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Theorem apexp1 11134
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 6083 . . . . . . 7 |- ( w = 1 -> ( A ^ w ) = ( A ^ 1 ) )
2 oveq2 6083 . . . . . . 7 |- ( w = 1 -> ( B ^ w ) = ( B ^ 1 ) )
31, 2breq12d 4138 . . . . . 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 6083 . . . . . . 7 |- ( w = k -> ( A ^ w ) = ( A ^ k ) )
7 oveq2 6083 . . . . . . 7 |- ( w = k -> ( B ^ w ) = ( B ^ k ) )
86, 7breq12d 4138 . . . . . 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 6083 . . . . . . 7 |- ( w = ( k + 1 ) -> ( A ^ w ) = ( A ^ ( k + 1 ) ) )
12 oveq2 6083 . . . . . . 7 |- ( w = ( k + 1 ) -> ( B ^ w ) = ( B ^ ( k + 1 ) ) )
1311, 12breq12d 4138 . . . . . 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 6083 . . . . . . 7 |- ( w = N -> ( A ^ w ) = ( A ^ N ) )
17 oveq2 6083 . . . . . . 7 |- ( w = N -> ( B ^ w ) = ( B ^ N ) )
1816, 17breq12d 4138 . . . . . 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 11084 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A ^ 1 ) = A )
23 simpr 110 . . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> B e. CC )
2423exp1d 11084 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( B ^ 1 ) = B )
2522, 24breq12d 4138 . . . . 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 540 . . . . . . . 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 497 . . . . . . . . . 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 9549 . . . . . . . . . . 11 |- ( k e. NN -> k e. NN0 )
3433ad3antrrr 496 . . . . . . . . . 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 11090 . . . . . . . . 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 497 . . . . . . . . . 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 11090 . . . . . . . . 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 4156 . . . . . . . 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 11089 . . . . . . . . 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 11089 . . . . . . . . 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 8932 . . . . . . . . 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 1279 . . . . . . . 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 810 . . . . . 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 9299 . . 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 1225 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 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4125  (class class class)co 6075   CCcc 8167   1c1 8170   + caddc 8172   x. cmul 8174   # cap 8899   NNcn 9283   NN0cn0 9542   ^cexp 10953
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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287
This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-seqfrec 10863  df-exp 10954
This theorem is referenced by:  logbgcd1irraplemap  15994
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