Theorem u1lemaa 600

Description: Lemma for Sasaki implication study. Equation 4.10 of [MegPav2000] p. 23. This is the second part of the equation. (Contributed by NM, 14-Dec-1997.)

Assertion

Ref	Expression
u1lemaa	((a →1 b) ∩ a) = (a ∩ b)

Proof of Theorem u1lemaa

Step	Hyp	Ref	Expression
1		df-i1 44	. . 3 (a →1 b) = (a⊥ ∪ (a ∩ b))
2	1	ran 78	. 2 ((a →1 b) ∩ a) = ((a⊥ ∪ (a ∩ b)) ∩ a)
3		comid 187	. . . . 5 a C a
4	3	comcom2 183	. . . 4 a C a⊥
5		comanr1 464	. . . 4 a C (a ∩ b)
6	4, 5	fh1r 473	. . 3 ((a⊥ ∪ (a ∩ b)) ∩ a) = ((a⊥ ∩ a) ∪ ((a ∩ b) ∩ a))
7		ax-a2 31	. . . . 5 ((a⊥ ∩ a) ∪ ((a ∩ b) ∩ a)) = (((a ∩ b) ∩ a) ∪ (a⊥ ∩ a))
8		an32 83	. . . . . . 7 ((a ∩ b) ∩ a) = ((a ∩ a) ∩ b)
9		anidm 111	. . . . . . . 8 (a ∩ a) = a
10	9	ran 78	. . . . . . 7 ((a ∩ a) ∩ b) = (a ∩ b)
11	8, 10	ax-r2 36	. . . . . 6 ((a ∩ b) ∩ a) = (a ∩ b)
12		ancom 74	. . . . . . 7 (a⊥ ∩ a) = (a ∩ a⊥ )
13		dff 101	. . . . . . . 8 0 = (a ∩ a⊥ )
14	13	ax-r1 35	. . . . . . 7 (a ∩ a⊥ ) = 0
15	12, 14	ax-r2 36	. . . . . 6 (a⊥ ∩ a) = 0
16	11, 15	2or 72	. . . . 5 (((a ∩ b) ∩ a) ∪ (a⊥ ∩ a)) = ((a ∩ b) ∪ 0)
17	7, 16	ax-r2 36	. . . 4 ((a⊥ ∩ a) ∪ ((a ∩ b) ∩ a)) = ((a ∩ b) ∪ 0)
18		or0 102	. . . 4 ((a ∩ b) ∪ 0) = (a ∩ b)
19	17, 18	ax-r2 36	. . 3 ((a⊥ ∩ a) ∪ ((a ∩ b) ∩ a)) = (a ∩ b)
20	6, 19	ax-r2 36	. 2 ((a⊥ ∪ (a ∩ b)) ∩ a) = (a ∩ b)
21	2, 20	ax-r2 36	1 ((a →1 b) ∩ a) = (a ∩ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →₁ wi1 12

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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  u1lemnona  665  u12lembi  726  u1lem5  761  negantlem2  849  kb10iii  893  oas  925  oau  929  oaur  930  oa6to4  958  oa8to5  972