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Theorem sseqfn 26910
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 12358 . . . 4  |-  ( M  e. Word  S  ->  M  Fn  ( 0..^ ( # `  M ) ) )
31, 2syl 16 . . 3  |-  ( ph  ->  M  Fn  ( 0..^ ( # `  M
) ) )
4 fvex 5802 . . . . . 6  |-  ( x `
 ( ( # `  x )  -  1 ) )  e.  _V
5 df-lsw 12341 . . . . . 6  |- lastS  =  ( x  e.  _V  |->  ( x `  ( (
# `  x )  -  1 ) ) )
64, 5fnmpti 5640 . . . . 5  |- lastS  Fn  _V
76a1i 11 . . . 4  |-  ( ph  -> lastS 
Fn  _V )
8 lencl 12360 . . . . . 6  |-  ( M  e. Word  S  ->  ( # `
 M )  e. 
NN0 )
98nn0zd 10849 . . . . 5  |-  ( M  e. Word  S  ->  ( # `
 M )  e.  ZZ )
10 seqfn 11928 . . . . 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 3477 . . . . 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 5620 . . . 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 1219 . . 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 11695 . . . 4  |-  ( ( 0..^ ( # `  M
) )  i^i  ( ZZ>=
`  ( # `  M
) ) )  =  (/)
1716a1i 11 . . 3  |-  ( ph  ->  ( ( 0..^ (
# `  M )
)  i^i  ( ZZ>= `  ( # `  M ) ) )  =  (/) )
18 fnun 5618 . . 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 1218 . 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 26908 . . 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 10999 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
25 elnn0uz 11002 . . . . . 6  |-  ( (
# `  M )  e.  NN0  <->  ( # `  M
)  e.  ( ZZ>= ` 
0 ) )
26 fzouzsplit 11694 . . . . . 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 2504 . . 3  |-  ( ph  ->  NN0  =  ( ( 0..^ ( # `  M
) )  u.  ( ZZ>=
`  ( # `  M
) ) ) )
3023, 29fneq12d 5604 . 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 1370    e. wcel 1758   _Vcvv 3071    u. cun 3427    i^i cin 3428    C_ wss 3429   (/)c0 3738   {csn 3978    X. cxp 4939   `'ccnv 4940   ran crn 4942   "cima 4944    o. ccom 4945    Fn wfn 5514   -->wf 5515   ` cfv 5519  (class class class)co 6193    |-> cmpt2 6195   0cc0 9386   1c1 9387    - cmin 9699   NN0cn0 10683   ZZcz 10750   ZZ>=cuz 10965  ..^cfzo 11658    seqcseq 11916   #chash 12213  Word cword 12332   lastS clsw 12333   concat cconcat 12334   <"cs1 12335  seqstrcsseq 26903
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 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-map 7319  df-pm 7320  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-fzo 11659  df-seq 11917  df-hash 12214  df-word 12340  df-lsw 12341  df-s1 12343  df-sseq 26904
This theorem is referenced by:  sseqfres  26913
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