Theorem u1lemle2 715

Description: Sasaki implication to l.e. (Contributed by NM, 11-Jan-1998.)

Hypothesis

Ref	Expression
u1lemle2.1	(a →1 b) = 1

Assertion

Ref	Expression
u1lemle2	a ≤ b

Proof of Theorem u1lemle2

Step	Hyp	Ref	Expression
1		anidm 111	. . . . . . . . 9 (a ∩ a) = a
2	1	ran 78	. . . . . . . 8 ((a ∩ a) ∩ b) = (a ∩ b)
3	2	ax-r1 35	. . . . . . 7 (a ∩ b) = ((a ∩ a) ∩ b)
4		anass 76	. . . . . . 7 ((a ∩ a) ∩ b) = (a ∩ (a ∩ b))
5	3, 4	ax-r2 36	. . . . . 6 (a ∩ b) = (a ∩ (a ∩ b))
6		dff 101	. . . . . 6 0 = (a ∩ a⊥ )
7	5, 6	2or 72	. . . . 5 ((a ∩ b) ∪ 0) = ((a ∩ (a ∩ b)) ∪ (a ∩ a⊥ ))
8		ax-a2 31	. . . . . . . 8 (a⊥ ∪ (a ∩ b)) = ((a ∩ b) ∪ a⊥ )
9	8	lan 77	. . . . . . 7 (a ∩ (a⊥ ∪ (a ∩ b))) = (a ∩ ((a ∩ b) ∪ a⊥ ))
10		coman1 185	. . . . . . . 8 (a ∩ b) C a
11	10	comcom2 183	. . . . . . . 8 (a ∩ b) C a⊥
12	10, 11	fh2 470	. . . . . . 7 (a ∩ ((a ∩ b) ∪ a⊥ )) = ((a ∩ (a ∩ b)) ∪ (a ∩ a⊥ ))
13	9, 12	ax-r2 36	. . . . . 6 (a ∩ (a⊥ ∪ (a ∩ b))) = ((a ∩ (a ∩ b)) ∪ (a ∩ a⊥ ))
14	13	ax-r1 35	. . . . 5 ((a ∩ (a ∩ b)) ∪ (a ∩ a⊥ )) = (a ∩ (a⊥ ∪ (a ∩ b)))
15	7, 14	ax-r2 36	. . . 4 ((a ∩ b) ∪ 0) = (a ∩ (a⊥ ∪ (a ∩ b)))
16		df-i1 44	. . . . . . 7 (a →1 b) = (a⊥ ∪ (a ∩ b))
17	16	ax-r1 35	. . . . . 6 (a⊥ ∪ (a ∩ b)) = (a →1 b)
18		u1lemle2.1	. . . . . 6 (a →1 b) = 1
19	17, 18	ax-r2 36	. . . . 5 (a⊥ ∪ (a ∩ b)) = 1
20	19	lan 77	. . . 4 (a ∩ (a⊥ ∪ (a ∩ b))) = (a ∩ 1)
21	15, 20	ax-r2 36	. . 3 ((a ∩ b) ∪ 0) = (a ∩ 1)
22		or0 102	. . 3 ((a ∩ b) ∪ 0) = (a ∩ b)
23		an1 106	. . 3 (a ∩ 1) = a
24	21, 22, 23	3tr2 64	. 2 (a ∩ b) = a
25	24	df2le1 135	1 a ≤ b

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  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:  3vded11  814  3vded12  815