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Theorem sumrb 12186
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 11048 . . . 4  |-  seq  M
(  +  ,  F
)  e.  _V
4 climres 12049 . . . 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 12184 . . . . 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 4033 . . 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 12184 . . . . 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 4033 . . 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 11048 . . . 4  |-  seq  N
(  +  ,  F
)  e.  _V
26 climres 12049 . . . 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 10249 . . 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 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   ifcif 3565   class class class wbr 4023    e. cmpt 4077    |` cres 4691   ` cfv 5255   CCcc 8735   0cc0 8737    + caddc 8740   ZZcz 10024   ZZ>=cuz 10230    seq cseq 11046    ~~> cli 11958
This theorem is referenced by:  summo  12190  zsum  12191
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047  df-clim 11962
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