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Theorem d3oa 995
Description: Derivation of 3-OA from OA distributive law.
Hypothesis
Ref Expression
d3oa.1 f = ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))
Assertion
Ref Expression
d3oa ((a ->1 c) ^ f) =< (b ->1 c)
Proof of Theorem d3oa
StepHypRef Expression
1 1oai1 821 . . 3 ((a ->1 c) ^ ((a ^ b)' ->1 ((a ->1 c) ^ (b ->1 c)))) =< (b ->1 c)
2 2oath1i1 827 . . . 4 ((a ->1 c) ^ ((a ^ b)' ->2 ((a ->1 c) ^ (b ->1 c)))) = ((a ->1 c) ^ (b ->1 c))
3 lear 161 . . . 4 ((a ->1 c) ^ (b ->1 c)) =< (b ->1 c)
42, 3bltr 138 . . 3 ((a ->1 c) ^ ((a ^ b)' ->2 ((a ->1 c) ^ (b ->1 c)))) =< (b ->1 c)
51, 4le2or 168 . 2 (((a ->1 c) ^ ((a ^ b)' ->1 ((a ->1 c) ^ (b ->1 c)))) v ((a ->1 c) ^ ((a ^ b)' ->2 ((a ->1 c) ^ (b ->1 c))))) =< ((b ->1 c) v (b ->1 c))
6 id 59 . . . . 5 (((a ^ a) v ((a ->1 c) ^ (a ->1 c))) ^ ((b ^ a) v ((b ->1 c) ^ (a ->1 c)))) = (((a ^ a) v ((a ->1 c) ^ (a ->1 c))) ^ ((b ^ a) v ((b ->1 c) ^ (a ->1 c))))
7 id 59 . . . . 5 (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ a) v ((a ->1 c) ^ (a ->1 c))) ^ ((b ^ a) v ((b ->1 c) ^ (a ->1 c))))) = (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ a) v ((a ->1 c) ^ (a ->1 c))) ^ ((b ^ a) v ((b ->1 c) ^ (a ->1 c)))))
8 leid 148 . . . . 5 (a ->1 c) =< (a ->1 c)
9 df-i1 44 . . . . . . 7 ((a ^ b)' ->1 ((a ->1 c) ^ (b ->1 c))) = ((a ^ b)'' v ((a ^ b)' ^ ((a ->1 c) ^ (b ->1 c))))
10 ax-a1 30 . . . . . . . . . 10 (a ^ b) = (a ^ b)''
1110ax-r1 35 . . . . . . . . 9 (a ^ b)'' = (a ^ b)
1211bile 142 . . . . . . . 8 (a ^ b)'' =< (a ^ b)
13 lear 161 . . . . . . . 8 ((a ^ b)' ^ ((a ->1 c) ^ (b ->1 c))) =< ((a ->1 c) ^ (b ->1 c))
1412, 13le2or 168 . . . . . . 7 ((a ^ b)'' v ((a ^ b)' ^ ((a ->1 c) ^ (b ->1 c)))) =< ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))
159, 14bltr 138 . . . . . 6 ((a ^ b)' ->1 ((a ->1 c) ^ (b ->1 c))) =< ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))
16 leo 158 . . . . . 6 ((a ^ b) v ((a ->1 c) ^ (b ->1 c))) =< (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ a) v ((a ->1 c) ^ (a ->1 c))) ^ ((b ^ a) v ((b ->1 c) ^ (a ->1 c)))))
1715, 16letr 137 . . . . 5 ((a ^ b)' ->1 ((a ->1 c) ^ (b ->1 c))) =< (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ a) v ((a ->1 c) ^ (a ->1 c))) ^ ((b ^ a) v ((b ->1 c) ^ (a ->1 c)))))
18 df-i2 45 . . . . . . . 8 ((a ^ b)' ->2 ((a ->1 c) ^ (b ->1 c))) = (((a ->1 c) ^ (b ->1 c)) v ((a ^ b)'' ^ ((a ->1 c) ^ (b ->1 c))'))
19 ax-a2 31 . . . . . . . 8 (((a ->1 c) ^ (b ->1 c)) v ((a ^ b)'' ^ ((a ->1 c) ^ (b ->1 c))')) = (((a ^ b)'' ^ ((a ->1 c) ^ (b ->1 c))') v ((a ->1 c) ^ (b ->1 c)))
2018, 19ax-r2 36 . . . . . . 7 ((a ^ b)' ->2 ((a ->1 c) ^ (b ->1 c))) = (((a ^ b)'' ^ ((a ->1 c) ^ (b ->1 c))') v ((a ->1 c) ^ (b ->1 c)))
21 lea 160 . . . . . . . . 9 ((a ^ b)'' ^ ((a ->1 c) ^ (b ->1 c))') =< (a ^ b)''
2221, 11lbtr 139 . . . . . . . 8 ((a ^ b)'' ^ ((a ->1 c) ^ (b ->1 c))') =< (a ^ b)
23 leid 148 . . . . . . . 8 ((a ->1 c) ^ (b ->1 c)) =< ((a ->1 c) ^ (b ->1 c))
