Theorem ud4lem3 585

Description: Lemma for unified disjunction. (Contributed by NM, 23-Nov-1997.)

Assertion

Ref	Expression
ud4lem3	((a →4 b) →4 (a ∪ b)) = (a ∪ b)

Proof of Theorem ud4lem3

Step	Hyp	Ref	Expression
1		df-i4 47	. 2 ((a →4 b) →4 (a ∪ b)) = ((((a →4 b) ∩ (a ∪ b)) ∪ ((a →4 b)⊥ ∩ (a ∪ b))) ∪ (((a →4 b)⊥ ∪ (a ∪ b)) ∩ (a ∪ b)⊥ ))
2		ud4lem3a 583	. . . . . 6 ((a →4 b)⊥ ∩ (a ∪ b)) = (a →4 b)⊥
3	2	lor 70	. . . . 5 (((a →4 b) ∩ (a ∪ b)) ∪ ((a →4 b)⊥ ∩ (a ∪ b))) = (((a →4 b) ∩ (a ∪ b)) ∪ (a →4 b)⊥ )
4		comid 187	. . . . . . . 8 (a →4 b) C (a →4 b)
5	4	comcom2 183	. . . . . . 7 (a →4 b) C (a →4 b)⊥
6		df-i4 47	. . . . . . . 8 (a →4 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
7		comor1 461	. . . . . . . . . . . 12 (a ∪ b) C a
8		comor2 462	. . . . . . . . . . . 12 (a ∪ b) C b
9	7, 8	com2an 484	. . . . . . . . . . 11 (a ∪ b) C (a ∩ b)
10	7	comcom2 183	. . . . . . . . . . . 12 (a ∪ b) C a⊥
11	10, 8	com2an 484	. . . . . . . . . . 11 (a ∪ b) C (a⊥ ∩ b)
12	9, 11	com2or 483	. . . . . . . . . 10 (a ∪ b) C ((a ∩ b) ∪ (a⊥ ∩ b))
13	10, 8	com2or 483	. . . . . . . . . . 11 (a ∪ b) C (a⊥ ∪ b)
14	8	comcom2 183	. . . . . . . . . . 11 (a ∪ b) C b⊥
15	13, 14	com2an 484	. . . . . . . . . 10 (a ∪ b) C ((a⊥ ∪ b) ∩ b⊥ )
16	12, 15	com2or 483	. . . . . . . . 9 (a ∪ b) C (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
17	16	comcom 453	. . . . . . . 8 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) C (a ∪ b)
18	6, 17	bctr 181	. . . . . . 7 (a →4 b) C (a ∪ b)
19	5, 18	fh4r 476	. . . . . 6 (((a →4 b) ∩ (a ∪ b)) ∪ (a →4 b)⊥ ) = (((a →4 b) ∪ (a →4 b)⊥ ) ∩ ((a ∪ b) ∪ (a →4 b)⊥ ))
20		ancom 74	. . . . . . 7 (((a →4 b) ∪ (a →4 b)⊥ ) ∩ ((a ∪ b) ∪ (a →4 b)⊥ )) = (((a ∪ b) ∪ (a →4 b)⊥ ) ∩ ((a →4 b) ∪ (a →4 b)⊥ ))
21		ax-a2 31	. . . . . . . . . 10 ((a ∪ b) ∪ (a →4 b)⊥ ) = ((a →4 b)⊥ ∪ (a ∪ b))
22		ud4lem3b 584	. . . . . . . . . 10 ((a →4 b)⊥ ∪ (a ∪ b)) = (a ∪ b)
23	21, 22	ax-r2 36	. . . . . . . . 9 ((a ∪ b) ∪ (a →4 b)⊥ ) = (a ∪ b)
24		df-t 41	. . . . . . . . . 10 1 = ((a →4 b) ∪ (a →4 b)⊥ )
25	24	ax-r1 35	. . . . . . . . 9 ((a →4 b) ∪ (a →4 b)⊥ ) = 1
26	23, 25	2an 79	. . . . . . . 8 (((a ∪ b) ∪ (a →4 b)⊥ ) ∩ ((a →4 b) ∪ (a →4 b)⊥ )) = ((a ∪ b) ∩ 1)
27		an1 106	. . . . . . . 8 ((a ∪ b) ∩ 1) = (a ∪ b)
28	26, 27	ax-r2 36	. . . . . . 7 (((a ∪ b) ∪ (a →4 b)⊥ ) ∩ ((a →4 b) ∪ (a →4 b)⊥ )) = (a ∪ b)
29	20, 28	ax-r2 36	. . . . . 6 (((a →4 b) ∪ (a →4 b)⊥ ) ∩ ((a ∪ b) ∪ (a →4 b)⊥ )) = (a ∪ b)
30	19, 29	ax-r2 36	. . . . 5 (((a →4 b) ∩ (a ∪ b)) ∪ (a →4 b)⊥ ) = (a ∪ b)
31	3, 30	ax-r2 36	. . . 4 (((a →4 b) ∩ (a ∪ b)) ∪ ((a →4 b)⊥ ∩ (a ∪ b))) = (a ∪ b)
32	22	ran 78	. . . . 5 (((a →4 b)⊥ ∪ (a ∪ b)) ∩ (a ∪ b)⊥ ) = ((a ∪ b) ∩ (a ∪ b)⊥ )
33		dff 101	. . . . . 6 0 = ((a ∪ b) ∩ (a ∪ b)⊥ )
34	33	ax-r1 35	. . . . 5 ((a ∪ b) ∩ (a ∪ b)⊥ ) = 0
35	32, 34	ax-r2 36	. . . 4 (((a →4 b)⊥ ∪ (a ∪ b)) ∩ (a ∪ b)⊥ ) = 0
36	31, 35	2or 72	. . 3 ((((a →4 b) ∩ (a ∪ b)) ∪ ((a →4 b)⊥ ∩ (a ∪ b))) ∪ (((a →4 b)⊥ ∪ (a ∪ b)) ∩ (a ∪ b)⊥ )) = ((a ∪ b) ∪ 0)
37		or0 102	. . 3 ((a ∪ b) ∪ 0) = (a ∪ b)
38	36, 37	ax-r2 36	. 2 ((((a →4 b) ∩ (a ∪ b)) ∪ ((a →4 b)⊥ ∩ (a ∪ b))) ∪ (((a →4 b)⊥ ∪ (a ∪ b)) ∩ (a ∪ b)⊥ )) = (a ∪ b)
39	1, 38	ax-r2 36	1 ((a →4 b) →4 (a ∪ b)) = (a ∪ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   →₄ wi4 15

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

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  ud4  598