Theorem u2lemle2 716

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

Hypothesis

Ref	Expression
u2lemle2.1	(a →2 b) = 1

Assertion

Ref	Expression
u2lemle2	a ≤ b

Proof of Theorem u2lemle2

Step	Hyp	Ref	Expression
1		ax-a2 31	. . . . . . 7 (b ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ b)
2	1	lan 77	. . . . . 6 (a ∩ (b ∪ (a⊥ ∩ b⊥ ))) = (a ∩ ((a⊥ ∩ b⊥ ) ∪ b))
3		coman1 185	. . . . . . . . 9 (a⊥ ∩ b⊥ ) C a⊥
4	3	comcom7 460	. . . . . . . 8 (a⊥ ∩ b⊥ ) C a
5		coman2 186	. . . . . . . . 9 (a⊥ ∩ b⊥ ) C b⊥
6	5	comcom7 460	. . . . . . . 8 (a⊥ ∩ b⊥ ) C b
7	4, 6	fh2 470	. . . . . . 7 (a ∩ ((a⊥ ∩ b⊥ ) ∪ b)) = ((a ∩ (a⊥ ∩ b⊥ )) ∪ (a ∩ b))
8		ancom 74	. . . . . . . . . 10 ((a ∩ a⊥ ) ∩ b⊥ ) = (b⊥ ∩ (a ∩ a⊥ ))
9		anass 76	. . . . . . . . . 10 ((a ∩ a⊥ ) ∩ b⊥ ) = (a ∩ (a⊥ ∩ b⊥ ))
10		dff 101	. . . . . . . . . . . . 13 0 = (a ∩ a⊥ )
11	10	ax-r1 35	. . . . . . . . . . . 12 (a ∩ a⊥ ) = 0
12	11	lan 77	. . . . . . . . . . 11 (b⊥ ∩ (a ∩ a⊥ )) = (b⊥ ∩ 0)
13		an0 108	. . . . . . . . . . 11 (b⊥ ∩ 0) = 0
14	12, 13	ax-r2 36	. . . . . . . . . 10 (b⊥ ∩ (a ∩ a⊥ )) = 0
15	8, 9, 14	3tr2 64	. . . . . . . . 9 (a ∩ (a⊥ ∩ b⊥ )) = 0
16	15	ax-r5 38	. . . . . . . 8 ((a ∩ (a⊥ ∩ b⊥ )) ∪ (a ∩ b)) = (0 ∪ (a ∩ b))
17		ax-a2 31	. . . . . . . 8 (0 ∪ (a ∩ b)) = ((a ∩ b) ∪ 0)
18	16, 17	ax-r2 36	. . . . . . 7 ((a ∩ (a⊥ ∩ b⊥ )) ∪ (a ∩ b)) = ((a ∩ b) ∪ 0)
19	7, 18	ax-r2 36	. . . . . 6 (a ∩ ((a⊥ ∩ b⊥ ) ∪ b)) = ((a ∩ b) ∪ 0)
20	2, 19	ax-r2 36	. . . . 5 (a ∩ (b ∪ (a⊥ ∩ b⊥ ))) = ((a ∩ b) ∪ 0)
21	20	ax-r1 35	. . . 4 ((a ∩ b) ∪ 0) = (a ∩ (b ∪ (a⊥ ∩ b⊥ )))
22		df-i2 45	. . . . . . 7 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
23	22	ax-r1 35	. . . . . 6 (b ∪ (a⊥ ∩ b⊥ )) = (a →2 b)
24		u2lemle2.1	. . . . . 6 (a →2 b) = 1
25	23, 24	ax-r2 36	. . . . 5 (b ∪ (a⊥ ∩ b⊥ )) = 1
26	25	lan 77	. . . 4 (a ∩ (b ∪ (a⊥ ∩ b⊥ ))) = (a ∩ 1)
27	21, 26	ax-r2 36	. . 3 ((a ∩ b) ∪ 0) = (a ∩ 1)
28		or0 102	. . 3 ((a ∩ b) ∪ 0) = (a ∩ b)
29		an1 106	. . 3 (a ∩ 1) = a
30	27, 28, 29	3tr2 64	. 2 (a ∩ b) = a
31	30	df2le1 135	1 a ≤ b

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   →₂ wi2 13

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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  3vroa  831  imp3  841