Theorem dfi3b 499

Description: Alternate Kalmbach conditional. (Contributed by NM, 6-Aug-2001.)

Assertion

Ref	Expression
dfi3b	(a →3 b) = ((a⊥ ∪ b) ∩ ((a ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∩ b)))

Proof of Theorem dfi3b

Step	Hyp	Ref	Expression
1		ax-a2 31	. . 3 (((a⊥ ∩ (a⊥ ∩ b)) ∪ (b ∩ (a⊥ ∩ b))) ∪ (((a⊥ ∪ b) ∩ a) ∪ ((a⊥ ∪ b) ∩ (a⊥ ∩ b⊥ )))) = ((((a⊥ ∪ b) ∩ a) ∪ ((a⊥ ∪ b) ∩ (a⊥ ∩ b⊥ ))) ∪ ((a⊥ ∩ (a⊥ ∩ b)) ∪ (b ∩ (a⊥ ∩ b))))
2		ax-a3 32	. . . 4 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) = ((a⊥ ∩ b) ∪ ((a⊥ ∩ b⊥ ) ∪ (a ∩ (a⊥ ∪ b))))
3		oridm 110	. . . . . . 7 ((a⊥ ∩ b) ∪ (a⊥ ∩ b)) = (a⊥ ∩ b)
4	3	ax-r1 35	. . . . . 6 (a⊥ ∩ b) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b))
5		anidm 111	. . . . . . . . . 10 (a⊥ ∩ a⊥ ) = a⊥
6	5	ax-r1 35	. . . . . . . . 9 a⊥ = (a⊥ ∩ a⊥ )
7	6	ran 78	. . . . . . . 8 (a⊥ ∩ b) = ((a⊥ ∩ a⊥ ) ∩ b)
8		anass 76	. . . . . . . 8 ((a⊥ ∩ a⊥ ) ∩ b) = (a⊥ ∩ (a⊥ ∩ b))
9	7, 8	ax-r2 36	. . . . . . 7 (a⊥ ∩ b) = (a⊥ ∩ (a⊥ ∩ b))
10		anidm 111	. . . . . . . . . 10 (b ∩ b) = b
11	10	ax-r1 35	. . . . . . . . 9 b = (b ∩ b)
12	11	lan 77	. . . . . . . 8 (a⊥ ∩ b) = (a⊥ ∩ (b ∩ b))
13		an12 81	. . . . . . . 8 (a⊥ ∩ (b ∩ b)) = (b ∩ (a⊥ ∩ b))
14	12, 13	ax-r2 36	. . . . . . 7 (a⊥ ∩ b) = (b ∩ (a⊥ ∩ b))
15	9, 14	2or 72	. . . . . 6 ((a⊥ ∩ b) ∪ (a⊥ ∩ b)) = ((a⊥ ∩ (a⊥ ∩ b)) ∪ (b ∩ (a⊥ ∩ b)))
16	4, 15	ax-r2 36	. . . . 5 (a⊥ ∩ b) = ((a⊥ ∩ (a⊥ ∩ b)) ∪ (b ∩ (a⊥ ∩ b)))
17		lea 160	. . . . . . . . . . 11 (a⊥ ∩ b⊥ ) ≤ a⊥
18		leo 158	. . . . . . . . . . 11 a⊥ ≤ (a⊥ ∪ b)
19	17, 18	letr 137	. . . . . . . . . 10 (a⊥ ∩ b⊥ ) ≤ (a⊥ ∪ b)
20	19	df2le2 136	. . . . . . . . 9 ((a⊥ ∩ b⊥ ) ∩ (a⊥ ∪ b)) = (a⊥ ∩ b⊥ )
21	20	ax-r1 35	. . . . . . . 8 (a⊥ ∩ b⊥ ) = ((a⊥ ∩ b⊥ ) ∩ (a⊥ ∪ b))
22		ancom 74	. . . . . . . 8 ((a⊥ ∩ b⊥ ) ∩ (a⊥ ∪ b)) = ((a⊥ ∪ b) ∩ (a⊥ ∩ b⊥ ))
23	21, 22	ax-r2 36	. . . . . . 7 (a⊥ ∩ b⊥ ) = ((a⊥ ∪ b) ∩ (a⊥ ∩ b⊥ ))
24		ancom 74	. . . . . . 7 (a ∩ (a⊥ ∪ b)) = ((a⊥ ∪ b) ∩ a)
25	23, 24	2or 72	. . . . . 6 ((a⊥ ∩ b⊥ ) ∪ (a ∩ (a⊥ ∪ b))) = (((a⊥ ∪ b) ∩ (a⊥ ∩ b⊥ )) ∪ ((a⊥ ∪ b) ∩ a))
