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Theorem sumrb 13195
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 465 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  N  e.  ZZ )
3 seqex 11813 . . . 4  |-  seq M
(  +  ,  F
)  e.  _V
4 climres 13058 . . . 4  |-  ( ( N  e.  ZZ  /\  seq M (  +  ,  F )  e.  _V )  ->  ( (  seq M (  +  ,  F )  |`  ( ZZ>=
`  N ) )  ~~>  C  <->  seq M (  +  ,  F )  ~~>  C ) )
52, 3, 4sylancl 662 . . 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 465 . . . . 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 714 . . . . . 6  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  M )
)  /\  k  e.  A )  ->  B  e.  CC )
11 simpr 461 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  N  e.  ( ZZ>= `  M )
)
128, 10, 11sumrblem 13193 . . . . 5  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  M )
)  /\  A  C_  ( ZZ>=
`  N ) )  ->  (  seq M
(  +  ,  F
)  |`  ( ZZ>= `  N
) )  =  seq N (  +  ,  F ) )
137, 12mpdan 668 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  (  seq M (  +  ,  F )  |`  ( ZZ>=
`  N ) )  =  seq N (  +  ,  F ) )
1413breq1d 4307 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  ( (  seq M (  +  ,  F )  |`  ( ZZ>=
`  N ) )  ~~>  C  <->  seq N (  +  ,  F )  ~~>  C ) )
155, 14bitr3d 255 . 2  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  (  seq M (  +  ,  F )  ~~>  C  <->  seq N (  +  ,  F )  ~~>  C ) )
16 sumrb.6 . . . . . 6  |-  ( ph  ->  A  C_  ( ZZ>= `  M ) )
1716adantr 465 . . . . 5  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  A  C_  ( ZZ>=
`  M ) )
189adantlr 714 . . . . . 6  |-  ( ( ( ph  /\  M  e.  ( ZZ>= `  N )
)  /\  k  e.  A )  ->  B  e.  CC )
19 simpr 461 . . . . . 6  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  M  e.  ( ZZ>= `  N )
)
208, 18, 19sumrblem 13193 . . . . 5  |-  ( ( ( ph  /\  M  e.  ( ZZ>= `  N )
)  /\  A  C_  ( ZZ>=
`  M ) )  ->  (  seq N
(  +  ,  F
)  |`  ( ZZ>= `  M
) )  =  seq M (  +  ,  F ) )
2117, 20mpdan 668 . . . 4  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  (  seq N (  +  ,  F )  |`  ( ZZ>=
`  M ) )  =  seq M (  +  ,  F ) )
2221breq1d 4307 . . 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 465 . . . 4  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  M  e.  ZZ )
25 seqex 11813 . . . 4  |-  seq N
(  +  ,  F
)  e.  _V
26 climres 13058 . . . 4  |-  ( ( M  e.  ZZ  /\  seq N (  +  ,  F )  e.  _V )  ->  ( (  seq N (  +  ,  F )  |`  ( ZZ>=
`  M ) )  ~~>  C  <->  seq N (  +  ,  F )  ~~>  C ) )
2724, 25, 26sylancl 662 . . 3  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  ( (  seq N (  +  ,  F )  |`  ( ZZ>=
`  M ) )  ~~>  C  <->  seq N (  +  ,  F )  ~~>  C ) )
2822, 27bitr3d 255 . 2  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  (  seq M (  +  ,  F )  ~~>  C  <->  seq N (  +  ,  F )  ~~>  C ) )
29 uztric 10887 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  e.  (
ZZ>= `  M )  \/  M  e.  ( ZZ>= `  N ) ) )
3023, 1, 29syl2anc 661 . 2  |-  ( ph  ->  ( N  e.  (
ZZ>= `  M )  \/  M  e.  ( ZZ>= `  N ) ) )
3115, 28, 30mpjaodan 784 1  |-  ( ph  ->  (  seq M (  +  ,  F )  ~~>  C  <->  seq N (  +  ,  F )  ~~>  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2977    C_ wss 3333   ifcif 3796   class class class wbr 4297    e. cmpt 4355    |` cres 4847   ` cfv 5423   CCcc 9285   0cc0 9287    + caddc 9290   ZZcz 10651   ZZ>=cuz 10866    seqcseq 11811    ~~> cli 12967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-seq 11812  df-clim 12971
This theorem is referenced by:  summo  13199  zsum  13200
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