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Theorem isumrb 10420
Description: Rebase the starting point of a sum. (Contributed by Jim Kingdon, 5-Mar-2022.)
Hypotheses
Ref Expression
isummo.1  |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
isummo.2  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
isumrb.4  |-  ( ph  ->  M  e.  ZZ )
isumrb.5  |-  ( ph  ->  N  e.  ZZ )
isumrb.6  |-  ( ph  ->  A  C_  ( ZZ>= `  M ) )
isumrb.7  |-  ( ph  ->  A  C_  ( ZZ>= `  N ) )
isumrb.mdc  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  -> DECID  k  e.  A
)
isumrb.ndc  |-  ( (
ph  /\  k  e.  ( ZZ>= `  N )
)  -> DECID  k  e.  A
)
Assertion
Ref Expression
isumrb  |-  ( ph  ->  (  seq M (  +  ,  F ,  CC )  ~~>  C  <->  seq N (  +  ,  F ,  CC )  ~~>  C )
)
Distinct variable groups:    A, k    B, k    k, N    ph, k    k, M
Allowed substitution hints:    C( k)    F( k)

Proof of Theorem isumrb
StepHypRef Expression
1 isumrb.5 . . . . 5  |-  ( ph  ->  N  e.  ZZ )
21adantr 270 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  N  e.  ZZ )
3 iseqex 9593 . . . 4  |-  seq M
(  +  ,  F ,  CC )  e.  _V
4 climres 10361 . . . 4  |-  ( ( N  e.  ZZ  /\  seq M (  +  ,  F ,  CC )  e.  _V )  ->  (
(  seq M (  +  ,  F ,  CC )  |`  ( ZZ>= `  N
) )  ~~>  C  <->  seq M (  +  ,  F ,  CC )  ~~>  C )
)
52, 3, 4sylancl 404 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  ( (  seq M (  +  ,  F ,  CC )  |`  ( ZZ>= `  N )
)  ~~>  C  <->  seq M (  +  ,  F ,  CC )  ~~>  C )
)
6 isumrb.7 . . . . 5  |-  ( ph  ->  A  C_  ( ZZ>= `  N ) )
7 isummo.1 . . . . . 6  |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
8 isummo.2 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
98adantlr 461 . . . . . 6  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  M )
)  /\  k  e.  A )  ->  B  e.  CC )
10 isumrb.mdc . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  -> DECID  k  e.  A
)
1110adantlr 461 . . . . . 6  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  M )
)  /\  k  e.  ( ZZ>= `  M )
)  -> DECID  k  e.  A
)
12 simpr 108 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  N  e.  ( ZZ>= `  M )
)
137, 9, 11, 12isumrblem 10418 . . . . 5  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  M )
)  /\  A  C_  ( ZZ>=
`  N ) )  ->  (  seq M
(  +  ,  F ,  CC )  |`  ( ZZ>=
`  N ) )  =  seq N (  +  ,  F ,  CC ) )
146, 13mpidan 414 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  (  seq M (  +  ,  F ,  CC )  |`  ( ZZ>= `  N )
)  =  seq N
(  +  ,  F ,  CC ) )
1514breq1d 3815 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  ( (  seq M (  +  ,  F ,  CC )  |`  ( ZZ>= `  N )
)  ~~>  C  <->  seq N (  +  ,  F ,  CC )  ~~>  C )
)
165, 15bitr3d 188 . 2  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  (  seq M (  +  ,  F ,  CC )  ~~>  C 
<->  seq N (  +  ,  F ,  CC ) 
~~>  C ) )
17 isumrb.6 . . . . 5  |-  ( ph  ->  A  C_  ( ZZ>= `  M ) )
188adantlr 461 . . . . . 6  |-  ( ( ( ph  /\  M  e.  ( ZZ>= `  N )
