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Theorem cvgratnnlemseq 10916
Description: Lemma for cvgratnn 10921. (Contributed by Jim Kingdon, 21-Nov-2022.)
Hypotheses
Ref Expression
cvgratnn.3  |-  ( ph  ->  A  e.  RR )
cvgratnn.4  |-  ( ph  ->  A  <  1 )
cvgratnn.gt0  |-  ( ph  ->  0  <  A )
cvgratnn.6  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e.  CC )
cvgratnn.7  |-  ( (
ph  /\  k  e.  NN )  ->  ( abs `  ( F `  (
k  +  1 ) ) )  <_  ( A  x.  ( abs `  ( F `  k
) ) ) )
cvgratnn.m  |-  ( ph  ->  M  e.  NN )
cvgratnn.n  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
Assertion
Ref Expression
cvgratnnlemseq  |-  ( ph  ->  ( (  seq 1
(  +  ,  F
) `  N )  -  (  seq 1
(  +  ,  F
) `  M )
)  =  sum_ i  e.  ( ( M  + 
1 ) ... N
) ( F `  i ) )
Distinct variable groups:    A, k    k, F    k, N    ph, k    i, F, k    i, M    i, N    ph, i
Allowed substitution hints:    A( i)    M( k)

Proof of Theorem cvgratnnlemseq
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnuz 9052 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
2 1zzd 8775 . . . . . . 7  |-  ( ph  ->  1  e.  ZZ )
3 cvgratnn.6 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e.  CC )
41, 2, 3serf 9896 . . . . . 6  |-  ( ph  ->  seq 1 (  +  ,  F ) : NN --> CC )
54adantr 270 . . . . 5  |-  ( (
ph  /\  M  <  N )  ->  seq 1
(  +  ,  F
) : NN --> CC )
6 cvgratnn.m . . . . . 6  |-  ( ph  ->  M  e.  NN )
76adantr 270 . . . . 5  |-  ( (
ph  /\  M  <  N )  ->  M  e.  NN )
85, 7ffvelrnd 5435 . . . 4  |-  ( (
ph  /\  M  <  N )  ->  (  seq 1 (  +  ,  F ) `  M
)  e.  CC )
9 eqid 2088 . . . . . . 7  |-  ( ZZ>= `  ( M  +  1
) )  =  (
ZZ>= `  ( M  + 
1 ) )
106nnzd 8865 . . . . . . . 8  |-  ( ph  ->  M  e.  ZZ )
1110peano2zd 8869 . . . . . . 7  |-  ( ph  ->  ( M  +  1 )  e.  ZZ )
12 fveq2 5305 . . . . . . . . 9  |-  ( k  =  x  ->  ( F `  k )  =  ( F `  x ) )
1312eleq1d 2156 . . . . . . . 8  |-  ( k  =  x  ->  (
( F `  k
)  e.  CC  <->  ( F `  x )  e.  CC ) )
143ralrimiva 2446 . . . . . . . . 9  |-  ( ph  ->  A. k  e.  NN  ( F `  k )  e.  CC )
1514adantr 270 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  A. k  e.  NN  ( F `  k )  e.  CC )
166peano2nnd 8435 . . . . . . . . 9  |-  ( ph  ->  ( M  +  1 )  e.  NN )
17 eluznn 9085 . . . . . . . . 9  |-  ( ( ( M  +  1 )  e.  NN  /\  x  e.  ( ZZ>= `  ( M  +  1
) ) )  ->  x  e.  NN )
1816, 17sylan 277 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  x  e.  NN )
1913, 15, 18rspcdva 2727 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( F `  x )  e.  CC )
209, 11, 19serf 9896 . . . . . 6  |-  ( ph  ->  seq ( M  + 
1 ) (  +  ,  F ) : ( ZZ>= `  ( M  +  1 ) ) --> CC )
2120adantr 270 . . . . 5  |-  ( (
ph  /\  M  <  N )  ->  seq ( M  +  1 ) (  +  ,  F
