Theorem oagen1 1014

Description: "Generalized" OA. (Contributed by NM, 19-Nov-1998.)

Hypothesis

Ref	Expression
oagen1.1	d ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))

Assertion

Ref	Expression
oagen1	((a →2 b) ∩ (d ∪ ((a →2 b) ∩ (a →2 c)))) = ((a →2 b) ∩ (a →2 c))

Proof of Theorem oagen1

Step	Hyp	Ref	Expression
1		oagen1.1	. . . . . . 7 d ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
2		df-i0 43	. . . . . . 7 ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))) = ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))
3	1, 2	lbtr 139	. . . . . 6 d ≤ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))
4	3	leror 152	. . . . 5 (d ∪ ((a →2 b) ∩ (a →2 c))) ≤ (((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c))) ∪ ((a →2 b) ∩ (a →2 c)))
5		ax-a3 32	. . . . . 6 (((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c))) ∪ ((a →2 b) ∩ (a →2 c))) = ((b ∪ c)⊥ ∪ (((a →2 b) ∩ (a →2 c)) ∪ ((a →2 b) ∩ (a →2 c))))
6		oridm 110	. . . . . . 7 (((a →2 b) ∩ (a →2 c)) ∪ ((a →2 b) ∩ (a →2 c))) = ((a →2 b) ∩ (a →2 c))
7	6	lor 70	. . . . . 6 ((b ∪ c)⊥ ∪ (((a →2 b) ∩ (a →2 c)) ∪ ((a →2 b) ∩ (a →2 c)))) = ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))
8	5, 7	ax-r2 36	. . . . 5 (((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c))) ∪ ((a →2 b) ∩ (a →2 c))) = ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))
9	4, 8	lbtr 139	. . . 4 (d ∪ ((a →2 b) ∩ (a →2 c))) ≤ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))
10	9	lelan 167	. . 3 ((a →2 b) ∩ (d ∪ ((a →2 b) ∩ (a →2 c)))) ≤ ((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c))))
11		oath1 1004	. . 3 ((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))) = ((a →2 b) ∩ (a →2 c))
12	10, 11	lbtr 139	. 2 ((a →2 b) ∩ (d ∪ ((a →2 b) ∩ (a →2 c)))) ≤ ((a →2 b) ∩ (a →2 c))
13		lea 160	. . 3 ((a →2 b) ∩ (a →2 c)) ≤ (a →2 b)
14		leor 159	. . 3 ((a →2 b) ∩ (a →2 c)) ≤ (d ∪ ((a →2 b) ∩ (a →2 c)))
15	13, 14	ler2an 173	. 2 ((a →2 b) ∩ (a →2 c)) ≤ ((a →2 b) ∩ (d ∪ ((a →2 b) ∩ (a →2 c))))
16	12, 15	lebi 145	1 ((a →2 b) ∩ (d ∪ ((a →2 b) ∩ (a →2 c)))) = ((a →2 b) ∩ (a →2 c))

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₀ wi0 11   →₂ wi2 13

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  ax-3oa 998

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i0 43  df-i1 44  df-i2 45  df-le1 130  df-le2 131

This theorem is referenced by:  oagen1b  1015  oadist  1019  oadistb  1020