Theorem u3lemc4 703

Description: Lemma for Kalmbach implication study. (Contributed by NM, 24-Dec-1997.)

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 46	. 2 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
2		ulemc3.1	. . . . . . . 8 a C b
3	2	comcom3 454	. . . . . . 7 a⊥ C b
4	2	comcom4 455	. . . . . . 7 a⊥ C b⊥
5	3, 4	fh1 469	. . . . . 6 (a⊥ ∩ (b ∪ b⊥ )) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
6	5	ax-r1 35	. . . . 5 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) = (a⊥ ∩ (b ∪ b⊥ ))
7		df-t 41	. . . . . . . 8 1 = (b ∪ b⊥ )
8	7	ax-r1 35	. . . . . . 7 (b ∪ b⊥ ) = 1
9	8	lan 77	. . . . . 6 (a⊥ ∩ (b ∪ b⊥ )) = (a⊥ ∩ 1)
10		an1 106	. . . . . 6 (a⊥ ∩ 1) = a⊥
11	9, 10	ax-r2 36	. . . . 5 (a⊥ ∩ (b ∪ b⊥ )) = a⊥
12	6, 11	ax-r2 36	. . . 4 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) = a⊥
13		comid 187	. . . . . . 7 a C a
14	13	comcom2 183	. . . . . 6 a C a⊥
15	14, 2	fh1 469	. . . . 5 (a ∩ (a⊥ ∪ b)) = ((a ∩ a⊥ ) ∪ (a ∩ b))
16		ax-a2 31	. . . . . 6 ((a ∩ a⊥ ) ∪ (a ∩ b)) = ((a ∩ b) ∪ (a ∩ a⊥ ))
17		dff 101	. . . . . . . . 9 0 = (a ∩ a⊥ )
18	17	ax-r1 35	. . . . . . . 8 (a ∩ a⊥ ) = 0
19	18	lor 70	. . . . . . 7 ((a ∩ b) ∪ (a ∩ a⊥ )) = ((a ∩ b) ∪ 0)
20		or0 102	. . . . . . 7 ((a ∩ b) ∪ 0) = (a ∩ b)
21	19, 20	ax-r2 36	. . . . . 6 ((a ∩ b) ∪ (a ∩ a⊥ )) = (a ∩ b)
22	16, 21	ax-r2 36	. . . . 5 ((a ∩ a⊥ ) ∪ (a ∩ b)) = (a ∩ b)
23	15, 22	ax-r2 36	. . . 4 (a ∩ (a⊥ ∪ b)) = (a ∩ b)
24	12, 23	2or 72	. . 3 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ (a ∩ b))
25	14, 2	fh4 472	. . . 4 (a⊥ ∪ (a ∩ b)) = ((a⊥ ∪ a) ∩ (a⊥ ∪ b))
26		ancom 74	. . . . 5 ((a⊥ ∪ a) ∩ (a⊥ ∪ b)) = ((a⊥ ∪ b) ∩ (a⊥ ∪ a))
27		ax-a2 31	. . . . . . . 8 (a⊥ ∪ a) = (a ∪ a⊥ )
28		df-t 41	. . . . . . . . 9 1 = (a ∪ a⊥ )
29	28	ax-r1 35	. . . . . . . 8 (a ∪ a⊥ ) = 1
30	27, 29	ax-r2 36	. . . . . . 7 (a⊥ ∪ a) = 1
31	30	lan 77	. . . . . 6 ((a⊥ ∪ b) ∩ (a⊥ ∪ a)) = ((a⊥ ∪ b) ∩ 1)
32		an1 106	. . . . . 6 ((a⊥ ∪ b) ∩ 1) = (a⊥ ∪ b)
33	31, 32	ax-r2 36	. . . . 5 ((a⊥ ∪ b) ∩ (a⊥ ∪ a)) = (a⊥ ∪ b)
34	26, 33	ax-r2 36	. . . 4 ((a⊥ ∪ a) ∩ (a⊥ ∪ b)) = (a⊥ ∪ b)
35	25, 34	ax-r2 36	. . 3 (a⊥ ∪ (a ∩ b)) = (a⊥ ∪ b)
36	24, 35	ax-r2 36	. 2 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ b)
37	1, 36	ax-r2 36	1 (a →3 b) = (a⊥ ∪ b)

Syntax hints:   = wb 1   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   →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:  u3lemle1  712  u3lem1  736  u3lem2  746  u3lem5  763  u3lem6  767  u3lem7  774  u3lem8  783  u3lem9  784