Theorem i3th1 543

Description: Theorem for Kalmbach implication. (Contributed by NM, 16-Nov-1997.)

Assertion

Ref	Expression
i3th1	(a →3 (a →3 (b →3 a))) = 1

Proof of Theorem i3th1

Step	Hyp	Ref	Expression
1		df2i3 498	. . 3 (b →3 a) = ((b⊥ ∩ a⊥ ) ∪ ((b⊥ ∪ a) ∩ (b ∪ (b⊥ ∩ a))))
2	1	lor 70	. 2 (a⊥ ∪ (b →3 a)) = (a⊥ ∪ ((b⊥ ∩ a⊥ ) ∪ ((b⊥ ∪ a) ∩ (b ∪ (b⊥ ∩ a)))))
3		lem4 511	. 2 (a →3 (a →3 (b →3 a))) = (a⊥ ∪ (b →3 a))
4		ax-a3 32	. . . . 5 ((a⊥ ∪ b) ∪ (a ∩ b⊥ )) = (a⊥ ∪ (b ∪ (a ∩ b⊥ )))
5		anor1 88	. . . . . 6 (a ∩ b⊥ ) = (a⊥ ∪ b)⊥
6	5	lor 70	. . . . 5 ((a⊥ ∪ b) ∪ (a ∩ b⊥ )) = ((a⊥ ∪ b) ∪ (a⊥ ∪ b)⊥ )
7		ax-a3 32	. . . . . . . 8 ((a⊥ ∪ (a⊥ ∩ b)) ∪ ((b⊥ ∪ a) ∩ (b ∪ (b⊥ ∩ a)))) = (a⊥ ∪ ((a⊥ ∩ b) ∪ ((b⊥ ∪ a) ∩ (b ∪ (b⊥ ∩ a)))))
8		ax-a2 31	. . . . . . . . . . . . 13 (b⊥ ∪ a) = (a ∪ b⊥ )
9		anor2 89	. . . . . . . . . . . . . . 15 (a⊥ ∩ b) = (a ∪ b⊥ )⊥
10	9	con2 67	. . . . . . . . . . . . . 14 (a⊥ ∩ b)⊥ = (a ∪ b⊥ )
11	10	ax-r1 35	. . . . . . . . . . . . 13 (a ∪ b⊥ ) = (a⊥ ∩ b)⊥
12	8, 11	ax-r2 36	. . . . . . . . . . . 12 (b⊥ ∪ a) = (a⊥ ∩ b)⊥
13		ancom 74	. . . . . . . . . . . . 13 (b⊥ ∩ a) = (a ∩ b⊥ )
14	13	lor 70	. . . . . . . . . . . 12 (b ∪ (b⊥ ∩ a)) = (b ∪ (a ∩ b⊥ ))
15	12, 14	2an 79	. . . . . . . . . . 11 ((b⊥ ∪ a) ∩ (b ∪ (b⊥ ∩ a))) = ((a⊥ ∩ b)⊥ ∩ (b ∪ (a ∩ b⊥ )))
16	15	lor 70	. . . . . . . . . 10 ((a⊥ ∩ b) ∪ ((b⊥ ∪ a) ∩ (b ∪ (b⊥ ∩ a)))) = ((a⊥ ∩ b) ∪ ((a⊥ ∩ b)⊥ ∩ (b ∪ (a ∩ b⊥ ))))
17		oml5 449	. . . . . . . . . 10 ((a⊥ ∩ b) ∪ ((a⊥ ∩ b)⊥ ∩ (b ∪ (a ∩ b⊥ )))) = (b ∪ (a ∩ b⊥ ))
