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Theorem pwm1geoserap1 11284
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 11283 . 2  |-  ( ph  -> 
sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( A ^ k
)  =  ( ( 1  -  ( A ^ N ) )  /  ( 1  -  A ) ) )
5 eqcom 2141 . . 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 7789 . . . . . 6  |-  ( ph  ->  1  e.  CC )
71, 3expcld 10431 . . . . . 6  |-  ( ph  ->  ( A ^ N
)  e.  CC )
8 apsym 8375 . . . . . . . 8  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A #  1  <->  1 #  A ) )
91, 6, 8syl2anc 408 . . . . . . 7  |-  ( ph  ->  ( A #  1  <->  1 #  A ) )
102, 9mpbid 146 . . . . . 6  |-  ( ph  ->  1 #  A )
116, 7, 6, 1, 10div2subapd 8604 . . . . 5  |-  ( ph  ->  ( ( 1  -  ( A ^ N
) )  /  (
1  -  A ) )  =  ( ( ( A ^ N
)  -  1 )  /  ( A  - 
1 ) ) )
1211eqeq1d 2148 . . . 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 8035 . . . . . 6  |-  ( ( A ^ N )  e.  CC  ->  (
( A ^ N
)  -  1 )  e.  CC )
147, 13syl 14 . . . . 5  |-  ( ph  ->  ( ( A ^ N )  -  1 )  e.  CC )
15 0zd 9073 . . . . . . 7  |-  ( ph  ->  0  e.  ZZ )
163nn0zd 9178 . . . . . . . 8  |-  ( ph  ->  N  e.  ZZ )
17 peano2zm 9099 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
1816, 17syl 14 . . . . . . 7  |-  ( ph  ->  ( N  -  1 )  e.  ZZ )
1915, 18fzfigd 10211 . . . . . 6  |-  ( ph  ->  ( 0 ... ( N  -  1 ) )  e.  Fin )
201adantr 274 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  A  e.  CC )
21 elfznn0 9901 . . . . . . . 8  |-  ( k  e.  ( 0 ... ( N  -  1 ) )  ->  k  e.  NN0 )
2221adantl 275 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  k  e.  NN0 )
2320, 22expcld 10431 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( A ^ k )  e.  CC )
2419, 23fsumcl 11176 . . . . 5  |-  ( ph  -> 
sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( A ^ k
)  e.  CC )
25 peano2cnm 8035 . . . . . 6  |-  ( A  e.  CC  ->  ( A  -  1 )  e.  CC )
261, 25syl 14 . . . . 5  |-  ( ph  ->  ( A  -  1 )  e.  CC )
271, 6, 2subap0d 8413 . . . . 5  |-  ( ph  ->  ( A  -  1 ) #  0 )
2814, 24, 26, 27divmulap2d 8591 . . . 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 187 . . 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 191 . 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 146 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 103    <-> wb 104    = wceq 1331    e. wcel 1480   class class class wbr 3929  (class class class)co 5774   CCcc 7625   0cc0 7627   1c1 7628    x. cmul 7632    - cmin 7940   # cap 8350    / cdiv 8439   NN0cn0 8984   ZZcz 9061   ...cfz 9797   ^cexp 10299   sum_csu 11129
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745  ax-arch 7746  ax-caucvg 7747
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-inn 8728  df-2 8786  df-3 8787  df-4 8788  df-n0 8985  df-z 9062  df-uz 9334  df-q 9419  df-rp 9449  df-fz 9798  df-fzo 9927  df-seqfrec 10226  df-exp 10300  df-ihash 10529  df-cj 10621  df-re 10622  df-im 10623  df-rsqrt 10777  df-abs 10778  df-clim 11055  df-sumdc 11130
This theorem is referenced by: (None)
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