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Theorem sumrb 12499
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 11317 . . . 4  |-  seq  M
(  +  ,  F
)  e.  _V
4 climres 12361 . . . 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 12497 . . . . 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 4214 . . 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 12497 . . . . 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 4214 . . 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 11317 . . . 4  |-  seq  N
(  +  ,  F
)  e.  _V
26 climres 12361 . . . 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 10499 . . 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 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312   ifcif 3731   class class class wbr 4204    e. cmpt 4258    |` cres 4872   ` cfv 5446   CCcc 8980   0cc0 8982    + caddc 8985   ZZcz 10274   ZZ>=cuz 10480    seq cseq 11315    ~~> cli 12270
This theorem is referenced by:  summo  12503  zsum  12504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-seq 11316  df-clim 12274
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