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Theorem cvgratnnlemseq 11529
Description: Lemma for cvgratnn 11534. (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 9561 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
2 1zzd 9278 . . . . . . 7  |-  ( ph  ->  1  e.  ZZ )
3 cvgratnn.6 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e.  CC )
41, 2, 3serf 10471 . . . . . 6  |-  ( ph  ->  seq 1 (  +  ,  F ) : NN --> CC )
54adantr 276 . . . . 5  |-  ( (
ph  /\  M  <  N )  ->  seq 1
(  +  ,  F
) : NN --> CC )
6 cvgratnn.m . . . . . 6  |-  ( ph  ->  M  e.  NN )
76adantr 276 . . . . 5  |-  ( (
ph  /\  M  <  N )  ->  M  e.  NN )
85, 7ffvelcdmd 5652 . . . 4  |-  ( (
ph  /\  M  <  N )  ->  (  seq 1 (  +  ,  F ) `  M
)  e.  CC )
9 eqid 2177 . . . . . . 7  |-  ( ZZ>= `  ( M  +  1
) )  =  (
ZZ>= `  ( M  + 
1 ) )
106nnzd 9372 . . . . . . . 8  |-  ( ph  ->  M  e.  ZZ )
1110peano2zd 9376 . . . . . . 7  |-  ( ph  ->  ( M  +  1 )  e.  ZZ )
12 fveq2 5515 . . . . . . . . 9  |-  ( k  =  x  ->  ( F `  k )  =  ( F `  x ) )
1312eleq1d 2246 . . . . . . . 8  |-  ( k  =  x  ->  (
( F `  k
)  e.  CC  <->  ( F `  x )  e.  CC ) )
143ralrimiva 2550 . . . . . . . . 9  |-  ( ph  ->  A. k  e.  NN  ( F `  k )  e.  CC )
1514adantr 276 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  A. k  e.  NN  ( F `  k )  e.  CC )
166peano2nnd 8932 . . . . . . . . 9  |-  ( ph  ->  ( M  +  1 )  e.  NN )
17 eluznn 9598 . . . . . . . . 9  |-  ( ( ( M  +  1 )  e.  NN  /\  x  e.  ( ZZ>= `  ( M  +  1
) ) )  ->  x  e.  NN )
1816, 17sylan 283 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  x  e.  NN )
1913, 15, 18rspcdva 2846 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( F `  x )  e.  CC )
209, 11, 19serf 10471 . . . . . 6  |-  ( ph  ->  seq ( M  + 
1 ) (  +  ,  F ) : ( ZZ>= `  ( M  +  1 ) ) --> CC )
2120adantr 276 . . . . 5  |-  ( (
ph  /\  M  <  N )  ->  seq ( M  +  1 ) (  +  ,  F
) : ( ZZ>= `  ( M  +  1
) ) --> CC )
2211adantr 276 . . . . . 6  |-  ( (
ph  /\  M  <  N )  ->  ( M  +  1 )  e.  ZZ )
23 cvgratnn.n . . . . . . . 8  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
24 eluzelz 9535 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
2523, 24syl 14 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
2625adantr 276 . . . . . 6  |-  ( (
ph  /\  M  <  N )  ->  N  e.  ZZ )
27 zltp1le 9305 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  <->  ( M  +  1 )  <_  N ) )
2810, 25, 27syl2anc 411 . . . . . . 7  |-  ( ph  ->  ( M  <  N  <->  ( M  +  1 )  <_  N ) )
2928biimpa 296 . . . . . 6  |-  ( (
ph  /\  M  <  N )  ->  ( M  +  1 )  <_  N )
30 eluz2 9532 . . . . . 6  |-  ( N  e.  ( ZZ>= `  ( M  +  1 ) )  <->  ( ( M  +  1 )  e.  ZZ  /\  N  e.  ZZ  /\  ( M  +  1 )  <_  N ) )
3122, 26, 29, 30syl3anbrc 1181 . . . . 5  |-  ( (
ph  /\  M  <  N )  ->  N  e.  ( ZZ>= `  ( M  +  1 ) ) )
3221, 31ffvelcdmd 5652 . . . 4  |-  ( (
ph  /\  M  <  N )  ->  (  seq ( M  +  1
) (  +  ,  F ) `  N
)  e.  CC )
338, 32pncan2d 8268 . . 3  |-  ( (
ph  /\  M  <  N )  ->  ( (
(  seq 1 (  +  ,  F ) `  M )  +  (  seq ( M  + 
1 ) (  +  ,  F ) `  N ) )  -  (  seq 1 (  +  ,  F ) `  M ) )  =  (  seq ( M  +  1 ) (  +  ,  F ) `
