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Theorem iseqp1t 9883
Description: Value of the sequence builder function at a successor.

New proofs should use seq3p1 9884 instead (together with iseqsst 9886 or iseqseq3 9902 if need be).

(Contributed by Jim Kingdon, 30-Apr-2022.) (New usage is discouraged.)

Hypotheses
Ref Expression
iseqp1t.m  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
iseqp1t.f  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
iseqp1t.pl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
iseqp1t.t  |-  ( ph  ->  S  C_  T )
Assertion
Ref Expression
iseqp1t  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  T ) `  ( N  +  1 ) )  =  ( (  seq M (  .+  ,  F ,  T ) `
 N )  .+  ( F `  ( N  +  1 ) ) ) )
Distinct variable groups:    x,  .+ , y    x, F, y    x, M, y    x, N, y   
x, S, y    x, T, y    ph, x, y

Proof of Theorem iseqp1t
Dummy variables  a  b  w  z  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iseqp1t.m . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
2 eluzel2 9024 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
31, 2syl 14 . . . 4  |-  ( ph  ->  M  e.  ZZ )
4 fveq2 5305 . . . . . 6  |-  ( x  =  M  ->  ( F `  x )  =  ( F `  M ) )
54eleq1d 2156 . . . . 5  |-  ( x  =  M  ->  (
( F `  x
)  e.  S  <->  ( F `  M )  e.  S
) )
6 iseqp1t.f . . . . . 6  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
76ralrimiva 2446 . . . . 5  |-  ( ph  ->  A. x  e.  (
ZZ>= `  M ) ( F `  x )  e.  S )
8 uzid 9033 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
93, 8syl 14 . . . . 5  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
105, 7, 9rspcdva 2727 . . . 4  |-  ( ph  ->  ( F `  M
)  e.  S )
11 iseqp1t.t . . . 4  |-  ( ph  ->  S  C_  T )
12 iseqp1t.pl . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
136, 12iseqovex 9870 . . . 4  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  (
x ( z  e.  ( ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) y )  e.  S )
14 iseqvalcbv 9872 . . . 4  |- frec ( ( a  e.  ( ZZ>= `  M ) ,  b  e.  T  |->  <. (
a  +  1 ) ,  ( a ( c  e.  ( ZZ>= `  M ) ,  d  e.  S  |->  ( d 
.+  ( F `  ( c  +  1 ) ) ) ) b ) >. ) ,  <. M ,  ( F `  M )
>. )  = frec (
( x  e.  (
ZZ>= `  M ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x ( z  e.  (
ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `
 ( z  +  1 ) ) ) ) y ) >.
) ,  <. M , 
( F `  M
) >. )
153, 14, 6, 12, 11iseqvalt 9873 . . . 4  |-  ( ph  ->  seq M (  .+  ,  F ,  T )  =  ran frec ( (
a  e.  ( ZZ>= `  M ) ,  b  e.  T  |->  <. (
a  +  1 ) ,  ( a ( c  e.  ( ZZ>= `  M ) ,  d  e.  S  |->  ( d 
.+  ( F `  ( c  +  1 ) ) ) ) b ) >. ) ,  <. M ,  ( F `  M )
>. ) )
163, 10, 11, 13, 14, 15frecuzrdgsuct 9831 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F ,  T ) `  ( N  +  1 ) )  =  ( N ( z  e.  ( ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) (  seq M (  .+  ,  F ,  T ) `  N ) ) )
171, 16mpdan 412 . 2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  T ) `  ( N  +  1 ) )  =  ( N ( z  e.  (
ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `
