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Theorem iseqid3s 9562
Description: A sequence that consists of zeroes up to  N sums to zero at  N. In this case by "zero" we mean whatever the identity  Z is for the operation  .+). (Contributed by Jim Kingdon, 18-Aug-2021.)
Hypotheses
Ref Expression
iseqid3s.1  |-  ( ph  ->  ( Z  .+  Z
)  =  Z )
iseqid3s.2  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
iseqid3s.3  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( F `  x )  =  Z )
iseqid3s.z  |-  ( ph  ->  Z  e.  S )
iseqid3s.f  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
iseqid3s.cl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
Assertion
Ref Expression
iseqid3s  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  N
)  =  Z )
Distinct variable groups:    x, y,  .+    x, F, y    x, M, y    ph, x, y    x, Z, y    x, N, y   
x, S, y

Proof of Theorem iseqid3s
Dummy variables  k  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iseqid3s.2 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
2 eluzfz2 9127 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
3 fveq2 5209 . . . . . 6  |-  ( w  =  M  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  S ) `
 M ) )
43eqeq1d 2090 . . . . 5  |-  ( w  =  M  ->  (
(  seq M (  .+  ,  F ,  S ) `
 w )  =  Z  <->  (  seq M
(  .+  ,  F ,  S ) `  M
)  =  Z ) )
54imbi2d 228 . . . 4  |-  ( w  =  M  ->  (
( ph  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  Z )  <->  ( ph  ->  (  seq M (  .+  ,  F ,  S ) `
 M )  =  Z ) ) )
6 fveq2 5209 . . . . . 6  |-  ( w  =  k  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  S ) `
 k ) )
76eqeq1d 2090 . . . . 5  |-  ( w  =  k  ->  (
(  seq M (  .+  ,  F ,  S ) `
 w )  =  Z  <->  (  seq M
(  .+  ,  F ,  S ) `  k
)  =  Z ) )
87imbi2d 228 . . . 4  |-  ( w  =  k  ->  (
( ph  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  Z )  <->  ( ph  ->  (  seq M (  .+  ,  F ,  S ) `
 k )  =  Z ) ) )
9 fveq2 5209 . . . . . 6  |-  ( w  =  ( k  +  1 )  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  S ) `
 ( k  +  1 ) ) )
109eqeq1d 2090 . . . . 5  |-  ( w  =  ( k  +  1 )  ->  (
(  seq M (  .+  ,  F ,  S ) `
 w )  =  Z  <->  (  seq M
(  .+  ,  F ,  S ) `  (
k  +  1 ) )  =  Z ) )
1110imbi2d 228 . . . 4  |-  ( w  =  ( k  +  1 )  ->  (
( ph  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  Z )  <->  ( ph  ->  (  seq M (  .+  ,  F ,  S ) `
 ( k  +  1 ) )  =  Z ) ) )
12 fveq2 5209 . . . . . 6  |-  ( w  =  N  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  S ) `
 N ) )
1312eqeq1d 2090 . . . . 5  |-  ( w  =  N  ->  (
(  seq M (  .+  ,  F ,  S ) `
 w )  =  Z  <->  (  seq M
(  .+  ,  F ,  S ) `  N
)  =  Z ) )
1413imbi2d 228 . . . 4  |-  ( w  =  N  ->  (
( ph  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  Z )  <->  ( ph  ->  (  seq M (  .+  ,  F ,  S ) `
 N )  =  Z ) ) )
15 eluzel2 8705 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
161, 15syl 14 . . . . . . 7  |-  ( ph  ->  M  e.  ZZ )
17 iseqid3s.f . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
18 iseqid3s.cl . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
1916, 17, 18iseq1 9533 . . . . . 6  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  M
)  =  ( F `
 M ) )
20 iseqid3s.3 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( F `  x )  =  Z )
2120ralrimiva 2435 . . . . . . 7  |-  ( ph  ->  A. x  e.  ( M ... N ) ( F `  x
)  =  Z )
22 eluzfz1 9126 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
23 fveq2 5209 . . . . . . . . . 10  |-  ( x  =  M  ->  ( F `  x )  =  ( F `  M ) )
2423eqeq1d 2090 . . . . . . . . 9  |-  ( x  =  M  ->  (
( F `  x
)  =  Z  <->  ( F `  M )  =  Z ) )
2524rspcv 2698 . . . . . . . 8  |-  ( M  e.  ( M ... N )  ->  ( A. x  e.  ( M ... N ) ( F `  x )  =  Z  ->  ( F `  M )  =  Z ) )
261, 22, 253syl 17 . . . . . . 7  |-  ( ph  ->  ( A. x  e.  ( M ... N
) ( F `  x )  =  Z  ->  ( F `  M )  =  Z ) )
2721, 26mpd 13 . . . . . 6  |-  ( ph  ->  ( F `  M
)  =  Z )
2819, 27eqtrd 2114 . . . . 5  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  M