) : ( ZZ>= `  ( M  +  1
) ) --> CC )
2211adantr 270 . . . . . 6  |-  ( (
ph  /\  M  <  N )  ->  ( M  +  1 )  e.  ZZ )
23 cvgratnn.n . . . . . . . 8  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
24 eluzelz 9026 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
2523, 24syl 14 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
2625adantr 270 . . . . . 6  |-  ( (
ph  /\  M  <  N )  ->  N  e.  ZZ )
27 zltp1le 8802 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  <->  ( M  +  1 )  <_  N ) )
2810, 25, 27syl2anc 403 . . . . . . 7  |-  ( ph  ->  ( M  <  N  <->  ( M  +  1 )  <_  N ) )
2928biimpa 290 . . . . . 6  |-  ( (
ph  /\  M  <  N )  ->  ( M  +  1 )  <_  N )
30 eluz2 9023 . . . . . 6  |-  ( N  e.  ( ZZ>= `  ( M  +  1 ) )  <->  ( ( M  +  1 )  e.  ZZ  /\  N  e.  ZZ  /\  ( M  +  1 )  <_  N ) )
3122, 26, 29, 30syl3anbrc 1127 . . . . 5  |-  ( (
ph  /\  M  <  N )  ->  N  e.  ( ZZ>= `  ( M  +  1 ) ) )
3221, 31ffvelrnd 5435 . . . 4  |-  ( (
ph  /\  M  <  N )  ->  (  seq ( M  +  1
) (  +  ,  F ) `  N
)  e.  CC )
338, 32pncan2d 7793 . . 3  |-  ( (
ph  /\  M  <  N )  ->  ( (
(  seq 1 (  +  ,  F ) `  M )  +  (  seq ( M  + 
1 ) (  +  ,  F ) `  N ) )  -  (  seq 1 (  +  ,  F ) `  M ) )  =  (  seq ( M  +  1 ) (  +  ,  F ) `
 N ) )
34 addcl 7465 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
3534adantl 271 . . . . 5  |-  ( ( ( ph  /\  M  <  N )  /\  (
x  e.  CC  /\  y  e.  CC )
)  ->  ( x  +  y )  e.  CC )
36 addass 7470 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  +  y )  +  z )  =  ( x  +  ( y  +  z ) ) )
3736adantl 271 . . . . 5  |-  ( ( ( ph  /\  M  <  N )  /\  (
x  e.  CC  /\  y  e.  CC  /\  z  e.  CC ) )  -> 
( ( x  +  y )  +  z )  =  ( x  +  ( y  +  z ) ) )
386, 1syl6eleq 2180 . . . . . 6  |-  ( ph  ->  M  e.  ( ZZ>= ` 
1 ) )
3938adantr 270 . . . . 5  |-  ( (
ph  /\  M  <  N )  ->  M  e.  ( ZZ>= `  1 )
)
4014ad2antrr 472 . . . . . 6  |-  ( ( ( ph  /\  M  <  N )  /\  x  e.  ( ZZ>= `  1 )
)  ->  A. k  e.  NN  ( F `  k )  e.  CC )
41 simpr 108 . . . . . . 7  |-  ( ( ( ph  /\  M  <  N )  /\  x  e.  ( ZZ>= `  1 )
)  ->  x  e.  ( ZZ>= `  1 )
)
4241, 1syl6eleqr 2181 . . . . . 6  |-  ( ( ( ph  /\  M  <  N )  /\  x  e.  ( ZZ>= `  1 )
)  ->  x  e.  NN )
4313, 40, 42rspcdva 2727 . . . . 5  |-  ( ( ( ph  /\  M  <  N )  /\  x  e.  ( ZZ>= `  1 )
)  ->  ( F `  x )  e.  CC )
4435, 37, 31, 39, 43seq3split 9903 . . . 4  |-  ( (
ph  /\  M  <  N )  ->  (  seq 1 (  +  ,  F ) `  N
)  =  ( (  seq 1 (  +  ,  F ) `  M )  +  (  seq ( M  + 
1 ) (  +  ,  F ) `  N ) ) )
4544oveq1d 5667 . . 3  |-  ( (
ph  /\  M  <  N )  ->  ( (  seq 1 (  +  ,  F ) `  N
)  -  (  seq 1 (  +  ,  F ) `  M
) )  =  ( ( (  seq 1
(  +  ,  F
) `  M )  +  (  seq ( M  +  1 ) (  +  ,  F
) `  N )
)  -  (  seq 1 (  +  ,  F ) `  M
