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Theorem seqid 11091
Description: Discard the first few terms of a sequence that starts with all zeroes (or whatever the identity  Z is for operation  .+). (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
Hypotheses
Ref Expression
seqid.1  |-  ( (
ph  /\  x  e.  S )  ->  ( Z  .+  x )  =  x )
seqid.2  |-  ( ph  ->  Z  e.  S )
seqid.3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
seqid.4  |-  ( ph  ->  ( F `  N
)  e.  S )
seqid.5  |-  ( (
ph  /\  x  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  x )  =  Z )
Assertion
Ref Expression
seqid  |-  ( ph  ->  (  seq  M ( 
.+  ,  F )  |`  ( ZZ>= `  N )
)  =  seq  N
(  .+  ,  F
) )
Distinct variable groups:    x,  .+    x, F    x, M    x, N    x, S    x, Z    ph, x

Proof of Theorem seqid
StepHypRef Expression
1 seqid.3 . 2  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
2 eluzelz 10238 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
3 seq1 11059 . . . . 5  |-  ( N  e.  ZZ  ->  (  seq  N (  .+  ,  F ) `  N
)  =  ( F `
 N ) )
41, 2, 33syl 18 . . . 4  |-  ( ph  ->  (  seq  N ( 
.+  ,  F ) `
 N )  =  ( F `  N
) )
5 seqeq1 11049 . . . . . 6  |-  ( N  =  M  ->  seq  N (  .+  ,  F
)  =  seq  M
(  .+  ,  F
) )
65fveq1d 5527 . . . . 5  |-  ( N  =  M  ->  (  seq  N (  .+  ,  F ) `  N
)  =  (  seq 
M (  .+  ,  F ) `  N
) )
76eqeq1d 2291 . . . 4  |-  ( N  =  M  ->  (
(  seq  N (  .+  ,  F ) `  N )  =  ( F `  N )  <-> 
(  seq  M (  .+  ,  F ) `  N )  =  ( F `  N ) ) )
84, 7syl5ibcom 211 . . 3  |-  ( ph  ->  ( N  =  M  ->  (  seq  M
(  .+  ,  F
) `  N )  =  ( F `  N ) ) )
9 eluzel2 10235 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
101, 9syl 15 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
11 seqm1 11063 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>= `  ( M  +  1
) ) )  -> 
(  seq  M (  .+  ,  F ) `  N )  =  ( (  seq  M ( 
.+  ,  F ) `
 ( N  - 
1 ) )  .+  ( F `  N ) ) )
1210, 11sylan 457 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq  M (  .+  ,  F
) `  N )  =  ( (  seq 
M (  .+  ,  F ) `  ( N  -  1 ) )  .+  ( F `
 N ) ) )
13 seqid.2 . . . . . . . . 9  |-  ( ph  ->  Z  e.  S )
14 seqid.1 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  S )  ->  ( Z  .+  x )  =  x )
1514ralrimiva 2626 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  S  ( Z  .+  x )  =  x )
16 oveq2 5866 . . . . . . . . . . 11  |-  ( x  =  Z  ->  ( Z  .+  x )  =  ( Z  .+  Z
) )
17 id 19 . . . . . . . . . . 11  |-  ( x  =  Z  ->  x  =  Z )
1816, 17eqeq12d 2297 . . . . . . . . . 10  |-  ( x  =  Z  ->  (
( Z  .+  x
)  =  x  <->  ( Z  .+  Z )  =  Z ) )
1918rspcv 2880 . . . . . . . . 9  |-  ( Z  e.  S  ->  ( A. x  e.  S  ( Z  .+  x )  =  x  ->  ( Z  .+  Z )  =  Z ) )
2013, 15, 19sylc 56 . . . . . . . 8  |-  ( ph  ->  ( Z  .+  Z
)  =  Z )
2120adantr 451 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( Z  .+  Z )  =  Z )
22 eluzp1m1 10251 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>= `  ( M  +  1
) ) )  -> 
( N  -  1 )  e.  ( ZZ>= `  M ) )
2310, 22sylan 457 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( N  -  1 )  e.  ( ZZ>= `  M )
)
24 seqid.5 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  x )  =  Z )
2524adantlr 695 . . . . . . 7  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  /\  x  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  x )  =  Z )
2621, 23, 25seqid3 11090 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq  M (  .+  ,  F
) `  ( N  -  1 ) )  =  Z )
2726oveq1d 5873 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( (  seq  M (  .+  ,  F ) `  ( N  -  1 ) )  .+  ( F `
 N ) )  =  ( Z  .+  ( F `  N ) ) )
28 seqid.4 . . . . . . 7  |-  ( ph  ->  ( F `  N
)  e.  S )
2928adantr 451 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( F `  N )  e.  S
)
3015adantr 451 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  A. x  e.  S  ( Z  .+  x )  =  x )
31 oveq2 5866 . . . . . . . 8  |-  ( x  =  ( F `  N )  ->  ( Z  .+  x )  =  ( Z  .+  ( F `  N )
) )
32 id 19 . . . . . . . 8  |-  ( x  =  ( F `  N )  ->  x  =  ( F `  N ) )
3331, 32eqeq12d 2297 . . . . . . 7  |-  ( x  =  ( F `  N )  ->  (
( Z  .+  x
)  =  x  <->  ( Z  .+  ( F `  N
) )  =  ( F `  N ) ) )
3433rspcv 2880 . . . . . 6  |-  ( ( F `  N )  e.  S  ->  ( A. x  e.  S  ( Z  .+  x )  =  x  ->  ( Z  .+  ( F `  N ) )  =  ( F `  N
) ) )
3529, 30, 34sylc 56 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( Z  .+  ( F `  N
) )  =  ( F `  N ) )
3612, 27, 353eqtrd 2319 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq  M (  .+  ,  F
) `  N )  =  ( F `  N ) )
3736ex 423 . . 3  |-  ( ph  ->  ( N  e.  (
ZZ>= `  ( M  + 
1 ) )  -> 
(  seq  M (  .+  ,  F ) `  N )  =  ( F `  N ) ) )
38 uzp1 10261 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  =  M  \/  N  e.  ( ZZ>= `  ( M  +  1 ) ) ) )
391, 38syl 15 . . 3  |-  ( ph  ->  ( N  =  M  \/  N  e.  (
ZZ>= `  ( M  + 
1 ) ) ) )
408, 37, 39mpjaod 370 . 2  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 N )  =  ( F `  N
) )
41 eqidd 2284 . 2  |-  ( (
ph  /\  x  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  x )  =  ( F `  x ) )
421, 40, 41seqfeq2 11069 1  |-  ( ph  ->  (  seq  M ( 
.+  ,  F )  |`  ( ZZ>= `  N )
)  =  seq  N
(  .+  ,  F
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    |` cres 4691   ` cfv 5255  (class class class)co 5858   1c1 8738    + caddc 8740    - cmin 9037   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046
This theorem is referenced by:  seqcoll  11401  sumrblem  12184  logtayl  20007  leibpilem2  20237
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  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
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