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Theorem apexp1 10935
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 6008 . . . . . . 7 |- ( w = 1 -> ( A ^ w ) = ( A ^ 1 ) )
2 oveq2 6008 . . . . . . 7 |- ( w = 1 -> ( B ^ w ) = ( B ^ 1 ) )
31, 2breq12d 4095 . . . . . 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 6008 . . . . . . 7 |- ( w = k -> ( A ^ w ) = ( A ^ k ) )
7 oveq2 6008 . . . . . . 7 |- ( w = k -> ( B ^ w ) = ( B ^ k ) )
86, 7breq12d 4095 . . . . . 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 6008 . . . . . . 7 |- ( w = ( k + 1 ) -> ( A ^ w ) = ( A ^ ( k + 1 ) ) )
12 oveq2 6008 . . . . . . 7 |- ( w = ( k + 1 ) -> ( B ^ w ) = ( B ^ ( k + 1 ) ) )
1311, 12breq12d 4095 . . . . . 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 6008 . . . . . . 7 |- ( w = N -> ( A ^ w ) = ( A ^ N ) )
17 oveq2 6008 . . . . . . 7 |- ( w = N -> ( B ^ w ) = ( B ^ N ) )
1816, 17breq12d 4095 . . . . . 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 10885 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A ^ 1 ) = A )
23 simpr 110 . . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> B e. CC )
2423exp1d 10885 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( B ^ 1 ) = B )
2522, 24breq12d 4095 . . . . 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 9372 . . . . . . . . . . 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 10891 . . . . . . . . 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 10891 . . . . . . . . 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 4113 . . . . . . . 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 10890 . . . . . . . . 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 10890 . . . . . . . . 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 8757 . . . . . . . . 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 1272 . . . . . . . 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 803 . . . . . 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 9122 . . 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 1218 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 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4082  (class class class)co 6000   CCcc 7993   1c1 7996   + caddc 7998   x. cmul 8000   # cap 8724   NNcn 9106   NN0cn0 9365   ^cexp 10755
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719  df-seqfrec 10665  df-exp 10756
This theorem is referenced by:  logbgcd1irraplemap  15637
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