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Theorem pwm1geoserap1 10902
Description: The n-th power of a number decreased by 1 expressed by the finite geometric series  1  +  A ^ 1  +  A ^ 2  +...  +  A ^ ( N  - 
1 ). (Contributed by AV, 14-Aug-2021.) (Revised by Jim Kingdon, 24-Oct-2022.)
Hypotheses
Ref Expression
pwm1geoser.1  |-  ( ph  ->  A  e.  CC )
pwm1geoser.3  |-  ( ph  ->  N  e.  NN0 )
pwm1geoserap1.ap  |-  ( ph  ->  A #  1 )
Assertion
Ref Expression
pwm1geoserap1  |-  ( ph  ->  ( ( A ^ N )  -  1 )  =  ( ( A  -  1 )  x.  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( A ^ k
) ) )
Distinct variable groups:    A, k    k, N    ph, k

Proof of Theorem pwm1geoserap1
StepHypRef Expression
1 pwm1geoser.1 . . 3  |-  ( ph  ->  A  e.  CC )
2 pwm1geoserap1.ap . . 3  |-  ( ph  ->  A #  1 )
3 pwm1geoser.3 . . 3  |-  ( ph  ->  N  e.  NN0 )
41, 2, 3geoserap 10901 . 2  |-  ( ph  -> 
sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( A ^ k
)  =  ( ( 1  -  ( A ^ N ) )  /  ( 1  -  A ) ) )
5 eqcom 2090 . . 3  |-  ( sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( A ^ k )  =  ( ( 1  -  ( A ^ N ) )  / 
( 1  -  A
) )  <->  ( (
1  -  ( A ^ N ) )  /  ( 1  -  A ) )  = 
sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( A ^ k
) )
6 1cnd 7504 . . . . . 6  |-  ( ph  ->  1  e.  CC )
71, 3expcld 10086 . . . . . 6  |-  ( ph  ->  ( A ^ N
)  e.  CC )
8 apsym 8083 . . . . . . . 8  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A #  1  <->  1 #  A ) )
91, 6, 8syl2anc 403 . . . . . . 7  |-  ( ph  ->  ( A #  1  <->  1 #  A ) )
102, 9mpbid 145 . . . . . 6  |-  ( ph  ->  1 #  A )
116, 7, 6, 1, 10div2subapd 8303 . . . . 5  |-  ( ph  ->  ( ( 1  -  ( A ^ N
) )  /  (
1  -  A ) )  =  ( ( ( A ^ N
)  -  1 )  /  ( A  - 
1 ) ) )
1211eqeq1d 2096 . . . 4  |-  ( ph  ->  ( ( ( 1  -  ( A ^ N ) )  / 
( 1  -  A
) )  =  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( A ^ k )  <-> 
( ( ( A ^ N )  - 
1 )  /  ( A  -  1 ) )  =  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( A ^
k ) ) )
13 peano2cnm 7748 . . . . . 6  |-  ( ( A ^ N )  e.  CC  ->  (
( A ^ N
)  -  1 )  e.  CC )
147, 13syl 14 . . . . 5  |-  ( ph  ->  ( ( A ^ N )  -  1 )  e.  CC )
15 0zd 8762 . . . . . . 7  |-  ( ph  ->  0  e.  ZZ )
163nn0zd 8866 . . . . . . . 8  |-  ( ph  ->  N  e.  ZZ )
17 peano2zm 8788 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
1816, 17syl 14 . . . . . . 7  |-  ( ph  ->  ( N  -  1 )  e.  ZZ )
1915, 18fzfigd 9838 . . . . . 6  |-  ( ph  ->  ( 0 ... ( N  -  1 ) )  e.  Fin )
201adantr 270 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  A  e.  CC )
21 elfznn0 9528 . . . . . . . 8  |-  ( k  e.  ( 0 ... ( N  -  1 ) )  ->  k  e.  NN0 )
2221adantl 271 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  k  e.  NN0 )
2320, 22expcld 10086 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( A ^ k )  e.  CC )
2419, 23fsumcl 10794 . . . . 5  |-  ( ph  -> 
sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( A ^ k
)  e.  CC )
25 peano2cnm 7748 . . . . . 6  |-  ( A  e.  CC  ->  ( A  -  1 )  e.  CC )
261, 25syl 14 . . . . 5  |-  ( ph  ->  ( A  -  1 )  e.  CC )
271, 6, 2subap0d 8119 . . . . 5  |-  ( ph  ->  ( A  -  1 ) #  0 )
2814, 24, 26, 27divmulap2d 8291 . . . 4  |-  ( ph  ->  ( ( ( ( A ^ N )  -  1 )  / 
( A  -  1 ) )  =  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( A ^ k )  <-> 
( ( A ^ N )  -  1 )  =  ( ( A  -  1 )  x.  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( A ^ k
) ) ) )
2912, 28bitrd 186 . . 3  |-  ( ph  ->  ( ( ( 1  -  ( A ^ N ) )  / 
( 1  -  A
) )  =  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( A ^ k )  <-> 
( ( A ^ N )  -  1 )  =  ( ( A  -  1 )  x.  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( A ^ k
) ) ) )
305, 29syl5bb 190 . 2  |-  ( ph  ->  ( sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( A ^ k
)  =  ( ( 1  -  ( A ^ N ) )  /  ( 1  -  A ) )  <->  ( ( A ^ N )  - 
1 )  =  ( ( A  -  1 )  x.  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( A ^
k ) ) ) )
314, 30mpbid 145 1  |-  ( ph  ->  ( ( A ^ N )  -  1 )  =  ( ( A  -  1 )  x.  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( A ^ k
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   class class class wbr 3845  (class class class)co 5652   CCcc 7348   0cc0 7350   1c1 7351    x. cmul 7355    - cmin 7653   # cap 8058    / cdiv 8139   NN0cn0 8673   ZZcz 8750   ...cfz 9424   ^cexp 9954   sum_csu 10742
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462  ax-pre-mulext 7463  ax-arch 7464  ax-caucvg 7465
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-isom 5024  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-frec 6156  df-1o 6181  df-oadd 6185  df-er 6292  df-en 6458  df-dom 6459  df-fin 6460  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059  df-div 8140  df-inn 8423  df-2 8481  df-3 8482  df-4 8483  df-n0 8674  df-z 8751  df-uz 9020  df-q 9105  df-rp 9135  df-fz 9425  df-fzo 9554  df-iseq 9853  df-seq3 9854  df-exp 9955  df-ihash 10184  df-cj 10276  df-re 10277  df-im 10278  df-rsqrt 10431  df-abs 10432  df-clim 10667  df-isum 10743
This theorem is referenced by: (None)
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