) ) )
46 eqidd 2089 . . . 4  |-  ( ( ( ph  /\  M  <  N )  /\  i  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( F `  i )  =  ( F `  i ) )
47 fveq2 5305 . . . . . 6  |-  ( k  =  i  ->  ( F `  k )  =  ( F `  i ) )
4847eleq1d 2156 . . . . 5  |-  ( k  =  i  ->  (
( F `  k
)  e.  CC  <->  ( F `  i )  e.  CC ) )
4914ad2antrr 472 . . . . 5  |-  ( ( ( ph  /\  M  <  N )  /\  i  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  A. k  e.  NN  ( F `  k )  e.  CC )
5016ad2antrr 472 . . . . . 6  |-  ( ( ( ph  /\  M  <  N )  /\  i  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( M  +  1 )  e.  NN )
51 simpr 108 . . . . . 6  |-  ( ( ( ph  /\  M  <  N )  /\  i  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  i  e.  ( ZZ>= `  ( M  +  1 ) ) )
52 eluznn 9085 . . . . . 6  |-  ( ( ( M  +  1 )  e.  NN  /\  i  e.  ( ZZ>= `  ( M  +  1
) ) )  -> 
i  e.  NN )
5350, 51, 52syl2anc 403 . . . . 5  |-  ( ( ( ph  /\  M  <  N )  /\  i  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  i  e.  NN )
5448, 49, 53rspcdva 2727 . . . 4  |-  ( ( ( ph  /\  M  <  N )  /\  i  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( F `  i )  e.  CC )
5546, 31, 54fsum3ser 10787 . . 3  |-  ( (
ph  /\  M  <  N )  ->  sum_ i  e.  ( ( M  + 
1 ) ... N
) ( F `  i )  =  (  seq ( M  + 
1 ) (  +  ,  F ) `  N ) )
5633, 45, 553eqtr4d 2130 . 2  |-  ( (
ph  /\  M  <  N )  ->  ( (  seq 1 (  +  ,  F ) `  N
)  -  (  seq 1 (  +  ,  F ) `  M
) )  =  sum_ i  e.  ( ( M  +  1 ) ... N ) ( F `  i ) )
57 simpr 108 . . . . . . 7  |-  ( (
ph  /\  M  =  N )  ->  M  =  N )
586nnred 8433 . . . . . . . . 9  |-  ( ph  ->  M  e.  RR )
5958ltp1d 8389 . . . . . . . 8  |-  ( ph  ->  M  <  ( M  +  1 ) )
6059adantr 270 . . . . . . 7  |-  ( (
ph  /\  M  =  N )  ->  M  <  ( M  +  1 ) )
6157, 60eqbrtrrd 3867 . . . . . 6  |-  ( (
ph  /\  M  =  N )  ->  N  <  ( M  +  1 ) )
6211adantr 270 . . . . . . 7  |-  ( (
ph  /\  M  =  N )  ->  ( M  +  1 )  e.  ZZ )
6325adantr 270 . . . . . . 7  |-  ( (
ph  /\  M  =  N )  ->  N  e.  ZZ )
64 fzn 9454 . . . . . . 7  |-  ( ( ( M  +  1 )  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <  ( M  +  1 )  <-> 
( ( M  + 
1 ) ... N
)  =  (/) ) )
6562, 63, 64syl2anc 403 . . . . . 6  |-  ( (
ph  /\  M  =  N )  ->  ( N  <  ( M  + 
1 )  <->  ( ( M  +  1 ) ... N )  =  (/) ) )
6661, 65mpbid 145 . . . . 5  |-  ( (
ph  /\  M  =  N )  ->  (
( M  +  1 ) ... N )  =  (/) )
6766sumeq1d 10751 . . . 4  |-  ( (
ph  /\  M  =  N )  ->  sum_ i  e.  ( ( M  + 
1 ) ... N
) ( F `  i )  =  sum_ i  e.  (/)  ( F `
 i ) )
68 sum0 10776 . . . 4  |-  sum_ i  e.  (/)  ( F `  i )  =  0
6967, 68syl6eq 2136 . . 3  |-  ( (
ph  /\  M  =  N )  ->  sum_ i  e.  ( ( M  + 
1 ) ... N
) ( F `  i )  =  0 )
704, 6ffvelrnd 5435 . . . . 5  |-  ( ph  ->  (  seq 1 (  +  ,  F ) `
