪ (a ∩ (a ∩ b))) = ((a⊥ ∩ (a ∩ b)) ∪ (a ∩ b))
8		lear 153	. . . . . 6 (a⊥ ∩ (a ∩ b)) ≤ (a ∩ b)
9	8	df-le2 123	. . . . 5 ((a⊥ ∩ (a ∩ b)) ∪ (a ∩ b)) = (a ∩ b)
10	1, 7, 9	3tr 62	. . . 4 ((a ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b))) = (a ∩ b)
11	10	ax-r5 37	. . 3 (((a ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b))) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ )) = ((a ∩ b) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ ))
12		leo 150	. . . . 5 (a ∩ b) ≤ ((a ∩ b) ∪ a⊥ )
13		lea 152	. . . . . 6 ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ ) ≤ (a⊥ ∪ (a ∩ b))
14		ax-a2 30	. . . . . 6 (a⊥ ∪ (a ∩ b)) = ((a ∩ b) ∪ a⊥ )
15	13, 14	lbtr 131	. . . . 5 ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ ) ≤ ((a ∩ b) ∪ a⊥ )
16	12, 15	lel2or 162	. . . 4 ((a ∩ b) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ )) ≤ ((a ∩ b) ∪ a⊥ )
17		leo 150	. . . . . 6 a⊥ ≤ (a⊥ ∪ (a ∩ b))
18		lea 152	. . . . . . 7 (a ∩ b) ≤ a
19	18	lecon 146	. . . . . 6 a⊥ ≤ (a ∩ b)⊥
20	17, 19	ler2an 165	. . . . 5 a⊥ ≤ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ )
21	20	lelor 158	. . . 4 ((a ∩ b) ∪ a⊥ ) ≤ ((a ∩ b) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ ))
22	16, 21	lebi 137	. . 3 ((a ∩ b) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ )) = ((a ∩ b) ∪ a⊥ )
23		ax-a2 30	. . 3 ((a ∩ b) ∪ a⊥ ) = (a⊥ ∪ (a ∩ b))
24	11, 22, 23	3tr 62	. 2 (((a ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b))) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ )) = (a⊥ ∪ (a ∩ b))
25		df-i4 46	. 2 (a →4 (a ∩ b)) = (((a ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b))) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ ))
26		df-i1 43	. 2 (a →1 b) = (a⊥ ∪ (a ∩ b))
27	24, 25, 26	3tr1 60	1 (a →4 (a ∩ b)) = (a →1 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 13   →₄ wi4 16

This theorem is referenced by:  nom43 320

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i1 43  df-i4 46  df-le1 122  df-le2 123

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