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Theorem i2or 344
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
StepHypRef Expression
1 df-i2 45 . . . 4 (a ->2 c) = (c v (a' ^ c'))
2 lea 160 . . . . . . 7 (a ^ b) =< a
32lecon 154 . . . . . 6 a' =< (a ^ b)'
43leran 153 . . . . 5 (a' ^ c') =< ((a ^ b)' ^ c')
54lelor 166 . . . 4 (c v (a' ^ c')) =< (c v ((a ^ b)' ^ c'))
61, 5bltr 138 . . 3 (a ->2 c) =< (c v ((a ^ b)' ^ c'))
7 df-i2 45 . . . 4 (b ->2 c) = (c v (b' ^ c'))
8 lear 161 . . . . . . 7 (a ^ b) =< b
98lecon 154 . . . . . 6 b' =< (a ^ b)'
109leran 153 . . . . 5 (b' ^ c') =< ((a ^ b)' ^ c')
1110lelor 166 . . . 4 (c v (b' ^ c')) =< (c v ((a ^ b)' ^ c'))
127, 11bltr 138 . . 3 (b ->2 c) =< (c v ((a ^ b)' ^ c'))
136, 12lel2or 170 . 2 ((a ->2 c) v (b ->2 c)) =< (c v ((a ^ b)' ^ c'))
14 df-i2 45 . . 3 ((a ^ b) ->2 c) = (c v ((a ^ b)' ^ c'))
1514ax-r1 35 . 2 (c v ((a ^ b)' ^ c')) = ((a ^ b) ->2 c)
1613, 15lbtr 139 1 ((a ->2 c) v (b ->2 c)) =< ((a ^ b) ->2 c)
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   v 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
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