Theorem 4oagen1b 1043

Description: "Generalized" OA. (Contributed by NM, 29-Dec-1998.)

Hypotheses

Ref	Expression
4oa.1	e = (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))))
4oa.2	f = (((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) ∪ e)
4oagen1b.1	g ≤ f
4oagen1b.2	h ≤ (a →1 d)

Assertion

Ref	Expression
4oagen1b	(h ∩ (g ∪ ((a →1 d) ∩ (b →1 d)))) = (h ∩ (b →1 d))

Proof of Theorem 4oagen1b

Step	Hyp	Ref	Expression
1		4oa.1	. . . 4 e = (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))))
2		4oa.2	. . . 4 f = (((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) ∪ e)
3		4oagen1b.1	. . . 4 g ≤ f
4	1, 2, 3	4oagen1 1042	. . 3 ((a →1 d) ∩ (g ∪ ((a →1 d) ∩ (b →1 d)))) = ((a →1 d) ∩ (b →1 d))
5	4	lan 77	. 2 (h ∩ ((a →1 d) ∩ (g ∪ ((a →1 d) ∩ (b →1 d))))) = (h ∩ ((a →1 d) ∩ (b →1 d)))
6		anass 76	. . . 4 ((h ∩ (a →1 d)) ∩ (g ∪ ((a →1 d) ∩ (b →1 d)))) = (h ∩ ((a →1 d) ∩ (g ∪ ((a →1 d) ∩ (b →1 d)))))
7	6	ax-r1 35	. . 3 (h ∩ ((a →1 d) ∩ (g ∪ ((a →1 d) ∩ (b →1 d))))) = ((h ∩ (a →1 d)) ∩ (g ∪ ((a →1 d) ∩ (b →1 d))))
8		4oagen1b.2	. . . . 5 h ≤ (a →1 d)
9	8	df2le2 136	. . . 4 (h ∩ (a →1 d)) = h
10	9	ran 78	. . 3 ((h ∩ (a →1 d)) ∩ (g ∪ ((a →1 d) ∩ (b →1 d)))) = (h ∩ (g ∪ ((a →1 d) ∩ (b →1 d))))
11	7, 10	ax-r2 36	. 2 (h ∩ ((a →1 d) ∩ (g ∪ ((a →1 d) ∩ (b →1 d))))) = (h ∩ (g ∪ ((a →1 d) ∩ (b →1 d))))
12		anass 76	. . . 4 ((h ∩ (a →1 d)) ∩ (b →1 d)) = (h ∩ ((a →1 d) ∩ (b →1 d)))
13	12	ax-r1 35	. . 3 (h ∩ ((a →1 d) ∩ (b →1 d))) = ((h ∩ (a →1 d)) ∩ (b →1 d))
14	9	ran 78	. . 3 ((h ∩ (a →1 d)) ∩ (b →1 d)) = (h ∩ (b →1 d))
15	13, 14	ax-r2 36	. 2 (h ∩ ((a →1 d) ∩ (b →1 d))) = (h ∩ (b →1 d))
16	5, 11, 15	3tr2 64	1 (h ∩ (g ∪ ((a →1 d) ∩ (b →1 d)))) = (h ∩ (b →1 d))

Syntax hints:   = wb 1   ≤ wle 2   ∪ wo 6   ∩ wa 7   →₁ wi1 12

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439  ax-4oa 1033

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  4oadist  1044