 ( z  +  1 ) ) ) ) (  seq M
(  .+  ,  F ,  T ) `  N
) ) )
18 eqid 2088 . . . . 5  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
1918, 3, 6, 12, 11iseqfclt 9879 . . . 4  |-  ( ph  ->  seq M (  .+  ,  F ,  T ) : ( ZZ>= `  M
) --> S )
2019, 1ffvelrnd 5435 . . 3  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  T ) `  N
)  e.  S )
21 fveq2 5305 . . . . . 6  |-  ( x  =  ( N  + 
1 )  ->  ( F `  x )  =  ( F `  ( N  +  1
) ) )
2221eleq1d 2156 . . . . 5  |-  ( x  =  ( N  + 
1 )  ->  (
( F `  x
)  e.  S  <->  ( F `  ( N  +  1 ) )  e.  S
) )
23 peano2uz 9071 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ( ZZ>= `  M )
)
241, 23syl 14 . . . . 5  |-  ( ph  ->  ( N  +  1 )  e.  ( ZZ>= `  M ) )
2522, 7, 24rspcdva 2727 . . . 4  |-  ( ph  ->  ( F `  ( N  +  1 ) )  e.  S )
2612, 20, 25caovcld 5798 . . 3  |-  ( ph  ->  ( (  seq M
(  .+  ,  F ,  T ) `  N
)  .+  ( F `  ( N  +  1 ) ) )  e.  S )
27 oveq1 5659 . . . . . 6  |-  ( z  =  N  ->  (
z  +  1 )  =  ( N  + 
1 ) )
2827fveq2d 5309 . . . . 5  |-  ( z  =  N  ->  ( F `  ( z  +  1 ) )  =  ( F `  ( N  +  1
) ) )
2928oveq2d 5668 . . . 4  |-  ( z  =  N  ->  (
w  .+  ( F `  ( z  +  1 ) ) )  =  ( w  .+  ( F `  ( N  +  1 ) ) ) )
30 oveq1 5659 . . . 4  |-  ( w  =  (  seq M
(  .+  ,  F ,  T ) `  N
)  ->  ( w  .+  ( F `  ( N  +  1 ) ) )  =  ( (  seq M ( 
.+  ,  F ,  T ) `  N
)  .+  ( F `  ( N  +  1 ) ) ) )
31 eqid 2088 . . . 4  |-  ( z  e.  ( ZZ>= `  M
) ,  w  e.  S  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) )  =  ( z  e.  ( ZZ>= `  M ) ,  w  e.  S  |->  ( w 
.+  ( F `  ( z  +  1 ) ) ) )
3229, 30, 31ovmpt2g 5779 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  (  seq M (  .+  ,  F ,  T ) `  N )  e.  S  /\  ( (  seq M
(  .+  ,  F ,  T ) `  N
)  .+  ( F `  ( N  +  1 ) ) )  e.  S )  ->  ( N ( z  e.  ( ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) (  seq M (  .+  ,  F ,  T ) `  N ) )  =  ( (  seq M
(  .+  ,  F ,  T ) `  N
)  .+  ( F `  ( N  +  1 ) ) ) )
331, 20, 26, 32syl3anc 1174 . 2  |-  ( ph  ->  ( N ( z  e.  ( ZZ>= `  M
) ,  w  e.  S  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) (  seq M (  .+  ,  F ,  T ) `  N ) )  =  ( (  seq M
(  .+  ,  F ,  T ) `  N
)  .+  ( F `  ( N  +  1 ) ) ) )
3417, 33eqtrd 2120 1  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  T ) `  ( N  +  1 ) )  =  ( (  seq M (  .+  ,  F ,  T ) `
 N )  .+  ( F `  ( N  +  1 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438    C_ wss 2999   <.cop 3449   ` cfv 5015  (class class class)co 5652    |-> cmpt2 5654  freccfrec 6155   1c1 7351    + caddc 7353   ZZcz 8750   ZZ>=cuz 9019    seqcseq4 9851
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-addcom 7445  ax-addass 7447  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-0id 7453  ax-rnegex 7454  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-ltadd 7461
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-inn 8423  df-n0 8674  df-z 8751  df-uz 9020  df-iseq 9853
This theorem is referenced by:  iseqsst  9886
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