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Theorem d4oa 996
Description: Variant of proper 4-OA proved from OA distributive law.
Hypotheses
Ref Expression
d4oa.2 e = ((a ^ b) v ((a ->1 d) ^ (b ->1 d)))
d4oa.1 f = (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
Assertion
Ref Expression
d4oa ((a ->1 d) ^ (e v f)) =< (b ->1 d)

Proof of Theorem d4oa
StepHypRef Expression
1 ax-a2 31 . . . 4 (e v f) = (f v e)
21lan 77 . . 3 ((a ->1 d) ^ (e v f)) = ((a ->1 d) ^ (f v e))
3 id 59 . . . 4 (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))) = (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
4 d4oa.2 . . . . 5 e = ((a ^ b) v ((a ->1 d) ^ (b ->1 d)))
5 d4oa.1 . . . . 5 f = (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
64, 52or 72 . . . 4 (e v f) = (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))))
7 leid 148 . . . 4 (a ->1 d) =< (a ->1 d)
8 leor 159 . . . 4 f =< (e v f)
9 leo 158 . . . 4 e =< (e v f)
10 leor 159 . . . . 5 ((a ->1 d) ^ (b ->1 d)) =< ((a ^ b) v ((a ->1 d) ^ (b ->1 d)))
114ax-r1 35 . . . . 5 ((a ^ b) v ((a ->1 d) ^ (b ->1 d))) = e
1210, 11lbtr 139 . . . 4 ((a ->1 d) ^ (b ->1 d)) =< e
133, 6, 7, 8, 9, 12ax-oadist 994 . . 3 ((a ->1 d) ^ (f v e)) = (((a ->1 d) ^ f) v ((a ->1 d) ^ e))
142, 13ax-r2 36 . 2 ((a ->1 d) ^ (e v f)) = (((a ->1 d) ^ f) v ((a ->1 d) ^ e))
155lan 77 . . . . . 6 ((a ->1 d) ^ f) = ((a ->1 d) ^ (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))))
16 anass 76 . . . . . . 7 (((a ->1 d) ^ ((a ^ c) v ((a ->1 d) ^ (c ->1 d)))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))) = ((a ->1 d) ^ (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))))
1716ax-r1 35 . . . . . 6 ((a ->1 d) ^ (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))) = (((a ->1 d) ^ ((a ^ c) v ((a ->1 d) ^ (c ->1 d)))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
1815, 17ax-r2 36 . . . . 5 ((a ->1 d) ^ f) = (((a ->1 d) ^ ((a ^ c) v ((a ->1 d) ^ (c ->1 d)))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
19 id 59 . . . . . . 7 ((a ^ c) v ((a ->1 d) ^ (c ->1 d))) = ((a ^ c) v ((a ->1 d) ^ (c ->1 d)))
2019d3oa 995 . . . . . 6 ((a ->1 d) ^ ((a ^ c) v ((a ->1 d) ^ (c ->1 d)))) =< (c ->1 d)
2120leran 153 . . . . 5 (((a ->1 d) ^ ((a ^ c) v ((a ->1 d) ^ (c ->1 d)))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))) =< ((c ->1 d) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
2218, 21bltr 138 . . . 4 ((a ->1 d) ^ f) =< ((c ->1 d) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
23 ancom 74 . . . . . 6 (b ^ c) = (c ^ b)
24 ancom 74 . . . . . 6 ((b ->1 d) ^ (c ->1 d)) = ((c ->1 d) ^ (b ->1 d))
2523, 242or 72 . . . . 5 ((b ^ c) v ((b ->1 d) ^ (c ->1 d))) = ((c ^ b) v ((c ->1 d) ^ (b ->1 d)))
2625d3oa 995 . . . 4 ((c ->1 d) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))) =< (b ->1 d)
2722, 26letr 137 . . 3 ((a ->1 d) ^ f) =< (b ->1 d)
284d3oa 995 . . 3 ((a ->1 d) ^ e) =< (b ->1 d)
2927, 28lel2or 170 . 2 (((a ->1 d) ^ f) v ((a ->1 d) ^ e)) =< (b ->1 d)
3014, 29bltr 138 1 ((a ->1 d) ^ (e v f)) =< (b ->1 d)
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2   v wo 6   ^ wa 7   ->1 wi1 12
This theorem is referenced by:  d6oa  997
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
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