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Theorem apexp1 10631
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 5850 . . . . . . 7 |- ( w = 1 -> ( A ^ w ) = ( A ^ 1 ) )
2 oveq2 5850 . . . . . . 7 |- ( w = 1 -> ( B ^ w ) = ( B ^ 1 ) )
31, 2breq12d 3995 . . . . . 6 |- ( w = 1 -> ( ( A ^ w ) # ( B ^ w ) <-> ( A ^ 1 ) # ( B ^ 1 ) ) )
43imbi1d 230 . . . . 5 |- ( w = 1 -> ( ( ( A ^ w ) # ( B ^ w ) -> A # B ) <-> ( ( A ^ 1 ) # ( B ^ 1 ) -> A # B ) ) )
54imbi2d 229 . . . 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 5850 . . . . . . 7 |- ( w = k -> ( A ^ w ) = ( A ^ k ) )
7 oveq2 5850 . . . . . . 7 |- ( w = k -> ( B ^ w ) = ( B ^ k ) )
86, 7breq12d 3995 . . . . . 6 |- ( w = k -> ( ( A ^ w ) # ( B ^ w ) <-> ( A ^ k ) # ( B ^ k ) ) )
98imbi1d 230 . . . . 5 |- ( w = k -> ( ( ( A ^ w ) # ( B ^ w ) -> A # B ) <-> ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) )
109imbi2d 229 . . . 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 5850 . . . . . . 7 |- ( w = ( k + 1 ) -> ( A ^ w ) = ( A ^ ( k + 1 ) ) )
12 oveq2 5850 . . . . . . 7 |- ( w = ( k + 1 ) -> ( B ^ w ) = ( B ^ ( k + 1 ) ) )
1311, 12breq12d 3995 . . . . . 6 |- ( w = ( k + 1 ) -> ( ( A ^ w ) # ( B ^ w ) <-> ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) )
1413imbi1d 230 . . . . 5 |- ( w = ( k + 1 ) -> ( ( ( A ^ w ) # ( B ^ w ) -> A # B ) <-> ( ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) -> A # B ) ) )
1514imbi2d 229 . . . 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 5850 . . . . . . 7 |- ( w = N -> ( A ^ w ) = ( A ^ N ) )
17 oveq2 5850 . . . . . . 7 |- ( w = N -> ( B ^ w ) = ( B ^ N ) )
1816, 17breq12d 3995 . . . . . 6 |- ( w = N -> ( ( A ^ w ) # ( B ^ w ) <-> ( A ^ N ) # ( B ^ N ) ) )
1918imbi1d 230 . . . . 5 |- ( w = N -> ( ( ( A ^ w ) # ( B ^ w ) -> A # B ) <-> ( ( A ^ N ) # ( B ^ N ) -> A # B ) ) )
2019imbi2d 229 . . . 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 108 . . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> A e. CC )
2221exp1d 10583 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A ^ 1 ) = A )
23 simpr 109 . . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> B e. CC )
2423exp1d 10583 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( B ^ 1 ) = B )
2522, 24breq12d 3995 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 1 ) # ( B ^ 1 ) <-> A # B ) )
2625biimpd 143 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 1 ) # ( B ^ 1 ) -> A # B ) )
27 simpr 109 . . . . . . . 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 524 . . . . . . . 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 109 . . . . . . 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 109 . . . . . . . . 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 485 . . . . . . . . . 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 9121 . . . . . . . . . . 11 |- ( k e. NN -> k e. NN0 )
3433ad3antrrr 484 . . . . . . . . . 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 10589 . . . . . . . . 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 485 . . . . . . . . . 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 10589 . . . . . . . . 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 4013 . . . . . . . 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 10588 . . . . . . . . 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 10588 . . . . . . . . 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 8512 . . . . . . . . 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 1229 . . . . . . . 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 788 . . . . . 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 368 . . . . 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 8873 . . 3 |- ( N e. NN -> ( ( A e. CC /\ B e. CC ) -> ( ( A ^ N ) # ( B ^ N ) -> A # B ) ) )
4847impcom 124 . 2 |- ( ( ( A e. CC /\ B e. CC ) /\ N e. NN ) -> ( ( A ^ N ) # ( B ^ N ) -> A # B ) )
49483impa 1184 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 103   \/ wo 698   /\ w3a 968   = wceq 1343   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   CCcc 7751   1c1 7754   + caddc 7756   x. cmul 7758   # cap 8479   NNcn 8857   NN0cn0 9114   ^cexp 10454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467  df-seqfrec 10381  df-exp 10455
This theorem is referenced by:  logbgcd1irraplemap  13527
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