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Theorem sseqfn 27997
Description: A strong recursive sequence is a function over the nonnegative integers. (Contributed by Thierry Arnoux, 23-Apr-2019.)
Hypotheses
Ref Expression
sseqval.1  |-  ( ph  ->  S  e.  _V )
sseqval.2  |-  ( ph  ->  M  e. Word  S )
sseqval.3  |-  W  =  (Word  S  i^i  ( `' # " ( ZZ>= `  ( # `  M ) ) ) )
sseqval.4  |-  ( ph  ->  F : W --> S )
Assertion
Ref Expression
sseqfn  |-  ( ph  ->  ( Mseqstr F )  Fn  NN0 )

Proof of Theorem sseqfn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sseqval.2 . . . 4  |-  ( ph  ->  M  e. Word  S )
2 wrdfn 12526 . . . 4  |-  ( M  e. Word  S  ->  M  Fn  ( 0..^ ( # `  M ) ) )
31, 2syl 16 . . 3  |-  ( ph  ->  M  Fn  ( 0..^ ( # `  M
) ) )
4 fvex 5876 . . . . . 6  |-  ( x `
 ( ( # `  x )  -  1 ) )  e.  _V
5 df-lsw 12509 . . . . . 6  |- lastS  =  ( x  e.  _V  |->  ( x `  ( (
# `  x )  -  1 ) ) )
64, 5fnmpti 5709 . . . . 5  |- lastS  Fn  _V
76a1i 11 . . . 4  |-  ( ph  -> lastS 
Fn  _V )
8 lencl 12528 . . . . . 6  |-  ( M  e. Word  S  ->  ( # `
 M )  e. 
NN0 )
98nn0zd 10964 . . . . 5  |-  ( M  e. Word  S  ->  ( # `
 M )  e.  ZZ )
10 seqfn 12087 . . . . 5  |-  ( (
# `  M )  e.  ZZ  ->  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) )  Fn  ( ZZ>= `  ( # `  M ) ) )
111, 9, 103syl 20 . . . 4  |-  ( ph  ->  seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M concat  <" ( F `
 M ) "> ) } ) )  Fn  ( ZZ>= `  ( # `  M ) ) )
12 ssv 3524 . . . . 5  |-  ran  seq ( # `  M ) ( ( x  e. 
_V ,  y  e. 
_V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) )  C_  _V
1312a1i 11 . . . 4  |-  ( ph  ->  ran  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x concat  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) )  C_  _V )
14 fnco 5689 . . . 4  |-  ( ( lastS 
Fn  _V  /\  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) )  Fn  ( ZZ>= `  ( # `  M ) )  /\  ran  seq ( # `  M ) ( ( x  e. 
_V ,  y  e. 
_V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) )  C_  _V )  ->  ( lastS  o.  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) )  Fn  ( ZZ>=
`  ( # `  M
) ) )
157, 11, 13, 14syl3anc 1228 . . 3  |-  ( ph  ->  ( lastS  o.  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) )  Fn  ( ZZ>=
`  ( # `  M
) ) )
16 fzouzdisj 11829 . . . 4  |-  ( ( 0..^ ( # `  M
) )  i^i  ( ZZ>=
`  ( # `  M
) ) )  =  (/)
1716a1i 11 . . 3  |-  ( ph  ->  ( ( 0..^ (
# `  M )
)  i^i  ( ZZ>= `  ( # `  M ) ) )  =  (/) )
18 fnun 5687 . . 3  |-  ( ( ( M  Fn  (
0..^ ( # `  M
) )  /\  ( lastS  o. 
seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M concat  <" ( F `
 M ) "> ) } ) ) )  Fn  ( ZZ>=
`  ( # `  M
) ) )  /\  ( ( 0..^ (
# `  M )
)  i^i  ( ZZ>= `  ( # `  M ) ) )  =  (/) )  ->  ( M  u.  ( lastS  o.  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x concat  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) ) )  Fn  ( ( 0..^ (
# `  M )
)  u.  ( ZZ>= `  ( # `  M ) ) ) )
193, 15, 17, 18syl21anc 1227 . 2  |-  ( ph  ->  ( M  u.  ( lastS  o. 
seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M concat  <" ( F `
 M ) "> ) } ) ) ) )  Fn  ( ( 0..^ (
# `  M )
)  u.  ( ZZ>= `  ( # `  M ) ) ) )
20 sseqval.1 . . . 4  |-  ( ph  ->  S  e.  _V )
21 sseqval.3 . . . 4  |-  W  =  (Word  S  i^i  ( `' # " ( ZZ>= `  ( # `  M ) ) ) )
22 sseqval.4 . . . 4  |-  ( ph  ->  F : W --> S )
2320, 1, 21, 22sseqval 27995 . . 3  |-  ( ph  ->  ( Mseqstr F )  =  ( M  u.  ( lastS  o.  seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M concat  <" ( F `
 M ) "> ) } ) ) ) ) )
24 nn0uz 11116 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
25 elnn0uz 11119 . . . . . 6  |-  ( (
# `  M )  e.  NN0  <->  ( # `  M
)  e.  ( ZZ>= ` 
0 ) )
26 fzouzsplit 11828 . . . . . 6  |-  ( (
# `  M )  e.  ( ZZ>= `  0 )  ->  ( ZZ>= `  0 )  =  ( ( 0..^ ( # `  M
) )  u.  ( ZZ>=
`  ( # `  M
) ) ) )
2725, 26sylbi 195 . . . . 5  |-  ( (
# `  M )  e.  NN0  ->  ( ZZ>= ` 
0 )  =  ( ( 0..^ ( # `  M ) )  u.  ( ZZ>= `  ( # `  M
) ) ) )
281, 8, 273syl 20 . . . 4  |-  ( ph  ->  ( ZZ>= `  0 )  =  ( ( 0..^ ( # `  M
) )  u.  ( ZZ>=
`  ( # `  M
) ) ) )
2924, 28syl5eq 2520 . . 3  |-  ( ph  ->  NN0  =  ( ( 0..^ ( # `  M
) )  u.  ( ZZ>=
`  ( # `  M
) ) ) )
3023, 29fneq12d 5673 . 2  |-  ( ph  ->  ( ( Mseqstr F )  Fn  NN0  <->  ( M  u.  ( lastS  o.  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x concat  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) ) )  Fn  ( ( 0..^ (
# `  M )
)  u.  ( ZZ>= `  ( # `  M ) ) ) ) )
3119, 30mpbird 232 1  |-  ( ph  ->  ( Mseqstr F )  Fn  NN0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3113    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   {csn 4027    X. cxp 4997   `'ccnv 4998   ran crn 5000   "cima 5002    o. ccom 5003    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   0cc0 9492   1c1 9493    - cmin 9805   NN0cn0 10795   ZZcz 10864   ZZ>=cuz 11082  ..^cfzo 11792    seqcseq 12075   #chash 12373  Word cword 12500   lastS clsw 12501   concat cconcat 12502   <"cs1 12503  seqstrcsseq 27990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-seq 12076  df-hash 12374  df-word 12508  df-lsw 12509  df-s1 12511  df-sseq 27991
This theorem is referenced by:  sseqfres  28000
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