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Theorem seq3id 10308
Description: Discarding the first few terms of a sequence that starts with all zeroes (or any element which is a left-identity for  .+) has no effect on its sum. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Jim Kingdon, 8-Apr-2023.)
Hypotheses
Ref Expression
iseqid.1  |-  ( (
ph  /\  x  e.  S )  ->  ( Z  .+  x )  =  x )
iseqid.2  |-  ( ph  ->  Z  e.  S )
iseqid.3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
iseqid.4  |-  ( ph  ->  ( F `  N
)  e.  S )
iseqid.5  |-  ( (
ph  /\  x  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  x )  =  Z )
iseqid.f  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
iseqid.cl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
Assertion
Ref Expression
seq3id  |-  ( ph  ->  (  seq M ( 
.+  ,  F )  |`  ( ZZ>= `  N )
)  =  seq N
(  .+  ,  F
) )
Distinct variable groups:    x,  .+ , y    x, F, y    x, M, y    x, N, y   
x, S, y    x, Z, y    ph, x, y

Proof of Theorem seq3id
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 iseqid.3 . 2  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
2 eluzelz 9355 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
31, 2syl 14 . . . . 5  |-  ( ph  ->  N  e.  ZZ )
4 simpr 109 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  N )
)  ->  x  e.  ( ZZ>= `  N )
)
51adantr 274 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  N )
)  ->  N  e.  ( ZZ>= `  M )
)
6 uztrn 9362 . . . . . . 7  |-  ( ( x  e.  ( ZZ>= `  N )  /\  N  e.  ( ZZ>= `  M )
)  ->  x  e.  ( ZZ>= `  M )
)
74, 5, 6syl2anc 409 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ZZ>= `  N )
)  ->  x  e.  ( ZZ>= `  M )
)
8 iseqid.f . . . . . 6  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
97, 8syldan 280 . . . . 5  |-  ( (
ph  /\  x  e.  ( ZZ>= `  N )
)  ->  ( F `  x )  e.  S
)
10 iseqid.cl . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
113, 9, 10seq3-1 10260 . . . 4  |-  ( ph  ->  (  seq N ( 
.+  ,  F ) `
 N )  =  ( F `  N
) )
12 seqeq1 10248 . . . . . 6  |-  ( N  =  M  ->  seq N (  .+  ,  F )  =  seq M (  .+  ,  F ) )
1312fveq1d 5427 . . . . 5  |-  ( N  =  M  ->  (  seq N (  .+  ,  F ) `  N
)  =  (  seq M (  .+  ,  F ) `  N
) )
1413eqeq1d 2149 . . . 4  |-  ( N  =  M  ->  (
(  seq N (  .+  ,  F ) `  N
)  =  ( F `
 N )  <->  (  seq M (  .+  ,  F ) `  N
)  =  ( F `
 N ) ) )
1511, 14syl5ibcom 154 . . 3  |-  ( ph  ->  ( N  =  M  ->  (  seq M
(  .+  ,  F
) `  N )  =  ( F `  N ) ) )
16 eluzel2 9351 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
171, 16syl 14 . . . . . . 7  |-  ( ph  ->  M  e.  ZZ )
1817adantr 274 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  M  e.  ZZ )
19 simpr 109 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  N  e.  ( ZZ>= `  ( M  +  1 ) ) )
208adantlr 469 . . . . . 6  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
2110adantlr 469 . . . . . 6  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
2218, 19, 20, 21seq3m1 10268 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq M (  .+  ,  F ) `  N
)  =  ( (  seq M (  .+  ,  F ) `  ( N  -  1 ) )  .+  ( F `
 N ) ) )
23 oveq2 5786 . . . . . . . . . 10  |-  ( x  =  Z  ->  ( Z  .+  x )  =  ( Z  .+  Z
) )
24 id 19 . . . . . . . . . 10  |-  ( x  =  Z  ->  x  =  Z )
2523, 24eqeq12d 2155 . . . . . . . . 9  |-  ( x  =  Z  ->  (
( Z  .+  x
)  =  x  <->  ( Z  .+  Z )  =  Z ) )
26 iseqid.1 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  S )  ->  ( Z  .+  x )  =  x )
2726ralrimiva 2506 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  S  ( Z  .+  x )  =  x )
28 iseqid.2 . . . . . . . . 9  |-  ( ph  ->  Z  e.  S )
2925, 27, 28rspcdva 2795 . . . . . . . 8  |-  ( ph  ->  ( Z  .+  Z
