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Theorem seqid 11972
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 10985 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
3 seq1 11940 . . . . 5  |-  ( N  e.  ZZ  ->  (  seq N (  .+  ,  F ) `  N
)  =  ( F `
 N ) )
41, 2, 33syl 20 . . . 4  |-  ( ph  ->  (  seq N ( 
.+  ,  F ) `
 N )  =  ( F `  N
) )
5 seqeq1 11930 . . . . . 6  |-  ( N  =  M  ->  seq N (  .+  ,  F )  =  seq M (  .+  ,  F ) )
65fveq1d 5804 . . . . 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 10981 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
101, 9syl 16 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
11 seqm1 11944 . . . . . 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 2830 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  S  ( Z  .+  x )  =  x )
16 oveq2 6211 . . . . . . . . . . 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 3175 . . . . . . . . 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 10999 . . . . . . . 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 11971 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq M (  .+  ,  F ) `  ( N  -  1 ) )  =  Z )
2726oveq1d 6218 . . . . 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 6211 . . . . . . . 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 3175 . . . . . 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 434 . . 3  |-  ( ph  ->  ( N  e.  (
ZZ>= `  ( M  + 
1 ) )  -> 
(  seq M (  .+  ,  F ) `  N
)  =  ( F `
 N ) ) )
38 uzp1 11009 . . . 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 2455 . 2  |-  ( (
ph  /\  x  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  x )  =  ( F `  x ) )
421, 40, 41seqfeq2 11950 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 1370    e. wcel 1758   A.wral 2799    |` cres 4953   ` cfv 5529  (class class class)co 6203   1c1 9398    + caddc 9400    - cmin 9710   ZZcz 10761   ZZ>=cuz 10976   ...cfz 11558    seqcseq 11927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-seq 11928
This theorem is referenced by:  seqcoll  12338  sumrblem  13310  logtayl  22248  leibpilem2  22479  prodrblem  27609
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