 N ) )
34 addcl 7935 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
3534adantl 277 . . . . 5  |-  ( ( ( ph  /\  M  <  N )  /\  (
x  e.  CC  /\  y  e.  CC )
)  ->  ( x  +  y )  e.  CC )
36 addass 7940 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  +  y )  +  z )  =  ( x  +  ( y  +  z ) ) )
3736adantl 277 . . . . 5  |-  ( ( ( ph  /\  M  <  N )  /\  (
x  e.  CC  /\  y  e.  CC  /\  z  e.  CC ) )  -> 
( ( x  +  y )  +  z )  =  ( x  +  ( y  +  z ) ) )
386, 1eleqtrdi 2270 . . . . . 6  |-  ( ph  ->  M  e.  ( ZZ>= ` 
1 ) )
3938adantr 276 . . . . 5  |-  ( (
ph  /\  M  <  N )  ->  M  e.  ( ZZ>= `  1 )
)
4014ad2antrr 488 . . . . . 6  |-  ( ( ( ph  /\  M  <  N )  /\  x  e.  ( ZZ>= `  1 )
)  ->  A. k  e.  NN  ( F `  k )  e.  CC )
41 simpr 110 . . . . . . 7  |-  ( ( ( ph  /\  M  <  N )  /\  x  e.  ( ZZ>= `  1 )
)  ->  x  e.  ( ZZ>= `  1 )
)
4241, 1eleqtrrdi 2271 . . . . . 6  |-  ( ( ( ph  /\  M  <  N )  /\  x  e.  ( ZZ>= `  1 )
)  ->  x  e.  NN )
4313, 40, 42rspcdva 2846 . . . . 5  |-  ( ( ( ph  /\  M  <  N )  /\  x  e.  ( ZZ>= `  1 )
)  ->  ( F `  x )  e.  CC )
4435, 37, 31, 39, 43seq3split 10476 . . . 4  |-  ( (
ph  /\  M  <  N )  ->  (  seq 1 (  +  ,  F ) `  N
)  =  ( (  seq 1 (  +  ,  F ) `  M )  +  (  seq ( M  + 
1 ) (  +  ,  F ) `  N ) ) )
4544oveq1d 5889 . . 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 2178 . . . 4  |-  ( ( ( ph  /\  M  <  N )  /\  i  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( F `  i )  =  ( F `  i ) )
47 fveq2 5515 . . . . . 6  |-  ( k  =  i  ->  ( F `  k )  =  ( F `  i ) )
4847eleq1d 2246 . . . . 5  |-  ( k  =  i  ->  (
( F `  k
)  e.  CC  <->  ( F `  i )  e.  CC ) )
4914ad2antrr 488 . . . . 5  |-  ( ( ( ph  /\  M  <  N )  /\  i  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  A. k  e.  NN  ( F `  k )  e.  CC )
5016ad2antrr 488 . . . . . 6  |-  ( ( ( ph  /\  M  <  N )  /\  i  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( M  +  1 )  e.  NN )
51 simpr 110 . . . . . 6  |-  ( ( ( ph  /\  M  <  N )  /\  i  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  i  e.  ( ZZ>= `  ( M  +  1 ) ) )
52 eluznn 9598 . . . . . 6  |-  ( ( ( M  +  1 )  e.  NN  /\  i  e.  ( ZZ>= `  ( M  +  1
) ) )  -> 
i  e.  NN )
5350, 51, 52syl2anc 411 . . . . 5  |-  ( ( ( ph  /\  M  <  N )  /\  i  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  i  e.  NN )
5448, 49, 53rspcdva 2846 . . . 4  |-  ( ( ( ph  /\  M  <  N )  /\  i  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( F `  i )  e.  CC )
5546, 31, 54fsum3ser 11400 . . 3  |-  ( (
ph  /\  M  <  N )  ->  sum_ i  e.  ( ( M  + 
1 ) ... N
) ( F `  i )  =  (  seq ( M  + 
1 ) (  +  ,  F ) `  N ) )
5633, 45, 553eqtr4d 2220 . 2  |-  ( (
ph  /\  M  <  N )  ->  ( (  seq 1 (  +  ,  F ) `  N
)  -  (  seq 1 (  +  ,  F ) `  M
) )  =  sum_ i  e.  ( ( M  +  1 ) ... N ) ( F `  i ) )
57 simpr 110 . . . . . . 7  |-  ( (
ph  /\  M  =  N )  ->  M  =  N )
586nnred 8930 . . . . . . . . 9  |-  ( ph  ->  M  e.  RR )
5958ltp1d 8885 . . . . . . . 8  |-  ( ph  ->  M  <  ( M  +  1 ) )
6059adantr 276 . . . . . . 7  |-  ( (
ph  /\  M  =  N )  ->  M  <  ( M  +  1 ) )
6157, 60eqbrtrrd 4027 . . . . . 6  |-  ( (
