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Theorem geosergap 11528
Description: The value of the finite geometric series  A ^ M  +  A ^ ( M  +  1 )  +...  +  A ^
( N  -  1 ). (Contributed by Mario Carneiro, 2-May-2016.) (Revised by Jim Kingdon, 24-Oct-2022.)
Hypotheses
Ref Expression
geoserg.1  |-  ( ph  ->  A  e.  CC )
geosergap.2  |-  ( ph  ->  A #  1 )
geoserg.3  |-  ( ph  ->  M  e.  NN0 )
geoserg.4  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
Assertion
Ref Expression
geosergap  |-  ( ph  -> 
sum_ k  e.  ( M..^ N ) ( A ^ k )  =  ( ( ( A ^ M )  -  ( A ^ N ) )  / 
( 1  -  A
) ) )
Distinct variable groups:    A, k    k, M    k, N    ph, k

Proof of Theorem geosergap
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 geoserg.3 . . . . . . 7  |-  ( ph  ->  M  e.  NN0 )
21nn0zd 9387 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
3 geoserg.4 . . . . . . 7  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
4 eluzelz 9551 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
53, 4syl 14 . . . . . 6  |-  ( ph  ->  N  e.  ZZ )
6 fzofig 10446 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M..^ N )  e.  Fin )
72, 5, 6syl2anc 411 . . . . 5  |-  ( ph  ->  ( M..^ N )  e.  Fin )
8 ax-1cn 7918 . . . . . 6  |-  1  e.  CC
9 geoserg.1 . . . . . 6  |-  ( ph  ->  A  e.  CC )
10 subcl 8170 . . . . . 6  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  -  A
)  e.  CC )
118, 9, 10sylancr 414 . . . . 5  |-  ( ph  ->  ( 1  -  A
)  e.  CC )
129adantr 276 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M..^ N ) )  ->  A  e.  CC )
13 elfzouz 10165 . . . . . . 7  |-  ( k  e.  ( M..^ N
)  ->  k  e.  ( ZZ>= `  M )
)
14 eluznn0 9613 . . . . . . 7  |-  ( ( M  e.  NN0  /\  k  e.  ( ZZ>= `  M ) )  -> 
k  e.  NN0 )
151, 13, 14syl2an 289 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M..^ N ) )  ->  k  e.  NN0 )
1612, 15expcld 10668 . . . . 5  |-  ( (
ph  /\  k  e.  ( M..^ N ) )  ->  ( A ^
k )  e.  CC )
177, 11, 16fsummulc1 11471 . . . 4  |-  ( ph  ->  ( sum_ k  e.  ( M..^ N ) ( A ^ k )  x.  ( 1  -  A ) )  = 
sum_ k  e.  ( M..^ N ) ( ( A ^ k
)  x.  ( 1  -  A ) ) )
18 1cnd 7987 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( M..^ N ) )  ->  1  e.  CC )
1916, 18, 12subdid 8385 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M..^ N ) )  ->  ( ( A ^ k )  x.  ( 1  -  A
) )  =  ( ( ( A ^
k )  x.  1 )  -  ( ( A ^ k )  x.  A ) ) )
2016mulridd 7988 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( M..^ N ) )  ->  ( ( A ^ k )  x.  1 )  =  ( A ^ k ) )
2112, 15expp1d 10669 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( M..^ N ) )  ->  ( A ^
( k  +  1 ) )  =  ( ( A ^ k
)  x.  A ) )
2221eqcomd 2193 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( M..^ N ) )  ->  ( ( A ^ k )  x.  A )  =  ( A ^ ( k  +  1 ) ) )
2320, 22oveq12d 5906 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M..^ N ) )  ->  ( ( ( A ^ k )  x.  1 )  -  ( ( A ^
k )  x.  A
) )  =  ( ( A ^ k
)  -  ( A ^ ( k  +  1 ) ) ) )
2419, 23eqtrd 2220 . . . . 5  |-  ( (
ph  /\  k  e.  ( M..^ N ) )  ->  ( ( A ^ k )  x.  ( 1  -  A
) )  =  ( ( A ^ k
)  -  ( A ^ ( k  +  1 ) ) ) )
2524sumeq2dv 11390 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( M..^ N ) ( ( A ^ k
)  x.  ( 1  -  A ) )  =  sum_ k  e.  ( M..^ N ) ( ( A ^ k
)  -  ( A ^ ( k  +  1 ) ) ) )
26 oveq2 5896 . . . . 5  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
27 oveq2 5896 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
28 oveq2 5896 . . . . 5  |-  ( j  =  M  ->  ( A ^ j )  =  ( A ^ M
) )
29 oveq2 5896 . . . . 5  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
309adantr 276 . . . . . 6  |-  ( (
ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )
