Theorem i2or 336

Description: Lemma for disjunction of ->2.

Assertion

Ref	Expression
i2or	((a ->2 c) v (b ->2 c)) =< ((a ^ b) ->2 c)

Proof of Theorem i2or

Step	Hyp	Ref	Expression
1		df-i2 44	. . . 4 (a ->2 c) = (c v (a_|_ ^ c_|_))
2		lea 152	. . . . . . 7 (a ^ b) =< a
3	2	lecon 146	. . . . . 6 a_|_ =< (a ^ b)_|_
4	3	leran 145	. . . . 5 (a_|_ ^ c_|_) =< ((a ^ b)_|_ ^ c_|_)
5	4	lelor 158	. . . 4 (c v (a_|_ ^ c_|_)) =< (c v ((a ^ b)_|_ ^ c_|_))
6	1, 5	bltr 130	. . 3 (a ->2 c) =< (c v ((a ^ b)_|_ ^ c_|_))
7		df-i2 44	. . . 4 (b ->2 c) = (c v (b_|_ ^ c_|_))
8		lear 153	. . . . . . 7 (a ^ b) =< b
9	8	lecon 146	. . . . . 6 b_|_ =< (a ^ b)_|_
10	9	leran 145	. . . . 5 (b_|_ ^ c_|_) =< ((a ^ b)_|_ ^ c_|_)
11	10	lelor 158	. . . 4 (c v (b_|_ ^ c_|_)) =< (c v ((a ^ b)_|_ ^ c_|_))
12	7, 11	bltr 130	. . 3 (b ->2 c) =< (c v ((a ^ b)_|_ ^ c_|_))
13	6, 12	lel2or 162	. 2 ((a ->2 c) v (b ->2 c)) =< (c v ((a ^ b)_|_ ^ c_|_))
14		df-i2 44	. . 3 ((a ^ b) ->2 c) = (c v ((a ^ b)_|_ ^ c_|_))
15	14	ax-r1 34	. 2 (c v ((a ^ b)_|_ ^ c_|_)) = ((a ^ b) ->2 c)
16	13, 15	lbtr 131	1 ((a ->2 c) v (b ->2 c)) =< ((a ^ b) ->2 c)

Syntax hints:   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->2 wi2 14

This theorem is referenced by:  orbile 825

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-i2 44  df-le1 122  df-le2 123

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