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Theorem 4oagen1 1042
Description: "Generalized" 4-OA.
Hypotheses
Ref Expression
4oa.1 e = (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
4oa.2 f = (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v e)
4oagen1.1 g =< f
Assertion
Ref Expression
4oagen1 ((a ->1 d) ^ (g v ((a ->1 d) ^ (b ->1 d)))) = ((a ->1 d) ^ (b ->1 d))
Proof of Theorem 4oagen1
StepHypRef Expression
1 4oagen1.1 . . . . . . 7 g =< f
2 4oa.2 . . . . . . . 8 f = (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v e)
3 or32 82 . . . . . . . 8 (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v e) = (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))
42, 3ax-r2 36 . . . . . . 7 f = (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))
51, 4lbtr 139 . . . . . 6 g =< (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))
65leror 152 . . . . 5 (g v ((a ->1 d) ^ (b ->1 d))) =< ((((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d))) v ((a ->1 d) ^ (b ->1 d)))
7 ax-a3 32 . . . . . 6 ((((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d))) v ((a ->1 d) ^ (b ->1 d))) = (((a ^ b) v e) v (((a ->1 d) ^ (b ->1 d)) v ((a ->1 d) ^ (b ->1 d))))
8 oridm 110 . . . . . . . 8 (((a ->1 d) ^ (b ->1 d)) v ((a ->1 d) ^ (b ->1 d))) = ((a ->1 d) ^ (b ->1 d))
98lor 70 . . . . . . 7 (((a ^ b) v e) v (((a ->1 d) ^ (b ->1 d)) v ((a ->1 d) ^ (b ->1 d)))) = (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))
104ax-r1 35 . . . . . . 7 (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d))) = f
119, 10ax-r2 36 . . . . . 6 (((a ^ b) v e) v (((a ->1 d) ^ (b ->1 d)) v ((a ->1 d) ^ (b ->1 d)))) = f
127, 11ax-r2 36 . . . . 5 ((((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d))) v ((a ->1 d) ^ (b ->1 d))) = f
136, 12lbtr 139 . . . 4 (g v ((a ->1 d) ^ (b ->1 d))) =< f
1413lelan 167 . . 3 ((a ->1 d) ^ (g v ((a ->1 d) ^ (b ->1 d)))) =< ((a ->1 d) ^ f)
15 4oa.1 . . . 4 e = (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
1615, 24oath1 1041 . . 3 ((a ->1 d) ^ f) = ((a ->1 d) ^ (b ->1 d))
1714, 16lbtr 139 . 2 ((a ->1 d) ^ (g v ((a ->1 d) ^ (b ->1 d)))) =< ((a ->1 d) ^ (b ->1 d))
18 lea 160 . . 3 ((a ->1 d) ^ (b ->1 d)) =< (a ->1 d)
19 leor 159 . . 3 ((a ->1 d) ^ (b ->1 d)) =< (g v ((a ->1 d) ^ (b ->1 d)))
2018, 19ler2an 173 . 2 ((a ->1 d) ^ (b ->1 d)) =< ((a ->1 d) ^ (g v ((a ->1 d) ^ (b ->1 d))))
2117, 20lebi 145 1 ((a ->1 d) ^ (g v ((a ->1 d) ^ (b ->1 d)))) = ((a ->1 d) ^ (b ->1 d))
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2   v wo 6   ^ wa 7   ->1 wi1 12
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-4oa 1033
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  4oagen1b  1043
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