)  =  Z )
3029adantr 274 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( Z  .+  Z )  =  Z )
31 eluzp1m1 9369 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>= `  ( M  +  1
) ) )  -> 
( N  -  1 )  e.  ( ZZ>= `  M ) )
3217, 31sylan 281 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( N  -  1 )  e.  ( ZZ>= `  M )
)
33 iseqid.5 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  x )  =  Z )
3433adantlr 469 . . . . . . 7  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  /\  x  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  x )  =  Z )
3528adantr 274 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  Z  e.  S )
3630, 32, 34, 35, 20, 21seq3id3 10307 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq M (  .+  ,  F ) `  ( N  -  1 ) )  =  Z )
3736oveq1d 5793 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( (  seq M (  .+  ,  F ) `  ( N  -  1 ) )  .+  ( F `
 N ) )  =  ( Z  .+  ( F `  N ) ) )
38 oveq2 5786 . . . . . . 7  |-  ( x  =  ( F `  N )  ->  ( Z  .+  x )  =  ( Z  .+  ( F `  N )
) )
39 id 19 . . . . . . 7  |-  ( x  =  ( F `  N )  ->  x  =  ( F `  N ) )
4038, 39eqeq12d 2155 . . . . . 6  |-  ( x  =  ( F `  N )  ->  (
( Z  .+  x
)  =  x  <->  ( Z  .+  ( F `  N
) )  =  ( F `  N ) ) )
4127adantr 274 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  A. x  e.  S  ( Z  .+  x )  =  x )
42 iseqid.4 . . . . . . 7  |-  ( ph  ->  ( F `  N
)  e.  S )
4342adantr 274 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( F `  N )  e.  S
)
4440, 41, 43rspcdva 2795 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( Z  .+  ( F `  N
) )  =  ( F `  N ) )
4522, 37, 443eqtrd 2177 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq M (  .+  ,  F ) `  N
)  =  ( F `
 N ) )
4645ex 114 . . 3  |-  ( ph  ->  ( N  e.  (
ZZ>= `  ( M  + 
1 ) )  -> 
(  seq M (  .+  ,  F ) `  N
)  =  ( F `
 N ) ) )
47 uzp1 9379 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  =  M  \/  N  e.  ( ZZ>= `  ( M  +  1 ) ) ) )
481, 47syl 14 . . 3  |-  ( ph  ->  ( N  =  M  \/  N  e.  (
ZZ>= `  ( M  + 
1 ) ) ) )
4915, 46, 48mpjaod 708 . 2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 N )  =  ( F `  N
) )
50 eqidd 2141 . 2  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  k )  =  ( F `  k ) )
511, 49, 8, 9, 10, 50seq3feq2 10270 1  |-  ( ph  ->  (  seq M ( 
.+  ,  F )  |`  ( ZZ>= `  N )
)  =  seq N
(  .+  ,  F
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 698    = wceq 1332    e. wcel 1481   A.wral 2417    |` cres 4545   ` cfv 5127  (class class class)co 5778   1c1 7641    + caddc 7643    - cmin 7953   ZZcz 9074   ZZ>=cuz 9346   ...cfz 9817    seqcseq 10245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4047  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456  ax-iinf 4506  ax-cnex 7731  ax-resscn 7732  ax-1cn 7733  ax-1re 7734  ax-icn 7735  ax-addcl 7736  ax-addrcl 7737  ax-mulcl 7738  ax-addcom 7740  ax-addass 7742  ax-distr 7744  ax-i2m1 7745  ax-0lt1 7746  ax-0id 7748  ax-rnegex 7749  ax-cnre 7751  ax-pre-ltirr 7752  ax-pre-ltwlin 7753  ax-pre-lttrn 7754  ax-pre-ltadd 7756
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-tr 4031  df-id 4219  df-iord 4292  df-on 4294  df-ilim 4295  df-suc 4297  df-iom 4509  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-riota 5734  df-ov 5781  df-oprab 5782  df-mpo 5783  df-1st 6042  df-2nd 6043  df-recs 6206  df-frec 6292  df-pnf 7822  df-mnf 7823  df-xr 7824  df-ltxr 7825  df-le 7826  df-sub 7955  df-neg 7956  df-inn 8741  df-n0 8998  df-z 9075  df-uz 9347  df-fz 9818  df-fzo 9947  df-seqfrec 10246
This theorem is referenced by:  seq3coll  10613  sumrbdclem  11174  prodrbdclem  11368
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