Theorem u3lemc4 685

Description: Lemma for Kalmbach implication study.

Hypothesis

Ref	Expression
ulemc3.1	a C b

Assertion

Ref	Expression
u3lemc4	(a →3 b) = (a⊥ ∪ b)

Proof of Theorem u3lemc4

Step	Hyp	Ref	Expression
1		df-i3 45	. 2 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
2		ulemc3.1	. . . . . . . 8 a C b
3	2	comcom3 436	. . . . . . 7 a⊥ C b
4	2	comcom4 437	. . . . . . 7 a⊥ C b⊥
5	3, 4	fh1 451	. . . . . 6 (a⊥ ∩ (b ∪ b⊥ )) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
6	5	ax-r1 34	. . . . 5 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) = (a⊥ ∩ (b ∪ b⊥ ))
7		df-t 40	. . . . . . . 8 1 = (b ∪ b⊥ )
8	7	ax-r1 34	. . . . . . 7 (b ∪ b⊥ ) = 1
9	8	lan 70	. . . . . 6 (a⊥ ∩ (b ∪ b⊥ )) = (a⊥ ∩ 1)
10		an1 98	. . . . . 6 (a⊥ ∩ 1) = a⊥
11	9, 10	ax-r2 35	. . . . 5 (a⊥ ∩ (b ∪ b⊥ )) = a⊥
12	6, 11	ax-r2 35	. . . 4 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) = a⊥
13		comid 179	. . . . . . 7 a C a
14	13	comcom2 175	. . . . . 6 a C a⊥
15	14, 2	fh1 451	. . . . 5 (a ∩ (a⊥ ∪ b)) = ((a ∩ a⊥ ) ∪ (a ∩ b))
16		ax-a2 30	. . . . . 6 ((a ∩ a⊥ ) ∪ (a ∩ b)) = ((a ∩ b) ∪ (a ∩ a⊥ ))
17		dff 93	. . . . . . . . 9 0 = (a ∩ a⊥ )
18	17	ax-r1 34	. . . . . . . 8 (a ∩ a⊥ ) = 0
19	18	lor 66	. . . . . . 7 ((a ∩ b) ∪ (a ∩ a⊥ )) = ((a ∩ b) ∪ 0)
20		or0 94	. . . . . . 7 ((a ∩ b) ∪ 0) = (a ∩ b)
21	19, 20	ax-r2 35	. . . . . 6 ((a ∩ b) ∪ (a ∩ a⊥ )) = (a ∩ b)
22	16, 21	ax-r2 35	. . . . 5 ((a ∩ a⊥ ) ∪ (a ∩ b)) = (a ∩ b)
23	15, 22	ax-r2 35	. . . 4 (a ∩ (a⊥ ∪ b)) = (a ∩ b)
24	12, 23	2or 67	. . 3 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ (a ∩ b))
25	14, 2	fh4 454	. . . 4 (a⊥ ∪ (a ∩ b)) = ((a⊥ ∪ a) ∩ (a⊥ ∪ b))
26		ancom 68	. . . . 5 ((a⊥ ∪ a) ∩ (a⊥ ∪ b)) = ((a⊥ ∪ b) ∩ (a⊥ ∪ a))
27		ax-a2 30	. . . . . . . 8 (a⊥ ∪ a) = (a ∪ a⊥ )
28		df-t 40	. . . . . . . . 9 1 = (a ∪ a⊥ )
29	28	ax-r1 34	. . . . . . . 8 (a ∪ a⊥ ) = 1
30	27, 29	ax-r2 35	. . . . . . 7 (a⊥ ∪ a) = 1
31	30	lan 70	. . . . . 6 ((a⊥ ∪ b) ∩ (a⊥ ∪ a)) = ((a⊥ ∪ b) ∩ 1)
32		an1 98	. . . . . 6 ((a⊥ ∪ b) ∩ 1) = (a⊥ ∪ b)
33	31, 32	ax-r2 35	. . . . 5 ((a⊥ ∪ b) ∩ (a⊥ ∪ a)) = (a⊥ ∪ b)
34	26, 33	ax-r2 35	. . . 4 ((a⊥ ∪ a) ∩ (a⊥ ∪ b)) = (a⊥ ∪ b)
35	25, 34	ax-r2 35	. . 3 (a⊥ ∪ (a ∩ b)) = (a⊥ ∪ b)
36	24, 35	ax-r2 35	. 2 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ b)
37	1, 36	ax-r2 35	1 (a →3 b) = (a⊥ ∪ b)

Syntax hints:   = wb 1   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9  0wf 10   →₃ wi3 15

This theorem is referenced by:  u3lemle1 694  u3lem1 718  u3lem2 728  u3lem5 745  u3lem6 749  u3lem7 756  u3lem8 765  u3lem9 766

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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