)  =  Z )
2928a1i 9 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  M
)  =  Z ) )
30 elfzouz 9238 . . . . . . . . . . 11  |-  ( k  e.  ( M..^ N
)  ->  k  e.  ( ZZ>= `  M )
)
3130adantl 271 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( M..^ N ) )  ->  k  e.  (
ZZ>= `  M ) )
3217adantlr 461 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  ( M..^ N ) )  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
3318adantlr 461 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  ( M..^ N ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
3431, 32, 33iseqp1 9538 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( M..^ N ) )  ->  (  seq M
(  .+  ,  F ,  S ) `  (
k  +  1 ) )  =  ( (  seq M (  .+  ,  F ,  S ) `
 k )  .+  ( F `  ( k  +  1 ) ) ) )
3534adantr 270 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( M..^ N ) )  /\  (  seq M (  .+  ,  F ,  S ) `  k )  =  Z )  ->  (  seq M (  .+  ,  F ,  S ) `  ( k  +  1 ) )  =  ( (  seq M ( 
.+  ,  F ,  S ) `  k
)  .+  ( F `  ( k  +  1 ) ) ) )
36 simpr 108 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( M..^ N ) )  /\  (  seq M (  .+  ,  F ,  S ) `  k )  =  Z )  ->  (  seq M (  .+  ,  F ,  S ) `  k )  =  Z )
37 fveq2 5209 . . . . . . . . . . . 12  |-  ( x  =  ( k  +  1 )  ->  ( F `  x )  =  ( F `  ( k  +  1 ) ) )
3837eqeq1d 2090 . . . . . . . . . . 11  |-  ( x  =  ( k  +  1 )  ->  (
( F `  x
)  =  Z  <->  ( F `  ( k  +  1 ) )  =  Z ) )
3921adantr 270 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( M..^ N ) )  ->  A. x  e.  ( M ... N ) ( F `  x
)  =  Z )
40 fzofzp1 9313 . . . . . . . . . . . 12  |-  ( k  e.  ( M..^ N
)  ->  ( k  +  1 )  e.  ( M ... N
) )
4140adantl 271 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( M..^ N ) )  ->  ( k  +  1 )  e.  ( M ... N ) )
4238, 39, 41rspcdva 2708 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( M..^ N ) )  ->  ( F `  ( k  +  1 ) )  =  Z )
4342adantr 270 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( M..^ N ) )  /\  (  seq M (  .+  ,  F ,  S ) `  k )  =  Z )  ->  ( F `  ( k  +  1 ) )  =  Z )
4436, 43oveq12d 5561 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( M..^ N ) )  /\  (  seq M (  .+  ,  F ,  S ) `  k )  =  Z )  ->  ( (  seq M (  .+  ,  F ,  S ) `  k )  .+  ( F `  ( k  +  1 ) ) )  =  ( Z 
.+  Z ) )
45 iseqid3s.1 . . . . . . . . 9  |-  ( ph  ->  ( Z  .+  Z
)  =  Z )
4645ad2antrr 472 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( M..^ N ) )  /\  (  seq M (  .+  ,  F ,  S ) `  k )  =  Z )  ->  ( Z  .+  Z )  =  Z )
4735, 44, 463eqtrd 2118 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( M..^ N ) )  /\  (  seq M (  .+  ,  F ,  S ) `  k )  =  Z )  ->  (  seq M (  .+  ,  F ,  S ) `  ( k  +  1 ) )  =  Z )
4847ex 113 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M..^ N ) )  ->  ( (  seq M (  .+  ,  F ,  S ) `  k )  =  Z  ->  (  seq M
(  .+  ,  F ,  S ) `  (
k  +  1 ) )  =  Z ) )
4948expcom 114 . . . . 5  |-  ( k  e.  ( M..^ N
)  ->  ( ph  ->  ( (  seq M
(  .+  ,  F ,  S ) `  k
)  =  Z  -> 
(  seq M (  .+  ,  F ,  S ) `
 ( k  +  1 ) )  =  Z ) ) )
5049a2d 26 . . . 4  |-  ( k  e.  ( M..^ N
)  ->  ( ( ph  ->  (  seq M
(  .+  ,  F ,  S ) `  k
)  =  Z )  ->  ( ph  ->  (  seq M (  .+  ,  F ,  S ) `
 ( k  +  1 ) )  =  Z ) ) )
515, 8, 11, 14, 29, 50fzind2 9325 . . 3  |-  ( N  e.  ( M ... N )  ->  ( ph  ->  (  seq M
(  .+  ,  F ,  S ) `  N
)  =  Z ) )
521, 2, 513syl 17 . 2  |-  ( ph  ->  ( ph  ->  (  seq M (  .+  ,  F ,  S ) `  N )  =  Z ) )
5352pm2.43i 48 1  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  N
)  =  Z )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   A.wral 2349   ` cfv 4932  (class class class)co 5543   1c1 7044    + caddc 7046   ZZcz 8432   ZZ>=cuz 8700   ...cfz 9105  ..^cfzo 9229    seqcseq 9521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-ltadd 7154
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433  df-uz 8701  df-fz 9106  df-fzo 9230  df-iseq 9522
This theorem is referenced by:  iseqid  9563  iser0  9568
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