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Theorem seqid 12134
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 11091 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
3 seq1 12102 . . . . 5  |-  ( N  e.  ZZ  ->  (  seq N (  .+  ,  F ) `  N
)  =  ( F `
 N ) )
41, 2, 33syl 20 . . . 4  |-  ( ph  ->  (  seq N ( 
.+  ,  F ) `
 N )  =  ( F `  N
) )
5 seqeq1 12092 . . . . . 6  |-  ( N  =  M  ->  seq N (  .+  ,  F )  =  seq M (  .+  ,  F ) )
65fveq1d 5850 . . . . 5  |-  ( N  =  M  ->  (  seq N (  .+  ,  F ) `  N
)  =  (  seq M (  .+  ,  F ) `  N
) )
76eqeq1d 2456 . . . 4  |-  ( N  =  M  ->  (
(  seq N (  .+  ,  F ) `  N
)  =  ( F `
 N )  <->  (  seq M (  .+  ,  F ) `  N
)  =  ( F `
 N ) ) )
84, 7syl5ibcom 220 . . 3  |-  ( ph  ->  ( N  =  M  ->  (  seq M
(  .+  ,  F
) `  N )  =  ( F `  N ) ) )
9 eluzel2 11087 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
101, 9syl 16 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
11 seqm1 12106 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>= `  ( M  +  1
) ) )  -> 
(  seq M (  .+  ,  F ) `  N
)  =  ( (  seq M (  .+  ,  F ) `  ( N  -  1 ) )  .+  ( F `
 N ) ) )
1210, 11sylan 469 . . . . 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 2868 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  S  ( Z  .+  x )  =  x )
16 oveq2 6278 . . . . . . . . . . 11  |-  ( x  =  Z  ->  ( Z  .+  x )  =  ( Z  .+  Z
) )
17 id 22 . . . . . . . . . . 11  |-  ( x  =  Z  ->  x  =  Z )
1816, 17eqeq12d 2476 . . . . . . . . . 10  |-  ( x  =  Z  ->  (
( Z  .+  x
)  =  x  <->  ( Z  .+  Z )  =  Z ) )
1918rspcv 3203 . . . . . . . . 9  |-  ( Z  e.  S  ->  ( A. x  e.  S  ( Z  .+  x )  =  x  ->  ( Z  .+  Z )  =  Z ) )
2013, 15, 19sylc 60 . . . . . . . 8  |-  ( ph  ->  ( Z  .+  Z
)  =  Z )
2120adantr 463 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( Z  .+  Z )  =  Z )
22 eluzp1m1 11105 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>= `  ( M  +  1
) ) )  -> 
( N  -  1 )  e.  ( ZZ>= `  M ) )
2310, 22sylan 469 . . . . . . 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 712 . . . . . . 7  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  /\  x  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  x )  =  Z )
2621, 23, 25seqid3 12133 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq M (  .+  ,  F ) `  ( N  -  1 ) )  =  Z )
2726oveq1d 6285 . . . . 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 463 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( F `  N )  e.  S
)
3015adantr 463 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  A. x  e.  S  ( Z  .+  x )  =  x )
31 oveq2 6278 . . . . . . . 8  |-  ( x  =  ( F `  N )  ->  ( Z  .+  x )  =  ( Z  .+  ( F `  N )
) )
32 id 22 . . . . . . . 8  |-  ( x  =  ( F `  N )  ->  x  =  ( F `  N ) )
3331, 32eqeq12d 2476 . . . . . . 7  |-  ( x  =  ( F `  N )  ->  (
( Z  .+  x
)  =  x  <->  ( Z  .+  ( F `  N
) )  =  ( F `  N ) ) )
3433rspcv 3203 . . . . . 6  |-  ( ( F `  N )  e.  S  ->  ( A. x  e.  S  ( Z  .+  x )  =  x  ->  ( Z  .+  ( F `  N ) )  =  ( F `  N
) ) )
3529, 30, 34sylc 60 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( Z  .+  ( F `  N
) )  =  ( F `  N ) )
3612, 27, 353eqtrd 2499 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq M (  .+  ,  F ) `  N
)  =  ( F `
 N ) )
3736ex 432 . . 3  |-  ( ph  ->  ( N  e.  (
ZZ>= `  ( M  + 
1 ) )  -> 
(  seq M (  .+  ,  F ) `  N
)  =  ( F `
 N ) ) )
38 uzp1 11115 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  =  M  \/  N  e.  ( ZZ>= `  ( M  +  1 ) ) ) )
391, 38syl 16 . . 3  |-  ( ph  ->  ( N  =  M  \/  N  e.  (
ZZ>= `  ( M  + 
1 ) ) ) )
408, 37, 39mpjaod 379 . 2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 N )  =  ( F `  N
) )
41 eqidd 2455 . 2  |-  ( (
ph  /\  x  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  x )  =  ( F `  x ) )
421, 40, 41seqfeq2 12112 1  |-  ( ph  ->  (  seq M ( 
.+  ,  F )  |`  ( ZZ>= `  N )
)  =  seq N
(  .+  ,  F
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804    |` cres 4990   ` cfv 5570  (class class class)co 6270   1c1 9482    + caddc 9484    - cmin 9796   ZZcz 10860   ZZ>=cuz 11082   ...cfz 11675    seqcseq 12089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-seq 12090
This theorem is referenced by:  seqcoll  12496  sumrblem  13615  prodrblem  13818  logtayl  23209  leibpilem2  23469
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