)  /\  k  e.  A )  ->  B  e.  CC )
19 isumrb.ndc . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  N )
)  -> DECID  k  e.  A
)
2019adantlr 461 . . . . . 6  |-  ( ( ( ph  /\  M  e.  ( ZZ>= `  N )
)  /\  k  e.  ( ZZ>= `  N )
)  -> DECID  k  e.  A
)
21 simpr 108 . . . . . 6  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  M  e.  ( ZZ>= `  N )
)
227, 18, 20, 21isumrblem 10418 . . . . 5  |-  ( ( ( ph  /\  M  e.  ( ZZ>= `  N )
)  /\  A  C_  ( ZZ>=
`  M ) )  ->  (  seq N
(  +  ,  F ,  CC )  |`  ( ZZ>=
`  M ) )  =  seq M (  +  ,  F ,  CC ) )
2317, 22mpidan 414 . . . 4  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  (  seq N (  +  ,  F ,  CC )  |`  ( ZZ>= `  M )
)  =  seq M
(  +  ,  F ,  CC ) )
2423breq1d 3815 . . 3  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  ( (  seq N (  +  ,  F ,  CC )  |`  ( ZZ>= `  M )
)  ~~>  C  <->  seq M (  +  ,  F ,  CC )  ~~>  C )
)
25 isumrb.4 . . . . 5  |-  ( ph  ->  M  e.  ZZ )
2625adantr 270 . . . 4  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  M  e.  ZZ )
27 iseqex 9593 . . . 4  |-  seq N
(  +  ,  F ,  CC )  e.  _V
28 climres 10361 . . . 4  |-  ( ( M  e.  ZZ  /\  seq N (  +  ,  F ,  CC )  e.  _V )  ->  (
(  seq N (  +  ,  F ,  CC )  |`  ( ZZ>= `  M
) )  ~~>  C  <->  seq N (  +  ,  F ,  CC )  ~~>  C )
)
2926, 27, 28sylancl 404 . . 3  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  ( (  seq N (  +  ,  F ,  CC )  |`  ( ZZ>= `  M )
)  ~~>  C  <->  seq N (  +  ,  F ,  CC )  ~~>  C )
)
3024, 29bitr3d 188 . 2  |-  ( (
ph  /\  M  e.  ( ZZ>= `  N )
)  ->  (  seq M (  +  ,  F ,  CC )  ~~>  C 
<->  seq N (  +  ,  F ,  CC ) 
~~>  C ) )
31 uztric 8791 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  e.  (
ZZ>= `  M )  \/  M  e.  ( ZZ>= `  N ) ) )
3225, 1, 31syl2anc 403 . 2  |-  ( ph  ->  ( N  e.  (
ZZ>= `  M )  \/  M  e.  ( ZZ>= `  N ) ) )
3316, 30, 32mpjaodan 745 1  |-  ( ph  ->  (  seq M (  +  ,  F ,  CC )  ~~>  C  <->  seq N (  +  ,  F ,  CC )  ~~>  C )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662  DECID wdc 776    = wceq 1285    e. wcel 1434   _Vcvv 2610    C_ wss 2982   ifcif 3368   class class class wbr 3805    |-> cmpt 3859    |` cres 4393   ` cfv 4952   CCcc 7111   0cc0 7113    + caddc 7116   ZZcz 8502   ZZ>=cuz 8770    seqcseq 9591    ~~> cli 10336
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7199  ax-resscn 7200  ax-1cn 7201  ax-1re 7202  ax-icn 7203  ax-addcl 7204  ax-addrcl 7205  ax-mulcl 7206  ax-addcom 7208  ax-addass 7210  ax-distr 7212  ax-i2m1 7213  ax-0lt1 7214  ax-0id 7216  ax-rnegex 7217  ax-cnre 7219  ax-pre-ltirr 7220  ax-pre-ltwlin 7221  ax-pre-lttrn 7222  ax-pre-apti 7223  ax-pre-ltadd 7224
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-frec 6061  df-pnf 7287  df-mnf 7288  df-xr 7289  df-ltxr 7290  df-le 7291  df-sub 7418  df-neg 7419  df-inn 8177  df-n0 8426  df-z 8503  df-uz 8771  df-fz 9176  df-fzo 9300  df-iseq 9592  df-clim 10337
This theorem is referenced by: (None)
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