2422, 23le2or 168 . . . . . . 7 (((a ^ b)'' ^ ((a ->1 c) ^ (b ->1 c))') v ((a ->1 c) ^ (b ->1 c))) =< ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))
2520, 24bltr 138 . . . . . 6 ((a ^ b)' ->2 ((a ->1 c) ^ (b ->1 c))) =< ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))
2625, 16letr 137 . . . . 5 ((a ^ b)' ->2 ((a ->1 c) ^ (b ->1 c))) =< (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ a) v ((a ->1 c) ^ (a ->1 c))) ^ ((b ^ a) v ((b ->1 c) ^ (a ->1 c)))))
27 leo 158 . . . . . 6 ((a ->1 c) ^ (b ->1 c)) =< (((a ->1 c) ^ (b ->1 c)) v ((a ^ b)'' ^ ((a ->1 c) ^ (b ->1 c))'))
2818ax-r1 35 . . . . . 6 (((a ->1 c) ^ (b ->1 c)) v ((a ^ b)'' ^ ((a ->1 c) ^ (b ->1 c))')) = ((a ^ b)' ->2 ((a ->1 c) ^ (b ->1 c)))
2927, 28lbtr 139 . . . . 5 ((a ->1 c) ^ (b ->1 c)) =< ((a ^ b)' ->2 ((a ->1 c) ^ (b ->1 c)))
306, 7, 8, 17, 26, 29ax-oadist 994 . . . 4 ((a ->1 c) ^ (((a ^ b)' ->1 ((a ->1 c) ^ (b ->1 c))) v ((a ^ b)' ->2 ((a ->1 c) ^ (b ->1 c))))) = (((a ->1 c) ^ ((a ^ b)' ->1 ((a ->1 c) ^ (b ->1 c)))) v ((a ->1 c) ^ ((a ^ b)' ->2 ((a ->1 c) ^ (b ->1 c)))))
3130ax-r1 35 . . 3 (((a ->1 c) ^ ((a ^ b)' ->1 ((a ->1 c) ^ (b ->1 c)))) v ((a ->1 c) ^ ((a ^ b)' ->2 ((a ->1 c) ^ (b ->1 c))))) = ((a ->1 c) ^ (((a ^ b)' ->1 ((a ->1 c) ^ (b ->1 c))) v ((a ^ b)' ->2 ((a ->1 c) ^ (b ->1 c)))))
32 u12lem 771 . . . . . . 7 (((a ^ b)' ->1 ((a ->1 c) ^ (b ->1 c))) v ((a ^ b)' ->2 ((a ->1 c) ^ (b ->1 c)))) = ((a ^ b)' ->0 ((a ->1 c) ^ (b ->1 c)))
33 df-i0 43 . . . . . . 7 ((a ^ b)' ->0 ((a ->1 c) ^ (b ->1 c))) = ((a ^ b)'' v ((a ->1 c) ^ (b ->1 c)))
3432, 33ax-r2 36 . . . . . 6 (((a ^ b)' ->1 ((a ->1 c) ^ (b ->1 c))) v ((a ^ b)' ->2 ((a ->1 c) ^ (b ->1 c)))) = ((a ^ b)'' v ((a ->1 c) ^ (b ->1 c)))
3510ax-r5 38 . . . . . . 7 ((a ^ b) v ((a ->1 c) ^ (b ->1 c))) = ((a ^ b)'' v ((a ->1 c) ^ (b ->1 c)))
3635ax-r1 35 . . . . . 6 ((a ^ b)'' v ((a ->1 c) ^ (b ->1 c))) = ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))
3734, 36ax-r2 36 . . . . 5 (((a ^ b)' ->1 ((a ->1 c) ^ (b ->1 c))) v ((a ^ b)' ->2 ((a ->1 c) ^ (b ->1 c)))) = ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))
38 d3oa.1 . . . . . 6 f = ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))
3938ax-r1 35 . . . . 5 ((a ^ b) v ((a ->1 c) ^ (b ->1 c))) = f
4037, 39ax-r2 36 . . . 4 (((a ^ b)' ->1 ((a ->1 c) ^ (b ->1 c))) v ((a ^ b)' ->2 ((a ->1 c) ^ (b ->1 c)))) = f
4140lan 77 . . 3 ((a ->1 c) ^ (((a ^ b)' ->1 ((a ->1 c) ^ (b ->1 c))) v ((a ^ b)' ->2 ((a ->1 c) ^ (b ->1 c))))) = ((a ->1 c) ^ f)
4231, 41ax-r2 36 . 2 (((a ->1 c) ^ ((a ^ b)' ->1 ((a ->1 c) ^ (b ->1 c)))) v ((a ->1 c) ^ ((a ^ b)' ->2 ((a ->1 c) ^ (b ->1 c))))) = ((a ->1 c) ^ f)
43 oridm 110 . 2 ((b ->1 c) v (b ->1 c)) = (b ->1 c)
445, 42, 43le3tr2 141 1 ((a ->1 c) ^ f) =< (b ->1 c)
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->0 wi0 11   ->1 wi1 12   ->2 wi2 13
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439  ax-oadist 994
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i0 43  df-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  d4oa  996
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