ph  /\  M  =  N )  ->  N  <  ( M  +  1 ) )
6211adantr 276 . . . . . . 7  |-  ( (
ph  /\  M  =  N )  ->  ( M  +  1 )  e.  ZZ )
6325adantr 276 . . . . . . 7  |-  ( (
ph  /\  M  =  N )  ->  N  e.  ZZ )
64 fzn 10039 . . . . . . 7  |-  ( ( ( M  +  1 )  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <  ( M  +  1 )  <-> 
( ( M  + 
1 ) ... N
)  =  (/) ) )
6562, 63, 64syl2anc 411 . . . . . 6  |-  ( (
ph  /\  M  =  N )  ->  ( N  <  ( M  + 
1 )  <->  ( ( M  +  1 ) ... N )  =  (/) ) )
6661, 65mpbid 147 . . . . 5  |-  ( (
ph  /\  M  =  N )  ->  (
( M  +  1 ) ... N )  =  (/) )
6766sumeq1d 11369 . . . 4  |-  ( (
ph  /\  M  =  N )  ->  sum_ i  e.  ( ( M  + 
1 ) ... N
) ( F `  i )  =  sum_ i  e.  (/)  ( F `
 i ) )
68 sum0 11391 . . . 4  |-  sum_ i  e.  (/)  ( F `  i )  =  0
6967, 68eqtrdi 2226 . . 3  |-  ( (
ph  /\  M  =  N )  ->  sum_ i  e.  ( ( M  + 
1 ) ... N
) ( F `  i )  =  0 )
704, 6ffvelcdmd 5652 . . . . 5  |-  ( ph  ->  (  seq 1 (  +  ,  F ) `
 M )  e.  CC )
7170adantr 276 . . . 4  |-  ( (
ph  /\  M  =  N )  ->  (  seq 1 (  +  ,  F ) `  M
)  e.  CC )
7271subidd 8254 . . 3  |-  ( (
ph  /\  M  =  N )  ->  (
(  seq 1 (  +  ,  F ) `  M )  -  (  seq 1 (  +  ,  F ) `  M
) )  =  0 )
7357fveq2d 5519 . . . 4  |-  ( (
ph  /\  M  =  N )  ->  (  seq 1 (  +  ,  F ) `  M
)  =  (  seq 1 (  +  ,  F ) `  N
) )
7473oveq1d 5889 . . 3  |-  ( (
ph  /\  M  =  N )  ->  (
(  seq 1 (  +  ,  F ) `  M )  -  (  seq 1 (  +  ,  F ) `  M
) )  =  ( (  seq 1 (  +  ,  F ) `
 N )  -  (  seq 1 (  +  ,  F ) `  M ) ) )
7569, 72, 743eqtr2rd 2217 . 2  |-  ( (
ph  /\  M  =  N )  ->  (
(  seq 1 (  +  ,  F ) `  N )  -  (  seq 1 (  +  ,  F ) `  M
) )  =  sum_ i  e.  ( ( M  +  1 ) ... N ) ( F `  i ) )
76 eluzle 9538 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  <_  N )
7723, 76syl 14 . . 3  |-  ( ph  ->  M  <_  N )
78 zleloe 9298 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  <->  ( M  <  N  \/  M  =  N )
) )
7910, 25, 78syl2anc 411 . . 3  |-  ( ph  ->  ( M  <_  N  <->  ( M  <  N  \/  M  =  N )
) )
8077, 79mpbid 147 . 2  |-  ( ph  ->  ( M  <  N  \/  M  =  N
) )
8156, 75, 80mpjaodan 798 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 104    <-> wb 105    \/ wo 708    /\ w3a 978    = wceq 1353    e. wcel 2148   A.wral 2455   (/)c0 3422   class class class wbr 4003   -->wf 5212   ` cfv 5216  (class class class)co 5874   CCcc 7808   RRcr 7809   0cc0 7810   1c1 7811    + caddc 7813    x. cmul 7815    < clt 7990    <_ cle 7991    - cmin 8126   NNcn 8917   ZZcz 9251   ZZ>=cuz 9526   ...cfz 10006    seqcseq 10442   abscabs 11001   sum_csu 11356
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-frec 6391  df-1o 6416  df-oadd 6420  df-er 6534  df-en 6740  df-dom 6741  df-fin 6742  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-2 8976  df-3 8977  df-4 8978  df-n0 9175  df-z 9252  df-uz 9527  df-q 9618  df-rp 9652  df-fz 10007  df-fzo 10140  df-seqfrec 10443  df-exp 10517  df-ihash 10751  df-cj 10846  df-re 10847  df-im 10848  df-rsqrt 11002  df-abs 11003  df-clim 11282  df-sumdc 11357
This theorem is referenced by:  cvgratnnlemrate  11533
  Copyright terms: Public domain W3C validator