 M )  e.  CC )
7170adantr 270 . . . 4  |-  ( (
ph  /\  M  =  N )  ->  (  seq 1 (  +  ,  F ) `  M
)  e.  CC )
7271subidd 7779 . . 3  |-  ( (
ph  /\  M  =  N )  ->  (
(  seq 1 (  +  ,  F ) `  M )  -  (  seq 1 (  +  ,  F ) `  M
) )  =  0 )
7357fveq2d 5309 . . . 4  |-  ( (
ph  /\  M  =  N )  ->  (  seq 1 (  +  ,  F ) `  M
)  =  (  seq 1 (  +  ,  F ) `  N
) )
7473oveq1d 5667 . . 3  |-  ( (
ph  /\  M  =  N )  ->  (
(  seq 1 (  +  ,  F ) `  M )  -  (  seq 1 (  +  ,  F ) `  M
) )  =  ( (  seq 1 (  +  ,  F ) `
 N )  -  (  seq 1 (  +  ,  F ) `  M ) ) )
7569, 72, 743eqtr2rd 2127 . 2  |-  ( (
ph  /\  M  =  N )  ->  (
(  seq 1 (  +  ,  F ) `  N )  -  (  seq 1 (  +  ,  F ) `  M
) )  =  sum_ i  e.  ( ( M  +  1 ) ... N ) ( F `  i ) )
76 eluzle 9029 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  <_  N )
7723, 76syl 14 . . 3  |-  ( ph  ->  M  <_  N )
78 zleloe 8795 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  <->  ( M  <  N  \/  M  =  N )
) )
7910, 25, 78syl2anc 403 . . 3  |-  ( ph  ->  ( M  <_  N  <->  ( M  <  N  \/  M  =  N )
) )
8077, 79mpbid 145 . 2  |-  ( ph  ->  ( M  <  N  \/  M  =  N
) )
8156, 75, 80mpjaodan 747 1  |-  ( ph  ->  ( (  seq 1
(  +  ,  F
) `  N )  -  (  seq 1
(  +  ,  F
) `  M )
)  =  sum_ i  e.  ( ( M  + 
1 ) ... N
) ( F `  i ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664    /\ w3a 924    = wceq 1289    e. wcel 1438   A.wral 2359   (/)c0 3286   class class class wbr 3845   -->wf 5011   ` cfv 5015  (class class class)co 5652   CCcc 7346   RRcr 7347   0cc0 7348   1c1 7349    + caddc 7351    x. cmul 7353    < clt 7520    <_ cle 7521    - cmin 7651   NNcn 8420   ZZcz 8748   ZZ>=cuz 9017   ...cfz 9422    seqcseq 9848   abscabs 10426   sum_csu 10738
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 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-mulrcl 7442  ax-addcom 7443  ax-mulcom 7444  ax-addass 7445  ax-mulass 7446  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-1rid 7450  ax-0id 7451  ax-rnegex 7452  ax-precex 7453  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459  ax-pre-mulgt0 7460  ax-pre-mulext 7461  ax-arch 7462  ax-caucvg 7463
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 6290  df-en 6456  df-dom 6457  df-fin 6458  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-reap 8050  df-ap 8057  df-div 8138  df-inn 8421  df-2 8479  df-3 8480  df-4 8481  df-n0 8672  df-z 8749  df-uz 9018  df-q 9103  df-rp 9133  df-fz 9423  df-fzo 9550  df-iseq 9849  df-seq3 9850  df-exp 9951  df-ihash 10180  df-cj 10272  df-re 10273  df-im 10274  df-rsqrt 10427  df-abs 10428  df-clim 10663  df-isum 10739
This theorem is referenced by:  cvgratnnlemrate  10920
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