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Theorem sumrb 12435
Description: Rebase the starting point of a sum. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 9-Apr-2014.)
Hypotheses
Ref Expression
summo.1  |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
summo.2  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
sumrb.4  |-  ( ph  ->  M  e.  ZZ )
sumrb.5  |-  ( ph  ->  N  e.  ZZ )
sumrb.6  |-  ( ph  ->  A  C_  ( ZZ>= `  M ) )
sumrb.7  |-  ( ph  ->  A  C_  ( ZZ>= `  N ) )
Assertion
Ref Expression
sumrb  |-  ( ph  ->  (  seq  M (  +  ,  F )  ~~>  C  <->  seq  N (  +  ,  F )  ~~>  C ) )
Distinct variable groups:    A, k    k, F    k, N    ph, k    k, M
Allowed substitution hints:    B( k)    C( k)

Proof of Theorem sumrb
StepHypRef Expression
1 sumrb.5 . . . . 5  |-  ( ph  ->  N  e.  ZZ )
21adantr 452 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  N  e.  ZZ )
3 seqex 11253 . . . 4  |-  seq  M
(  +  ,  F
)  e.  _V
4 climres 12297 . . . 4  |-  ( ( N  e.  ZZ  /\  seq  M (  +  ,  F )  e.  _V )  ->  ( (  seq 
M (  +  ,  F )  |`  ( ZZ>=
`  N ) )  ~~>  C  <->  seq  M (  +  ,  F )  ~~>  C ) )
52, 3, 4sylancl 644 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  ( (  seq  M (  +  ,  F )  |`  ( ZZ>=
`  N ) )  ~~>  C  <->  seq  M (  +  ,  F )  ~~>  C ) )
6 sumrb.7 . . . . . 6  |-  ( ph  ->  A  C_  ( ZZ>= `  N ) )
76adantr 452 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  A  C_  ( ZZ>=
`  N ) )
8 summo.1 . . . . . 6  |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
9 summo.2 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
109adantlr 696 . . . . . 6  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  M )
)  /\  k  e.  A )  ->  B  e.  CC )
11 simpr 448 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  N  e.  ( ZZ>= `  M )
)
128, 10, 11sumrblem 12433 . . . . 5  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  M )
)  /\  A  C_  ( ZZ>=
`  N ) )  ->  (  seq  M
(  +  ,  F
)  |`  ( ZZ>= `  N
) )  =  seq  N (  +  ,  F
) )
137, 12mpdan 650 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  (  seq  M (  +  ,  F
)  |`  ( ZZ>= `  N
) )  =  seq  N (  +  ,  F
) )
1413breq1d 4164 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  ( (  seq  M (  +  ,  F )  |`  ( ZZ>=
`  N ) )  ~~>  C  <->  seq  N (  +  ,  F )  ~~>  C ) )
155, 14bitr3d 247 . 2  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  (  seq  M (  +  ,  F
)  ~~>  C  <->  seq  N (  +  ,  F )  ~~>  C ) )
16 sumrb.6 . . . . . 6  |-  ( ph  ->  A  C_  ( ZZ>= `  M ) )
1716adantr 452 . . . . 5  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  A  C_  ( ZZ>=
`  M ) )
189adantlr 696 . . . . . 6  |-  ( ( ( ph  /\  M  e.  ( ZZ>= `  N )
)  /\  k  e.  A )  ->  B  e.  CC )
19 simpr 448 . . . . . 6  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  M  e.  ( ZZ>= `  N )
)
208, 18, 19sumrblem 12433 . . . . 5  |-  ( ( ( ph  /\  M  e.  ( ZZ>= `  N )
)  /\  A  C_  ( ZZ>=
`  M ) )  ->  (  seq  N
(  +  ,  F
)  |`  ( ZZ>= `  M
) )  =  seq  M (  +  ,  F
) )
2117, 20mpdan 650 . . . 4  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  (  seq  N (  +  ,  F
)  |`  ( ZZ>= `  M
) )  =  seq  M (  +  ,  F
) )
2221breq1d 4164 . . 3  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  ( (  seq  N (  +  ,  F )  |`  ( ZZ>=
`  M ) )  ~~>  C  <->  seq  M (  +  ,  F )  ~~>  C ) )
23 sumrb.4 . . . . 5  |-  ( ph  ->  M  e.  ZZ )
2423adantr 452 . . . 4  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  M  e.  ZZ )
25 seqex 11253 . . . 4  |-  seq  N
(  +  ,  F
)  e.  _V
26 climres 12297 . . . 4  |-  ( ( M  e.  ZZ  /\  seq  N (  +  ,  F )  e.  _V )  ->  ( (  seq 
N (  +  ,  F )  |`  ( ZZ>=
`  M ) )  ~~>  C  <->  seq  N (  +  ,  F )  ~~>  C ) )
2724, 25, 26sylancl 644 . . 3  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  ( (  seq  N (  +  ,  F )  |`  ( ZZ>=
`  M ) )  ~~>  C  <->  seq  N (  +  ,  F )  ~~>  C ) )
2822, 27bitr3d 247 . 2  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  (  seq  M (  +  ,  F
)  ~~>  C  <->  seq  N (  +  ,  F )  ~~>  C ) )
29 uztric 10440 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  e.  (
ZZ>= `  M )  \/  M  e.  ( ZZ>= `  N ) ) )
3023, 1, 29syl2anc 643 . 2  |-  ( ph  ->  ( N  e.  (
ZZ>= `  M )  \/  M  e.  ( ZZ>= `  N ) ) )
3115, 28, 30mpjaodan 762 1  |-  ( ph  ->  (  seq  M (  +  ,  F )  ~~>  C  <->  seq  N (  +  ,  F )  ~~>  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2900    C_ wss 3264   ifcif 3683   class class class wbr 4154    e. cmpt 4208    |` cres 4821   ` cfv 5395   CCcc 8922   0cc0 8924    + caddc 8927   ZZcz 10215   ZZ>=cuz 10421    seq cseq 11251    ~~> cli 12206
This theorem is referenced by:  summo  12439  zsum  12440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-n0 10155  df-z 10216  df-uz 10422  df-fz 10977  df-seq 11252  df-clim 12210
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