Theorem u12lem 771

Description: Implication lemma. (Contributed by NM, 17-Nov-1998.)

Assertion

Ref	Expression
u12lem	((a →1 b) ∪ (a →2 b)) = (a →0 b)

Proof of Theorem u12lem

Step	Hyp	Ref	Expression
1		orordi 112	. . 3 ((a →1 b) ∪ (b ∪ (a⊥ ∩ b⊥ ))) = (((a →1 b) ∪ b) ∪ ((a →1 b) ∪ (a⊥ ∩ b⊥ )))
2		u1lemob 630	. . . . 5 ((a →1 b) ∪ b) = (a⊥ ∪ b)
3		df-i1 44	. . . . . . 7 (a →1 b) = (a⊥ ∪ (a ∩ b))
4	3	ax-r5 38	. . . . . 6 ((a →1 b) ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∪ (a ∩ b)) ∪ (a⊥ ∩ b⊥ ))
5		or32 82	. . . . . . 7 ((a⊥ ∪ (a ∩ b)) ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ b))
6		orabs 120	. . . . . . . 8 (a⊥ ∪ (a⊥ ∩ b⊥ )) = a⊥
7	6	ax-r5 38	. . . . . . 7 ((a⊥ ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ b)) = (a⊥ ∪ (a ∩ b))
8	5, 7	ax-r2 36	. . . . . 6 ((a⊥ ∪ (a ∩ b)) ∪ (a⊥ ∩ b⊥ )) = (a⊥ ∪ (a ∩ b))
9	4, 8	ax-r2 36	. . . . 5 ((a →1 b) ∪ (a⊥ ∩ b⊥ )) = (a⊥ ∪ (a ∩ b))
10	2, 9	2or 72	. . . 4 (((a →1 b) ∪ b) ∪ ((a →1 b) ∪ (a⊥ ∩ b⊥ ))) = ((a⊥ ∪ b) ∪ (a⊥ ∪ (a ∩ b)))
11		id 59	. . . . . . 7 (a⊥ ∪ b) = (a⊥ ∪ b)
12	11	bile 142	. . . . . 6 (a⊥ ∪ b) ≤ (a⊥ ∪ b)
13		lear 161	. . . . . . 7 (a ∩ b) ≤ b
14	13	lelor 166	. . . . . 6 (a⊥ ∪ (a ∩ b)) ≤ (a⊥ ∪ b)
15	12, 14	lel2or 170	. . . . 5 ((a⊥ ∪ b) ∪ (a⊥ ∪ (a ∩ b))) ≤ (a⊥ ∪ b)
16		leo 158	. . . . 5 (a⊥ ∪ b) ≤ ((a⊥ ∪ b) ∪ (a⊥ ∪ (a ∩ b)))
17	15, 16	lebi 145	. . . 4 ((a⊥ ∪ b) ∪ (a⊥ ∪ (a ∩ b))) = (a⊥ ∪ b)
18	10, 17	ax-r2 36	. . 3 (((a →1 b) ∪ b) ∪ ((a →1 b) ∪ (a⊥ ∩ b⊥ ))) = (a⊥ ∪ b)
19	1, 18	ax-r2 36	. 2 ((a →1 b) ∪ (b ∪ (a⊥ ∩ b⊥ ))) = (a⊥ ∪ b)
20		df-i2 45	. . 3 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
21	20	lor 70	. 2 ((a →1 b) ∪ (a →2 b)) = ((a →1 b) ∪ (b ∪ (a⊥ ∩ b⊥ )))
22		df-i0 43	. 2 (a →0 b) = (a⊥ ∪ b)
23	19, 21, 22	3tr1 63	1 ((a →1 b) ∪ (a →2 b)) = (a →0 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₀ wi0 11   →₁ wi1 12   →₂ wi2 13

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i0 43  df-i1 44  df-i2 45  df-le1 130  df-le2 131

This theorem is referenced by:  distoah2  941  distoah3  942  distoa  944  d3oa  995  oadist2b  1008  oadist12  1010  lem4.6.6i1j2  1093