Theorem i1or 337

Description: Lemma for disjunction of ->1.

Assertion

Ref	Expression
i1or	((c ->1 a) v (c ->1 b)) =< (c ->1 (a v b))

Proof of Theorem i1or

Step	Hyp	Ref	Expression
1		df-i1 43	. . . 4 (c ->1 a) = (c_|_ v (c ^ a))
2		leo 150	. . . . . 6 a =< (a v b)
3	2	lelan 159	. . . . 5 (c ^ a) =< (c ^ (a v b))
4	3	lelor 158	. . . 4 (c_|_ v (c ^ a)) =< (c_|_ v (c ^ (a v b)))
5	1, 4	bltr 130	. . 3 (c ->1 a) =< (c_|_ v (c ^ (a v b)))
6		df-i1 43	. . . 4 (c ->1 b) = (c_|_ v (c ^ b))
7		leor 151	. . . . . 6 b =< (a v b)
8	7	lelan 159	. . . . 5 (c ^ b) =< (c ^ (a v b))
9	8	lelor 158	. . . 4 (c_|_ v (c ^ b)) =< (c_|_ v (c ^ (a v b)))
10	6, 9	bltr 130	. . 3 (c ->1 b) =< (c_|_ v (c ^ (a v b)))
11	5, 10	lel2or 162	. 2 ((c ->1 a) v (c ->1 b)) =< (c_|_ v (c ^ (a v b)))
12		df-i1 43	. . 3 (c ->1 (a v b)) = (c_|_ v (c ^ (a v b)))
13	12	ax-r1 34	. 2 (c_|_ v (c ^ (a v b))) = (c ->1 (a v b))
14	11, 13	lbtr 131	1 ((c ->1 a) v (c ->1 b)) =< (c ->1 (a v b))

Syntax hints:   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->1 wi1 13

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-i1 43  df-le1 122  df-le2 123

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