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Theorem seqid 11138
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 10285 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
3 seq1 11106 . . . . 5  |-  ( N  e.  ZZ  ->  (  seq  N (  .+  ,  F ) `  N
)  =  ( F `
 N ) )
41, 2, 33syl 18 . . . 4  |-  ( ph  ->  (  seq  N ( 
.+  ,  F ) `
 N )  =  ( F `  N
) )
5 seqeq1 11096 . . . . . 6  |-  ( N  =  M  ->  seq  N (  .+  ,  F
)  =  seq  M
(  .+  ,  F
) )
65fveq1d 5565 . . . . 5  |-  ( N  =  M  ->  (  seq  N (  .+  ,  F ) `  N
)  =  (  seq 
M (  .+  ,  F ) `  N
) )
76eqeq1d 2324 . . . 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 10282 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
101, 9syl 15 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
11 seqm1 11110 . . . . . 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 2660 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  S  ( Z  .+  x )  =  x )
16 oveq2 5908 . . . . . . . . . . 11  |-  ( x  =  Z  ->  ( Z  .+  x )  =  ( Z  .+  Z
) )
17 id 19 . . . . . . . . . . 11  |-  ( x  =  Z  ->  x  =  Z )
1816, 17eqeq12d 2330 . . . . . . . . . 10  |-  ( x  =  Z  ->  (
( Z  .+  x
)  =  x  <->  ( Z  .+  Z )  =  Z ) )
1918rspcv 2914 . . . . . . . . 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 10298 . . . . . . . 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 11137 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq  M (  .+  ,  F
) `  ( N  -  1 ) )  =  Z )
2726oveq1d 5915 . . . . 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 5908 . . . . . . . 8  |-  ( x  =  ( F `  N )  ->  ( Z  .+  x )  =  ( Z  .+  ( F `  N )
) )
32 id 19 . . . . . . . 8  |-  ( x  =  ( F `  N )  ->  x  =  ( F `  N ) )
3331, 32eqeq12d 2330 . . . . . . 7  |-  ( x  =  ( F `  N )  ->  (
( Z  .+  x
)  =  x  <->  ( Z  .+  ( F `  N
) )  =  ( F `  N ) ) )
3433rspcv 2914 . . . . . 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 2352 . . . 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 10308 . . . 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 2317 . 2  |-  ( (
ph  /\  x  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  x )  =  ( F `  x ) )
421, 40, 41seqfeq2 11116 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 1633    e. wcel 1701   A.wral 2577    |` cres 4728   ` cfv 5292  (class class class)co 5900   1c1 8783    + caddc 8785    - cmin 9082   ZZcz 10071   ZZ>=cuz 10277   ...cfz 10829    seq cseq 11093
This theorem is referenced by:  seqcoll  11448  sumrblem  12231  logtayl  20060  leibpilem2  20290  prodrblem  24432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-n0 10013  df-z 10072  df-uz 10278  df-fz 10830  df-seq 11094
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