26		ax-a2 31	. . . . . 6 (((a⊥ ∪ b) ∩ (a⊥ ∩ b⊥ )) ∪ ((a⊥ ∪ b) ∩ a)) = (((a⊥ ∪ b) ∩ a) ∪ ((a⊥ ∪ b) ∩ (a⊥ ∩ b⊥ )))
27	25, 26	ax-r2 36	. . . . 5 ((a⊥ ∩ b⊥ ) ∪ (a ∩ (a⊥ ∪ b))) = (((a⊥ ∪ b) ∩ a) ∪ ((a⊥ ∪ b) ∩ (a⊥ ∩ b⊥ )))
28	16, 27	2or 72	. . . 4 ((a⊥ ∩ b) ∪ ((a⊥ ∩ b⊥ ) ∪ (a ∩ (a⊥ ∪ b)))) = (((a⊥ ∩ (a⊥ ∩ b)) ∪ (b ∩ (a⊥ ∩ b))) ∪ (((a⊥ ∪ b) ∩ a) ∪ ((a⊥ ∪ b) ∩ (a⊥ ∩ b⊥ ))))
29	2, 28	ax-r2 36	. . 3 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) = (((a⊥ ∩ (a⊥ ∩ b)) ∪ (b ∩ (a⊥ ∩ b))) ∪ (((a⊥ ∪ b) ∩ a) ∪ ((a⊥ ∪ b) ∩ (a⊥ ∩ b⊥ ))))
30		comor1 461	. . . . . 6 (a⊥ ∪ b) C a⊥
31	30	comcom7 460	. . . . 5 (a⊥ ∪ b) C a
32		comor2 462	. . . . . . 7 (a⊥ ∪ b) C b
33	32	comcom2 183	. . . . . 6 (a⊥ ∪ b) C b⊥
34	30, 33	com2an 484	. . . . 5 (a⊥ ∪ b) C (a⊥ ∩ b⊥ )
35	31, 34	fh1 469	. . . 4 ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b⊥ ))) = (((a⊥ ∪ b) ∩ a) ∪ ((a⊥ ∪ b) ∩ (a⊥ ∩ b⊥ )))
36		coman1 185	. . . . 5 (a⊥ ∩ b) C a⊥
37		coman2 186	. . . . 5 (a⊥ ∩ b) C b
38	36, 37	fh1r 473	. . . 4 ((a⊥ ∪ b) ∩ (a⊥ ∩ b)) = ((a⊥ ∩ (a⊥ ∩ b)) ∪ (b ∩ (a⊥ ∩ b)))
39	35, 38	2or 72	. . 3 (((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b⊥ ))) ∪ ((a⊥ ∪ b) ∩ (a⊥ ∩ b))) = ((((a⊥ ∪ b) ∩ a) ∪ ((a⊥ ∪ b) ∩ (a⊥ ∩ b⊥ ))) ∪ ((a⊥ ∩ (a⊥ ∩ b)) ∪ (b ∩ (a⊥ ∩ b))))
40	1, 29, 39	3tr1 63	. 2 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) = (((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b⊥ ))) ∪ ((a⊥ ∪ b) ∩ (a⊥ ∩ b)))
41		df-i3 46	. 2 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
42	31, 34	com2or 483	. . 3 (a⊥ ∪ b) C (a ∪ (a⊥ ∩ b⊥ ))
43	30, 32	com2an 484	. . 3 (a⊥ ∪ b) C (a⊥ ∩ b)
44	42, 43	fh1 469	. 2 ((a⊥ ∪ b) ∩ ((a ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∩ b))) = (((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b⊥ ))) ∪ ((a⊥ ∪ b) ∩ (a⊥ ∩ b)))
45	40, 41, 44	3tr1 63	1 (a →3 b) = ((a⊥ ∪ b) ∩ ((a ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∩ b)))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 14

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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  dfi4b  500  u3lem15  795  negantlem9  859