Theorem u12lem 753

Description: Implication lemma.

Assertion

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

Proof of Theorem u12lem

Step	Hyp	Ref	Expression
1		orordi 104	. . 3 ((a →1 b) ∪ (b ∪ (a⊥ ∩ b⊥ ))) = (((a →1 b) ∪ b) ∪ ((a →1 b) ∪ (a⊥ ∩ b⊥ )))
2		u1lemob 612	. . . . 5 ((a →1 b) ∪ b) = (a⊥ ∪ b)
3		df-i1 43	. . . . . . 7 (a →1 b) = (a⊥ ∪ (a ∩ b))
4	3	ax-r5 37	. . . . . 6 ((a →1 b) ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∪ (a ∩ b)) ∪ (a⊥ ∩ b⊥ ))
5		or32 75	. . . . . . 7 ((a⊥ ∪ (a ∩ b)) ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ b))
6		a5b 112	. . . . . . . 8 (a⊥ ∪ (a⊥ ∩ b⊥ )) = a⊥
7	6	ax-r5 37	. . . . . . 7 ((a⊥ ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ b)) = (a⊥ ∪ (a ∩ b))
8	5, 7	ax-r2 35	. . . . . 6 ((a⊥ ∪ (a ∩ b)) ∪ (a⊥ ∩ b⊥ )) = (a⊥ ∪ (a ∩ b))
9	4, 8	ax-r2 35	. . . . 5 ((a →1 b) ∪ (a⊥ ∩ b⊥ )) = (a⊥ ∪ (a ∩ b))
10	2, 9	2or 67	. . . 4 (((a →1 b) ∪ b) ∪ ((a →1 b) ∪ (a⊥ ∩ b⊥ ))) = ((a⊥ ∪ b) ∪ (a⊥ ∪ (a ∩ b)))
11		id 58	. . . . . . 7 (a⊥ ∪ b) = (a⊥ ∪ b)
12	11	bile 134	. . . . . 6 (a⊥ ∪ b) ≤ (a⊥ ∪ b)
13		lear 153	. . . . . . 7 (a ∩ b) ≤ b
14	13	lelor 158	. . . . . 6 (a⊥ ∪ (a ∩ b)) ≤ (a⊥ ∪ b)
15	12, 14	lel2or 162	. . . . 5 ((a⊥ ∪ b) ∪ (a⊥ ∪ (a ∩ b))) ≤ (a⊥ ∪ b)
16		leo 150	. . . . 5 (a⊥ ∪ b) ≤ ((a⊥ ∪ b) ∪ (a⊥ ∪ (a ∩ b)))
17	15, 16	lebi 137	. . . 4 ((a⊥ ∪ b) ∪ (a⊥ ∪ (a ∩ b))) = (a⊥ ∪ b)
18	10, 17	ax-r2 35	. . 3 (((a →1 b) ∪ b) ∪ ((a →1 b) ∪ (a⊥ ∩ b⊥ ))) = (a⊥ ∪ b)
19	1, 18	ax-r2 35	. 2 ((a →1 b) ∪ (b ∪ (a⊥ ∩ b⊥ ))) = (a⊥ ∪ b)
20		df-i2 44	. . 3 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
21	20	lor 66	. 2 ((a →1 b) ∪ (a →2 b)) = ((a →1 b) ∪ (b ∪ (a⊥ ∩ b⊥ )))
22		df-i0 42	. 2 (a →0 b) = (a⊥ ∪ b)
23	19, 21, 22	3tr1 60	1 ((a →1 b) ∪ (a →2 b)) = (a →0 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₀ wi0 12   →₁ wi1 13   →₂ wi2 14

This theorem is referenced by:  distoah2 921  distoah3 922  distoa 924  d3oa 975  oadist2b 988  oadist12 990

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i0 42  df-i1 43  df-i2 44  df-le1 122  df-le2 123

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