Theorem oi3ai3 503

Description: Theorem for Kalmbach implication. (Contributed by NM, 9-Nov-1997.)

Assertion

Ref	Expression
oi3ai3	((a ∩ b) ∪ (a →3 b)⊥ ) = ((a ∪ b) ∩ (a⊥ →3 b⊥ ))

Proof of Theorem oi3ai3

Step	Hyp	Ref	Expression
1		lea 160	. . . . . 6 (a ∩ b) ≤ a
2		leo 158	. . . . . 6 a ≤ (a ∪ b)
3	1, 2	letr 137	. . . . 5 (a ∩ b) ≤ (a ∪ b)
4	3	lecom 180	. . . 4 (a ∩ b) C (a ∪ b)
5		coman1 185	. . . . . 6 (a ∩ b) C a
6		ancom 74	. . . . . . . 8 (a ∩ b) = (b ∩ a)
7		coman1 185	. . . . . . . 8 (b ∩ a) C b
8	6, 7	bctr 181	. . . . . . 7 (a ∩ b) C b
9	8	comcom2 183	. . . . . 6 (a ∩ b) C b⊥
10	5, 9	com2an 484	. . . . 5 (a ∩ b) C (a ∩ b⊥ )
11	5	comcom2 183	. . . . . 6 (a ∩ b) C a⊥
12	5, 9	com2or 483	. . . . . 6 (a ∩ b) C (a ∪ b⊥ )
13	11, 12	com2an 484	. . . . 5 (a ∩ b) C (a⊥ ∩ (a ∪ b⊥ ))
14	10, 13	com2or 483	. . . 4 (a ∩ b) C ((a ∩ b⊥ ) ∪ (a⊥ ∩ (a ∪ b⊥ )))
15	4, 14	fh3 471	. . 3 ((a ∩ b) ∪ ((a ∪ b) ∩ ((a ∩ b⊥ ) ∪ (a⊥ ∩ (a ∪ b⊥ ))))) = (((a ∩ b) ∪ (a ∪ b)) ∩ ((a ∩ b) ∪ ((a ∩ b⊥ ) ∪ (a⊥ ∩ (a ∪ b⊥ )))))
16	3	df-le2 131	. . . 4 ((a ∩ b) ∪ (a ∪ b)) = (a ∪ b)
17		ax-a3 32	. . . . . 6 (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ (a⊥ ∩ (a ∪ b⊥ ))) = ((a ∩ b) ∪ ((a ∩ b⊥ ) ∪ (a⊥ ∩ (a ∪ b⊥ ))))
18	17	ax-r1 35	. . . . 5 ((a ∩ b) ∪ ((a ∩ b⊥ ) ∪ (a⊥ ∩ (a ∪ b⊥ )))) = (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ (a⊥ ∩ (a ∪ b⊥ )))
19		ax-a2 31	. . . . . 6 ((a ∩ b) ∪ (a ∩ b⊥ )) = ((a ∩ b⊥ ) ∪ (a ∩ b))
20	19	ax-r5 38	. . . . 5 (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ (a⊥ ∩ (a ∪ b⊥ ))) = (((a ∩ b⊥ ) ∪ (a ∩ b)) ∪ (a⊥ ∩ (a ∪ b⊥ )))
21	18, 20	ax-r2 36	. . . 4 ((a ∩ b) ∪ ((a ∩ b⊥ ) ∪ (a⊥ ∩ (a ∪ b⊥ )))) = (((a ∩ b⊥ ) ∪ (a ∩ b)) ∪ (a⊥ ∩ (a ∪ b⊥ )))
22	16, 21	2an 79	. . 3 (((a ∩ b) ∪ (a ∪ b)) ∩ ((a ∩ b) ∪ ((a ∩ b⊥ ) ∪ (a⊥ ∩ (a ∪ b⊥ ))))) = ((a ∪ b) ∩ (((a ∩ b⊥ ) ∪ (a ∩ b)) ∪ (a⊥ ∩ (a ∪ b⊥ ))))
23	15, 22	ax-r2 36	. 2 ((a ∩ b) ∪ ((a ∪ b) ∩ ((a ∩ b⊥ ) ∪ (a⊥ ∩ (a ∪ b⊥ ))))) = ((a ∪ b) ∩ (((a ∩ b⊥ ) ∪ (a ∩ b)) ∪ (a⊥ ∩ (a ∪ b⊥ ))))
24		ni32 502	. . 3 (a →3 b)⊥ = ((a ∪ b) ∩ ((a ∩ b⊥ ) ∪ (a⊥ ∩ (a ∪ b⊥ ))))
25	24	lor 70	. 2 ((a ∩ b) ∪ (a →3 b)⊥ ) = ((a ∩ b) ∪ ((a ∪ b) ∩ ((a ∩ b⊥ ) ∪ (a⊥ ∩ (a ∪ b⊥ )))))
26		i3n1 249	. . 3 (a⊥ →3 b⊥ ) = (((a ∩ b⊥ ) ∪ (a ∩ b)) ∪ (a⊥ ∩ (a ∪ b⊥ )))
27	26	lan 77	. 2 ((a ∪ b) ∩ (a⊥ →3 b⊥ )) = ((a ∪ b) ∩ (((a ∩ b⊥ ) ∪ (a ∩ b)) ∪ (a⊥ ∩ (a ∪ b⊥ ))))
28	23, 25, 27	3tr1 63	1 ((a ∩ b) ∪ (a →3 b)⊥ ) = ((a ∪ b) ∩ (a⊥ →3 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:  i3orlem6  557