Theorem u4lemab 613

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

Assertion

Ref	Expression
u4lemab	((a →4 b) ∩ b) = ((a ∩ b) ∪ (a⊥ ∩ b))

Proof of Theorem u4lemab

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) ∩ b) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∩ b)
3		comanr2 465	. . . . 5 b C (a ∩ b)
4		comanr2 465	. . . . 5 b C (a⊥ ∩ b)
5	3, 4	com2or 483	. . . 4 b C ((a ∩ b) ∪ (a⊥ ∩ b))
6		comanr2 465	. . . . 5 b⊥ C ((a⊥ ∪ b) ∩ b⊥ )
7	6	comcom6 459	. . . 4 b C ((a⊥ ∪ b) ∩ b⊥ )
8	5, 7	fh1r 473	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∩ b) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∩ b) ∪ (((a⊥ ∪ b) ∩ b⊥ ) ∩ b))
9		lear 161	. . . . . . 7 (a ∩ b) ≤ b
10		lear 161	. . . . . . 7 (a⊥ ∩ b) ≤ b
11	9, 10	lel2or 170	. . . . . 6 ((a ∩ b) ∪ (a⊥ ∩ b)) ≤ b
12	11	df2le2 136	. . . . 5 (((a ∩ b) ∪ (a⊥ ∩ b)) ∩ b) = ((a ∩ b) ∪ (a⊥ ∩ b))
13		an32 83	. . . . . 6 (((a⊥ ∪ b) ∩ b⊥ ) ∩ b) = (((a⊥ ∪ b) ∩ b) ∩ b⊥ )
14		anass 76	. . . . . . 7 (((a⊥ ∪ b) ∩ b) ∩ b⊥ ) = ((a⊥ ∪ b) ∩ (b ∩ b⊥ ))
15		dff 101	. . . . . . . . . 10 0 = (b ∩ b⊥ )
16	15	lan 77	. . . . . . . . 9 ((a⊥ ∪ b) ∩ 0) = ((a⊥ ∪ b) ∩ (b ∩ b⊥ ))
17	16	ax-r1 35	. . . . . . . 8 ((a⊥ ∪ b) ∩ (b ∩ b⊥ )) = ((a⊥ ∪ b) ∩ 0)
18		an0 108	. . . . . . . 8 ((a⊥ ∪ b) ∩ 0) = 0
19	17, 18	ax-r2 36	. . . . . . 7 ((a⊥ ∪ b) ∩ (b ∩ b⊥ )) = 0
20	14, 19	ax-r2 36	. . . . . 6 (((a⊥ ∪ b) ∩ b) ∩ b⊥ ) = 0
21	13, 20	ax-r2 36	. . . . 5 (((a⊥ ∪ b) ∩ b⊥ ) ∩ b) = 0
22	12, 21	2or 72	. . . 4 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∩ b) ∪ (((a⊥ ∪ b) ∩ b⊥ ) ∩ b)) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ 0)
23		or0 102	. . . 4 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ 0) = ((a ∩ b) ∪ (a⊥ ∩ b))
24	22, 23	ax-r2 36	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∩ b) ∪ (((a⊥ ∪ b) ∩ b⊥ ) ∩ b)) = ((a ∩ b) ∪ (a⊥ ∩ b))
25	8, 24	ax-r2 36	. 2 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∩ b) = ((a ∩ b) ∪ (a⊥ ∩ b))
26	2, 25	ax-r2 36	1 ((a →4 b) ∩ b) = ((a ∩ b) ∪ (a⊥ ∩ b))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →₄ wi4 15

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-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  u4lemnonb  678  u24lem  770