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Theorem seqid 12126
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 11094 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
3 seq1 12094 . . . . 5  |-  ( N  e.  ZZ  ->  (  seq N (  .+  ,  F ) `  N
)  =  ( F `
 N ) )
41, 2, 33syl 20 . . . 4  |-  ( ph  ->  (  seq N ( 
.+  ,  F ) `
 N )  =  ( F `  N
) )
5 seqeq1 12084 . . . . . 6  |-  ( N  =  M  ->  seq N (  .+  ,  F )  =  seq M (  .+  ,  F ) )
65fveq1d 5854 . . . . 5  |-  ( N  =  M  ->  (  seq N (  .+  ,  F ) `  N
)  =  (  seq M (  .+  ,  F ) `  N
) )
76eqeq1d 2443 . . . 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 11090 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
101, 9syl 16 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
11 seqm1 12098 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>= `  ( M  +  1
) ) )  -> 
(  seq M (  .+  ,  F ) `  N
)  =  ( (  seq M (  .+  ,  F ) `  ( N  -  1 ) )  .+  ( F `
 N ) ) )
1210, 11sylan 471 . . . . 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 2855 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  S  ( Z  .+  x )  =  x )
16 oveq2 6285 . . . . . . . . . . 11  |-  ( x  =  Z  ->  ( Z  .+  x )  =  ( Z  .+  Z
) )
17 id 22 . . . . . . . . . . 11  |-  ( x  =  Z  ->  x  =  Z )
1816, 17eqeq12d 2463 . . . . . . . . . 10  |-  ( x  =  Z  ->  (
( Z  .+  x
)  =  x  <->  ( Z  .+  Z )  =  Z ) )
1918rspcv 3190 . . . . . . . . 9  |-  ( Z  e.  S  ->  ( A. x  e.  S  ( Z  .+  x )  =  x  ->  ( Z  .+  Z )  =  Z ) )
2013, 15, 19sylc 60 . . . . . . . 8  |-  ( ph  ->  ( Z  .+  Z
)  =  Z )
2120adantr 465 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( Z  .+  Z )  =  Z )
22 eluzp1m1 11108 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>= `  ( M  +  1
) ) )  -> 
( N  -  1 )  e.  ( ZZ>= `  M ) )
2310, 22sylan 471 . . . . . . 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 714 . . . . . . 7  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  /\  x  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  x )  =  Z )
2621, 23, 25seqid3 12125 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq M (  .+  ,  F ) `  ( N  -  1 ) )  =  Z )
2726oveq1d 6292 . . . . 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 465 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( F `  N )  e.  S
)
3015adantr 465 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  A. x  e.  S  ( Z  .+  x )  =  x )
31 oveq2 6285 . . . . . . . 8  |-  ( x  =  ( F `  N )  ->  ( Z  .+  x )  =  ( Z  .+  ( F `  N )
) )
32 id 22 . . . . . . . 8  |-  ( x  =  ( F `  N )  ->  x  =  ( F `  N ) )
3331, 32eqeq12d 2463 . . . . . . 7  |-  ( x  =  ( F `  N )  ->  (
( Z  .+  x
)  =  x  <->  ( Z  .+  ( F `  N
) )  =  ( F `  N ) ) )
3433rspcv 3190 . . . . . 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 2486 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq M (  .+  ,  F ) `  N
)  =  ( F `
 N ) )
3736ex 434 . . 3  |-  ( ph  ->  ( N  e.  (
ZZ>= `  ( M  + 
1 ) )  -> 
(  seq M (  .+  ,  F ) `  N
)  =  ( F `
 N ) ) )
38 uzp1 11118 . . . 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 381 . 2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 N )  =  ( F `  N
) )
41 eqidd 2442 . 2  |-  ( (
ph  /\  x  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  x )  =  ( F `  x ) )
421, 40, 41seqfeq2 12104 1  |-  ( ph  ->  (  seq M ( 
.+  ,  F )  |`  ( ZZ>= `  N )
)  =  seq N
(  .+  ,  F
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1381    e. wcel 1802   A.wral 2791    |` cres 4987   ` cfv 5574  (class class class)co 6277   1c1 9491    + caddc 9493    - cmin 9805   ZZcz 10865   ZZ>=cuz 11085   ...cfz 11676    seqcseq 12081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11086  df-fz 11677  df-seq 12082
This theorem is referenced by:  seqcoll  12486  sumrblem  13507  logtayl  22906  leibpilem2  23137  prodrblem  29029
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