31 elfzuz 10035 . . . . . . 7  |-  ( j  e.  ( M ... N )  ->  j  e.  ( ZZ>= `  M )
)
32 eluznn0 9613 . . . . . . 7  |-  ( ( M  e.  NN0  /\  j  e.  ( ZZ>= `  M ) )  -> 
j  e.  NN0 )
331, 31, 32syl2an 289 . . . . . 6  |-  ( (
ph  /\  j  e.  ( M ... N ) )  ->  j  e.  NN0 )
3430, 33expcld 10668 . . . . 5  |-  ( (
ph  /\  j  e.  ( M ... N ) )  ->  ( A ^ j )  e.  CC )
3526, 27, 28, 29, 3, 34telfsumo 11488 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( M..^ N ) ( ( A ^ k
)  -  ( A ^ ( k  +  1 ) ) )  =  ( ( A ^ M )  -  ( A ^ N ) ) )
3617, 25, 353eqtrrd 2225 . . 3  |-  ( ph  ->  ( ( A ^ M )  -  ( A ^ N ) )  =  ( sum_ k  e.  ( M..^ N ) ( A ^ k
)  x.  ( 1  -  A ) ) )
379, 1expcld 10668 . . . . 5  |-  ( ph  ->  ( A ^ M
)  e.  CC )
38 eluznn0 9613 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  ( ZZ>= `  M ) )  ->  N  e.  NN0 )
391, 3, 38syl2anc 411 . . . . . 6  |-  ( ph  ->  N  e.  NN0 )
409, 39expcld 10668 . . . . 5  |-  ( ph  ->  ( A ^ N
)  e.  CC )
4137, 40subcld 8282 . . . 4  |-  ( ph  ->  ( ( A ^ M )  -  ( A ^ N ) )  e.  CC )
427, 16fsumcl 11422 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( M..^ N ) ( A ^ k )  e.  CC )
43 geosergap.2 . . . . . . 7  |-  ( ph  ->  A #  1 )
44 1cnd 7987 . . . . . . . 8  |-  ( ph  ->  1  e.  CC )
45 apneg 8582 . . . . . . . 8  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A #  1  <->  -u A #  -u 1 ) )
469, 44, 45syl2anc 411 . . . . . . 7  |-  ( ph  ->  ( A #  1  <->  -u A #  -u 1 ) )
4743, 46mpbid 147 . . . . . 6  |-  ( ph  -> 
-u A #  -u 1
)
489negcld 8269 . . . . . . 7  |-  ( ph  -> 
-u A  e.  CC )
4944negcld 8269 . . . . . . 7  |-  ( ph  -> 
-u 1  e.  CC )
50 apadd2 8580 . . . . . . 7  |-  ( (
-u A  e.  CC  /\  -u 1  e.  CC  /\  1  e.  CC )  ->  ( -u A #  -u 1  <->  ( 1  + 
-u A ) #  ( 1  +  -u 1
) ) )
5148, 49, 44, 50syl3anc 1248 . . . . . 6  |-  ( ph  ->  ( -u A #  -u 1  <->  ( 1  +  -u A
) #  ( 1  + 
-u 1 ) ) )
5247, 51mpbid 147 . . . . 5  |-  ( ph  ->  ( 1  +  -u A ) #  ( 1  +  -u 1 ) )
5344, 9negsubd 8288 . . . . 5  |-  ( ph  ->  ( 1  +  -u A )  =  ( 1  -  A ) )
54 1pneg1e0 9044 . . . . . 6  |-  ( 1  +  -u 1 )  =  0
5554a1i 9 . . . . 5  |-  ( ph  ->  ( 1  +  -u
1 )  =  0 )
5652, 53, 553brtr3d 4046 . . . 4  |-  ( ph  ->  ( 1  -  A
) #  0 )
5741, 42, 11, 56divmulap3d 8796 . . 3  |-  ( ph  ->  ( ( ( ( A ^ M )  -  ( A ^ N ) )  / 
( 1  -  A
) )  =  sum_ k  e.  ( M..^ N ) ( A ^ k )  <->  ( ( A ^ M )  -  ( A ^ N ) )  =  ( sum_ k  e.  ( M..^ N ) ( A ^ k )  x.  ( 1  -  A
) ) ) )
5836, 57mpbird 167 . 2  |-  ( ph  ->  ( ( ( A ^ M )  -  ( A ^ N ) )  /  ( 1  -  A ) )  =  sum_ k  e.  ( M..^ N ) ( A ^ k ) )
5958eqcomd 2193 1  |-  ( ph  -> 
sum_ k  e.  ( M..^ N ) ( A ^ k )  =  ( ( ( A ^ M )  -  ( A ^ N ) )  / 
( 1  -  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1363    e. wcel 2158   class class class wbr 4015   ` cfv 5228  (class class class)co 5888   Fincfn 6754   CCcc 7823   0cc0 7825   1c1 7826    + caddc 7828    x. cmul 7830    - cmin 8142   -ucneg 8143   # cap 8552    / cdiv 8643   NN0cn0 9190   ZZcz 9267   ZZ>=cuz 9542   ...cfz 10022  ..^cfzo 10156   ^cexp 10533   sum_csu 11375
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943  ax-arch 7944  ax-caucvg 7945
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-frec 6406  df-1o 6431  df-oadd 6435  df-er 6549  df-en 6755  df-dom 6756  df-fin 6757  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-n0 9191  df-z 9268  df-uz 9543  df-q 9634  df-rp 9668  df-fz 10023  df-fzo 10157  df-seqfrec 10460  df-exp 10534  df-ihash 10770  df-cj 10865  df-re 10866  df-im 10867  df-rsqrt 11021  df-abs 11022  df-clim 11301  df-sumdc 11376
This theorem is referenced by:  geoserap  11529  cvgratnnlemsumlt  11550  cvgcmp2nlemabs  15077
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