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Theorem sumrb 12202
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 451 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  N  e.  ZZ )
3 seqex 11064 . . . 4  |-  seq  M
(  +  ,  F
)  e.  _V
4 climres 12065 . . . 4  |-  ( ( N  e.  ZZ  /\  seq  M (  +  ,  F )  e.  _V )  ->  ( (  seq 
M (  +  ,  F )  |`  ( ZZ>=
`  N ) )  ~~>  C  <->  seq  M (  +  ,  F )  ~~>  C ) )
52, 3, 4sylancl 643 . . 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 451 . . . . 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 695 . . . . . 6  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  M )
)  /\  k  e.  A )  ->  B  e.  CC )
11 simpr 447 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  N  e.  ( ZZ>= `  M )
)
128, 10, 11sumrblem 12200 . . . . 5  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  M )
)  /\  A  C_  ( ZZ>=
`  N ) )  ->  (  seq  M
(  +  ,  F
)  |`  ( ZZ>= `  N
) )  =  seq  N (  +  ,  F
) )
137, 12mpdan 649 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  (  seq  M (  +  ,  F
)  |`  ( ZZ>= `  N
) )  =  seq  N (  +  ,  F
) )
1413breq1d 4049 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  ( (  seq  M (  +  ,  F )  |`  ( ZZ>=
`  N ) )  ~~>  C  <->  seq  N (  +  ,  F )  ~~>  C ) )
155, 14bitr3d 246 . 2  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  (  seq  M (  +  ,  F
)  ~~>  C  <->  seq  N (  +  ,  F )  ~~>  C ) )
16 sumrb.6 . . . . . 6  |-  ( ph  ->  A  C_  ( ZZ>= `  M ) )
1716adantr 451 . . . . 5  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  A  C_  ( ZZ>=
`  M ) )
189adantlr 695 . . . . . 6  |-  ( ( ( ph  /\  M  e.  ( ZZ>= `  N )
)  /\  k  e.  A )  ->  B  e.  CC )
19 simpr 447 . . . . . 6  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  M  e.  ( ZZ>= `  N )
)
208, 18, 19sumrblem 12200 . . . . 5  |-  ( ( ( ph  /\  M  e.  ( ZZ>= `  N )
)  /\  A  C_  ( ZZ>=
`  M ) )  ->  (  seq  N
(  +  ,  F
)  |`  ( ZZ>= `  M
) )  =  seq  M (  +  ,  F
) )
2117, 20mpdan 649 . . . 4  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  (  seq  N (  +  ,  F
)  |`  ( ZZ>= `  M
) )  =  seq  M (  +  ,  F
) )
2221breq1d 4049 . . 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 451 . . . 4  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  M  e.  ZZ )
25 seqex 11064 . . . 4  |-  seq  N
(  +  ,  F
)  e.  _V
26 climres 12065 . . . 4  |-  ( ( M  e.  ZZ  /\  seq  N (  +  ,  F )  e.  _V )  ->  ( (  seq 
N (  +  ,  F )  |`  ( ZZ>=
`  M ) )  ~~>  C  <->  seq  N (  +  ,  F )  ~~>  C ) )
2724, 25, 26sylancl 643 . . 3  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  ( (  seq  N (  +  ,  F )  |`  ( ZZ>=
`  M ) )  ~~>  C  <->  seq  N (  +  ,  F )  ~~>  C ) )
2822, 27bitr3d 246 . 2  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  (  seq  M (  +  ,  F
)  ~~>  C  <->  seq  N (  +  ,  F )  ~~>  C ) )
29 uztric 10265 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  e.  (
ZZ>= `  M )  \/  M  e.  ( ZZ>= `  N ) ) )
3023, 1, 29syl2anc 642 . 2  |-  ( ph  ->  ( N  e.  (
ZZ>= `  M )  \/  M  e.  ( ZZ>= `  N ) ) )
3115, 28, 30mpjaodan 761 1  |-  ( ph  ->  (  seq  M (  +  ,  F )  ~~>  C  <->  seq  N (  +  ,  F )  ~~>  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   ifcif 3578   class class class wbr 4039    e. cmpt 4093    |` cres 4707   ` cfv 5271   CCcc 8751   0cc0 8753    + caddc 8756   ZZcz 10040   ZZ>=cuz 10246    seq cseq 11062    ~~> cli 11974
This theorem is referenced by:  summo  12206  zsum  12207
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063  df-clim 11978
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