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Theorem apexp1 10664
Description: Exponentiation and apartness. (Contributed by Jim Kingdon, 9-Jul-2024.)
Assertion
Ref Expression
apexp1 |- ( ( A e. CC /\ B e. CC /\ N e. NN ) -> ( ( A ^ N ) # ( B ^ N ) -> A # B ) )
Proof of Theorem apexp1
Dummy variables k w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5873 . . . . . . 7 |- ( w = 1 -> ( A ^ w ) = ( A ^ 1 ) )
2 oveq2 5873 . . . . . . 7 |- ( w = 1 -> ( B ^ w ) = ( B ^ 1 ) )
31, 2breq12d 4011 . . . . . 6 |- ( w = 1 -> ( ( A ^ w ) # ( B ^ w ) <-> ( A ^ 1 ) # ( B ^ 1 ) ) )
43imbi1d 231 . . . . 5 |- ( w = 1 -> ( ( ( A ^ w ) # ( B ^ w ) -> A # B ) <-> ( ( A ^ 1 ) # ( B ^ 1 ) -> A # B ) ) )
54imbi2d 230 . . . 4 |- ( w = 1 -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A ^ w ) # ( B ^ w ) -> A # B ) ) <-> ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 1 ) # ( B ^ 1 ) -> A # B ) ) ) )
6 oveq2 5873 . . . . . . 7 |- ( w = k -> ( A ^ w ) = ( A ^ k ) )
7 oveq2 5873 . . . . . . 7 |- ( w = k -> ( B ^ w ) = ( B ^ k ) )
86, 7breq12d 4011 . . . . . 6 |- ( w = k -> ( ( A ^ w ) # ( B ^ w ) <-> ( A ^ k ) # ( B ^ k ) ) )
98imbi1d 231 . . . . 5 |- ( w = k -> ( ( ( A ^ w ) # ( B ^ w ) -> A # B ) <-> ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) )
109imbi2d 230 . . . 4 |- ( w = k -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A ^ w ) # ( B ^ w ) -> A # B ) ) <-> ( ( A e. CC /\ B e. CC ) -> ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) ) )
11 oveq2 5873 . . . . . . 7 |- ( w = ( k + 1 ) -> ( A ^ w ) = ( A ^ ( k + 1 ) ) )
12 oveq2 5873 . . . . . . 7 |- ( w = ( k + 1 ) -> ( B ^ w ) = ( B ^ ( k + 1 ) ) )
1311, 12breq12d 4011 . . . . . 6 |- ( w = ( k + 1 ) -> ( ( A ^ w ) # ( B ^ w ) <-> ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) )
1413imbi1d 231 . . . . 5 |- ( w = ( k + 1 ) -> ( ( ( A ^ w ) # ( B ^ w ) -> A # B ) <-> ( ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) -> A # B ) ) )
1514imbi2d 230 . . . 4 |- ( w = ( k + 1 ) -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A ^ w ) # ( B ^ w ) -> A # B ) ) <-> ( ( A e. CC /\ B e. CC ) -> ( ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) -> A # B ) ) ) )
16 oveq2 5873 . . . . . . 7 |- ( w = N -> ( A ^ w ) = ( A ^ N ) )
17 oveq2 5873 . . . . . . 7 |- ( w = N -> ( B ^ w ) = ( B ^ N ) )
1816, 17breq12d 4011 . . . . . 6 |- ( w = N -> ( ( A ^ w ) # ( B ^ w ) <-> ( A ^ N ) # ( B ^ N ) ) )
1918imbi1d 231 . . . . 5 |- ( w = N -> ( ( ( A ^ w ) # ( B ^ w ) -> A # B ) <-> ( ( A ^ N ) # ( B ^ N ) -> A # B ) ) )
2019imbi2d 230 . . . 4 |- ( w = N -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A ^ w ) # ( B ^ w ) -> A # B ) ) <-> ( ( A e. CC /\ B e. CC ) -> ( ( A ^ N ) # ( B ^ N ) -> A # B ) ) ) )
21 simpl 109 . . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> A e. CC )
2221exp1d 10616 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A ^ 1 ) = A )
23 simpr 110 . . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> B e. CC )
2423exp1d 10616 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( B ^ 1 ) = B )
2522, 24breq12d 4011 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 1 ) # ( B ^ 1 ) <-> A # B ) )
2625biimpd 144 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 1 ) # ( B ^ 1 ) -> A # B ) )
27 simpr 110 . . . . . . . 8 |- ( ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) /\ ( A ^ k ) # ( B ^ k ) ) -> ( A ^ k ) # ( B ^ k ) )
