Theorem u4lemaa 603

Description: Lemma for non-tollens implication study. (Contributed by NM, 14-Dec-1997.)

Assertion

Ref	Expression
u4lemaa	((a →4 b) ∩ a) = (a ∩ b)

Proof of Theorem u4lemaa

Step	Hyp	Ref	Expression
1		df-i4 47	. . 3 (a →4 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
2	1	ran 78	. 2 ((a →4 b) ∩ a) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∩ a)
3		comanr1 464	. . . . . 6 a C (a ∩ b)
4		comanr1 464	. . . . . . 7 a⊥ C (a⊥ ∩ b)
5	4	comcom6 459	. . . . . 6 a C (a⊥ ∩ b)
6	3, 5	com2or 483	. . . . 5 a C ((a ∩ b) ∪ (a⊥ ∩ b))
7	6	comcom 453	. . . 4 ((a ∩ b) ∪ (a⊥ ∩ b)) C a
8	3	comcom3 454	. . . . . . . 8 a⊥ C (a ∩ b)
9	8, 4	com2or 483	. . . . . . 7 a⊥ C ((a ∩ b) ∪ (a⊥ ∩ b))
10	9	comcom 453	. . . . . 6 ((a ∩ b) ∪ (a⊥ ∩ b)) C a⊥
11		comanr2 465	. . . . . . . 8 b C (a ∩ b)
12		comanr2 465	. . . . . . . 8 b C (a⊥ ∩ b)
13	11, 12	com2or 483	. . . . . . 7 b C ((a ∩ b) ∪ (a⊥ ∩ b))
14	13	comcom 453	. . . . . 6 ((a ∩ b) ∪ (a⊥ ∩ b)) C b
15	10, 14	com2or 483	. . . . 5 ((a ∩ b) ∪ (a⊥ ∩ b)) C (a⊥ ∪ b)
16	14	comcom2 183	. . . . 5 ((a ∩ b) ∪ (a⊥ ∩ b)) C b⊥
17	15, 16	com2an 484	. . . 4 ((a ∩ b) ∪ (a⊥ ∩ b)) C ((a⊥ ∪ b) ∩ b⊥ )
18	7, 17	fh2r 474	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∩ a) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∩ a) ∪ (((a⊥ ∪ b) ∩ b⊥ ) ∩ a))
19	3, 5	fh1r 473	. . . . . 6 (((a ∩ b) ∪ (a⊥ ∩ b)) ∩ a) = (((a ∩ b) ∩ a) ∪ ((a⊥ ∩ b) ∩ a))
20		an32 83	. . . . . . . . 9 ((a ∩ b) ∩ a) = ((a ∩ a) ∩ b)
21		anidm 111	. . . . . . . . . 10 (a ∩ a) = a
22	21	ran 78	. . . . . . . . 9 ((a ∩ a) ∩ b) = (a ∩ b)
23	20, 22	ax-r2 36	. . . . . . . 8 ((a ∩ b) ∩ a) = (a ∩ b)
24		ancom 74	. . . . . . . . 9 ((a⊥ ∩ b) ∩ a) = (a ∩ (a⊥ ∩ b))
25		anass 76	. . . . . . . . . . 11 ((a ∩ a⊥ ) ∩ b) = (a ∩ (a⊥ ∩ b))
26	25	ax-r1 35	. . . . . . . . . 10 (a ∩ (a⊥ ∩ b)) = ((a ∩ a⊥ ) ∩ b)
27		ancom 74	. . . . . . . . . . 11 ((a ∩ a⊥ ) ∩ b) = (b ∩ (a ∩ a⊥ ))
28		dff 101	. . . . . . . . . . . . . 14 0 = (a ∩ a⊥ )
29	28	ax-r1 35	. . . . . . . . . . . . 13 (a ∩ a⊥ ) = 0
30	29	lan 77	. . . . . . . . . . . 12 (b ∩ (a ∩ a⊥ )) = (b ∩ 0)
31		an0 108	. . . . . . . . . . . 12 (b ∩ 0) = 0
32	30, 31	ax-r2 36	. . . . . . . . . . 11 (b ∩ (a ∩ a⊥ )) = 0
33	27, 32	ax-r2 36	. . . . . . . . . 10 ((a ∩ a⊥ ) ∩ b) = 0
34	26, 33	ax-r2 36	. . . . . . . . 9 (a ∩ (a⊥ ∩ b)) = 0
35	24, 34	ax-r2 36	. . . . . . . 8 ((a⊥ ∩ b) ∩ a) = 0
36	23, 35	2or 72	. . . . . . 7 (((a ∩ b) ∩ a) ∪ ((a⊥ ∩ b) ∩ a)) = ((a ∩ b) ∪ 0)
37		or0 102	. . . . . . 7 ((a ∩ b) ∪ 0) = (a ∩ b)
38	36, 37	ax-r2 36	. . . . . 6 (((a ∩ b) ∩ a) ∪ ((a⊥ ∩ b) ∩ a)) = (a ∩ b)
39	19, 38	ax-r2 36	. . . . 5 (((a ∩ b) ∪ (a⊥ ∩ b)) ∩ a) = (a ∩ b)
40		anass 76	. . . . . 6 (((a⊥ ∪ b) ∩ b⊥ ) ∩ a) = ((a⊥ ∪ b) ∩ (b⊥ ∩ a))
41		ancom 74	. . . . . . . . 9 (b⊥ ∩ a) = (a ∩ b⊥ )
42		anor1 88	. . . . . . . . 9 (a ∩ b⊥ ) = (a⊥ ∪ b)⊥
43	41, 42	ax-r2 36	. . . . . . . 8 (b⊥ ∩ a) = (a⊥ ∪ b)⊥
44	43	lan 77	. . . . . . 7 ((a⊥ ∪ b) ∩ (b⊥ ∩ a)) = ((a⊥ ∪ b) ∩ (a⊥ ∪ b)⊥ )
45		dff 101	. . . . . . . 8 0 = ((a⊥ ∪ b) ∩ (a⊥ ∪ b)⊥ )
46	45	ax-r1 35	. . . . . . 7 ((a⊥ ∪ b) ∩ (a⊥ ∪ b)⊥ ) = 0
47	44, 46	ax-r2 36	. . . . . 6 ((a⊥ ∪ b) ∩ (b⊥ ∩ a)) = 0
48	40, 47	ax-r2 36	. . . . 5 (((a⊥ ∪ b) ∩ b⊥ ) ∩ a) = 0
49	39, 48	2or 72	. . . 4 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∩ a) ∪ (((a⊥ ∪ b) ∩ b⊥ ) ∩ a)) = ((a ∩ b) ∪ 0)
50	49, 37	ax-r2 36	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∩ a) ∪ (((a⊥ ∪ b) ∩ b⊥ ) ∩ a)) = (a ∩ b)
51	18, 50	ax-r2 36	. 2 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∩ a) = (a ∩ b)
52	2, 51	ax-r2 36	1 ((a →4 b) ∩ a) = (a ∩ b)

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →₄ wi4 15

This proof depends on 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 proof depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is used by:  u4lemnona  668  u4lem1  737  u4lem5  764