Theorem i2or 344

Description: Lemma for disjunction of →₂ . (Contributed by NM, 5-Jul-2000.)

Assertion

Ref	Expression
i2or	((a →2 c) ∪ (b →2 c)) ≤ ((a ∩ b) →2 c)

Proof of Theorem i2or

Step	Hyp	Ref	Expression
1		df-i2 45	. . . 4 (a →2 c) = (c ∪ (a⊥ ∩ c⊥ ))
2		lea 160	. . . . . . 7 (a ∩ b) ≤ a
3	2	lecon 154	. . . . . 6 a⊥ ≤ (a ∩ b)⊥
4	3	leran 153	. . . . 5 (a⊥ ∩ c⊥ ) ≤ ((a ∩ b)⊥ ∩ c⊥ )
5	4	lelor 166	. . . 4 (c ∪ (a⊥ ∩ c⊥ )) ≤ (c ∪ ((a ∩ b)⊥ ∩ c⊥ ))
6	1, 5	bltr 138	. . 3 (a →2 c) ≤ (c ∪ ((a ∩ b)⊥ ∩ c⊥ ))
7		df-i2 45	. . . 4 (b →2 c) = (c ∪ (b⊥ ∩ c⊥ ))
8		lear 161	. . . . . . 7 (a ∩ b) ≤ b
9	8	lecon 154	. . . . . 6 b⊥ ≤ (a ∩ b)⊥
10	9	leran 153	. . . . 5 (b⊥ ∩ c⊥ ) ≤ ((a ∩ b)⊥ ∩ c⊥ )
11	10	lelor 166	. . . 4 (c ∪ (b⊥ ∩ c⊥ )) ≤ (c ∪ ((a ∩ b)⊥ ∩ c⊥ ))
12	7, 11	bltr 138	. . 3 (b →2 c) ≤ (c ∪ ((a ∩ b)⊥ ∩ c⊥ ))
13	6, 12	lel2or 170	. 2 ((a →2 c) ∪ (b →2 c)) ≤ (c ∪ ((a ∩ b)⊥ ∩ c⊥ ))
14		df-i2 45	. . 3 ((a ∩ b) →2 c) = (c ∪ ((a ∩ b)⊥ ∩ c⊥ ))
15	14	ax-r1 35	. 2 (c ∪ ((a ∩ b)⊥ ∩ c⊥ )) = ((a ∩ b) →2 c)
16	13, 15	lbtr 139	1 ((a →2 c) ∪ (b →2 c)) ≤ ((a ∩ b) →2 c)

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →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-i2 45  df-le1 130  df-le2 131

This theorem is referenced by:  orbile  843