28 simpllr 534 . . . . . . . 8 |- ( ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) /\ ( A ^ k ) # ( B ^ k ) ) -> ( ( A ^ k ) # ( B ^ k ) -> A # B ) )
2927, 28mpd 13 . . . . . . 7 |- ( ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) /\ ( A ^ k ) # ( B ^ k ) ) -> A # B )
30 simpr 110 . . . . . . 7 |- ( ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) /\ A # B ) -> A # B )
31 simpr 110 . . . . . . . . 9 |- ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) -> ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) )
3221ad3antlr 493 . . . . . . . . . 10 |- ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) -> A e. CC )
33 nnnn0 9154 . . . . . . . . . . 11 |- ( k e. NN -> k e. NN0 )
3433ad3antrrr 492 . . . . . . . . . 10 |- ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) -> k e. NN0 )
3532, 34expp1d 10622 . . . . . . . . 9 |- ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
3623ad3antlr 493 . . . . . . . . . 10 |- ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) -> B e. CC )
3736, 34expp1d 10622 . . . . . . . . 9 |- ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) -> ( B ^ ( k + 1 ) ) = ( ( B ^ k ) x. B ) )
3831, 35, 373brtr3d 4029 . . . . . . . 8 |- ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) -> ( ( A ^ k ) x. A ) # ( ( B ^ k ) x. B ) )
3932, 34expcld 10621 . . . . . . . . 9 |- ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) -> ( A ^ k ) e. CC )
4036, 34expcld 10621 . . . . . . . . 9 |- ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) -> ( B ^ k ) e. CC )
41 mulext 8545 . . . . . . . . 9 |- ( ( ( ( A ^ k ) e. CC /\ A e. CC ) /\ ( ( B ^ k ) e. CC /\ B e. CC ) ) -> ( ( ( A ^ k ) x. A ) # ( ( B ^ k ) x. B ) -> ( ( A ^ k ) # ( B ^ k ) \/ A # B ) ) )
4239, 32, 40, 36, 41syl22anc 1239 . . . . . . . 8 |- ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) -> ( ( ( A ^ k ) x. A ) # ( ( B ^ k ) x. B ) -> ( ( A ^ k ) # ( B ^ k ) \/ A # B ) ) )
4338, 42mpd 13 . . . . . . 7 |- ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) -> ( ( A ^ k ) # ( B ^ k ) \/ A # B ) )
4429, 30, 43mpjaodan 798 . . . . . 6 |- ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) -> A # B )
4544exp41 370 . . . . 5 |- ( k e. NN -> ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ k ) # ( B ^ k ) -> A # B ) -> ( ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) -> A # B ) ) ) )
4645a2d 26 . . . 4 |- ( k e. NN -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) -> ( ( A e. CC /\ B e. CC ) -> ( ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) -> A # B ) ) ) )
475, 10, 15, 20, 26, 46nnind 8906 . . 3 |- ( N e. NN -> ( ( A e. CC /\ B e. CC ) -> ( ( A ^ N ) # ( B ^ N ) -> A # B ) ) )
4847impcom 125 . 2 |- ( ( ( A e. CC /\ B e. CC ) /\ N e. NN ) -> ( ( A ^ N ) # ( B ^ N ) -> A # B ) )
49483impa 1194 1 |- ( ( A e. CC /\ B e. CC /\ N e. NN ) -> ( ( A ^ N ) # ( B ^ N ) -> A # B ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   \/ wo 708   /\ w3a 978   = wceq 1353   e. wcel 2146   class class class wbr 3998  (class class class)co 5865   CCcc 7784   1c1 7787   + caddc 7789   x. cmul 7791   # cap 8512   NNcn 8890   NN0cn0 9147   ^cexp 10487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-n0 9148  df-z 9225  df-uz 9500  df-seqfrec 10414  df-exp 10488
This theorem is referenced by:  logbgcd1irraplemap  13938
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