18	16, 17	ax-r2 36	. . . . . . . . 9 ((a⊥ ∩ b) ∪ ((b⊥ ∪ a) ∩ (b ∪ (b⊥ ∩ a)))) = (b ∪ (a ∩ b⊥ ))
19	18	lor 70	. . . . . . . 8 (a⊥ ∪ ((a⊥ ∩ b) ∪ ((b⊥ ∪ a) ∩ (b ∪ (b⊥ ∩ a))))) = (a⊥ ∪ (b ∪ (a ∩ b⊥ )))
20	7, 19	ax-r2 36	. . . . . . 7 ((a⊥ ∪ (a⊥ ∩ b)) ∪ ((b⊥ ∪ a) ∩ (b ∪ (b⊥ ∩ a)))) = (a⊥ ∪ (b ∪ (a ∩ b⊥ )))
21	20	ax-r1 35	. . . . . 6 (a⊥ ∪ (b ∪ (a ∩ b⊥ ))) = ((a⊥ ∪ (a⊥ ∩ b)) ∪ ((b⊥ ∪ a) ∩ (b ∪ (b⊥ ∩ a))))
22		orabs 120	. . . . . . 7 (a⊥ ∪ (a⊥ ∩ b)) = a⊥
23	22	ax-r5 38	. . . . . 6 ((a⊥ ∪ (a⊥ ∩ b)) ∪ ((b⊥ ∪ a) ∩ (b ∪ (b⊥ ∩ a)))) = (a⊥ ∪ ((b⊥ ∪ a) ∩ (b ∪ (b⊥ ∩ a))))
24	21, 23	ax-r2 36	. . . . 5 (a⊥ ∪ (b ∪ (a ∩ b⊥ ))) = (a⊥ ∪ ((b⊥ ∪ a) ∩ (b ∪ (b⊥ ∩ a))))
25	4, 6, 24	3tr2 64	. . . 4 ((a⊥ ∪ b) ∪ (a⊥ ∪ b)⊥ ) = (a⊥ ∪ ((b⊥ ∪ a) ∩ (b ∪ (b⊥ ∩ a))))
26		df-t 41	. . . 4 1 = ((a⊥ ∪ b) ∪ (a⊥ ∪ b)⊥ )
27		ancom 74	. . . . . . 7 (b⊥ ∩ a⊥ ) = (a⊥ ∩ b⊥ )
28	27	lor 70	. . . . . 6 (a⊥ ∪ (b⊥ ∩ a⊥ )) = (a⊥ ∪ (a⊥ ∩ b⊥ ))
29		orabs 120	. . . . . 6 (a⊥ ∪ (a⊥ ∩ b⊥ )) = a⊥
30	28, 29	ax-r2 36	. . . . 5 (a⊥ ∪ (b⊥ ∩ a⊥ )) = a⊥
31	30	ax-r5 38	. . . 4 ((a⊥ ∪ (b⊥ ∩ a⊥ )) ∪ ((b⊥ ∪ a) ∩ (b ∪ (b⊥ ∩ a)))) = (a⊥ ∪ ((b⊥ ∪ a) ∩ (b ∪ (b⊥ ∩ a))))
32	25, 26, 31	3tr1 63	. . 3 1 = ((a⊥ ∪ (b⊥ ∩ a⊥ )) ∪ ((b⊥ ∪ a) ∩ (b ∪ (b⊥ ∩ a))))
33		ax-a3 32	. . 3 ((a⊥ ∪ (b⊥ ∩ a⊥ )) ∪ ((b⊥ ∪ a) ∩ (b ∪ (b⊥ ∩ a)))) = (a⊥ ∪ ((b⊥ ∩ a⊥ ) ∪ ((b⊥ ∪ a) ∩ (b ∪ (b⊥ ∩ a)))))
34	32, 33	ax-r2 36	. 2 1 = (a⊥ ∪ ((b⊥ ∩ a⊥ ) ∪ ((b⊥ ∪ a) ∩ (b ∪ (b⊥ ∩ a)))))
35	2, 3, 34	3tr1 63	1 (a →3 (a →3 (b →3 a))) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →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:  u3lem14aa  792