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Theorem sseqfn 29049
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 12672 . . . 4  |-  ( M  e. Word  S  ->  M  Fn  ( 0..^ ( # `  M ) ) )
31, 2syl 17 . . 3  |-  ( ph  ->  M  Fn  ( 0..^ ( # `  M
) ) )
4 fvex 5891 . . . . . 6  |-  ( x `
 ( ( # `  x )  -  1 ) )  e.  _V
5 df-lsw 12652 . . . . . 6  |- lastS  =  ( x  e.  _V  |->  ( x `  ( (
# `  x )  -  1 ) ) )
64, 5fnmpti 5724 . . . . 5  |- lastS  Fn  _V
76a1i 11 . . . 4  |-  ( ph  -> lastS 
Fn  _V )
8 lencl 12674 . . . . . 6  |-  ( M  e. Word  S  ->  ( # `
 M )  e. 
NN0 )
98nn0zd 11038 . . . . 5  |-  ( M  e. Word  S  ->  ( # `
 M )  e.  ZZ )
10 seqfn 12222 . . . . 5  |-  ( (
# `  M )  e.  ZZ  ->  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) )  Fn  ( ZZ>= `  ( # `  M ) ) )
111, 9, 103syl 18 . . . 4  |-  ( ph  ->  seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++ 
<" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M ++  <" ( F `
 M ) "> ) } ) )  Fn  ( ZZ>= `  ( # `  M ) ) )
12 ssv 3490 . . . . 5  |-  ran  seq ( # `  M ) ( ( x  e. 
_V ,  y  e. 
_V  |->  ( x ++  <" ( F `  x
) "> )
) ,  ( NN0 
X.  { ( M ++ 
<" ( F `  M ) "> ) } ) )  C_  _V
1312a1i 11 . . . 4  |-  ( ph  ->  ran  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x ++  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) )  C_  _V )
14 fnco 5702 . . . 4  |-  ( ( lastS 
Fn  _V  /\  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) )  Fn  ( ZZ>= `  ( # `  M ) )  /\  ran  seq ( # `  M ) ( ( x  e. 
_V ,  y  e. 
_V  |->  ( x ++  <" ( F `  x
) "> )
) ,  ( NN0 
X.  { ( M ++ 
<" ( F `  M ) "> ) } ) )  C_  _V )  ->  ( lastS  o.  seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++ 
<" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M ++  <" ( F `
 M ) "> ) } ) ) )  Fn  ( ZZ>=
`  ( # `  M
) ) )
157, 11, 13, 14syl3anc 1264 . . 3  |-  ( ph  ->  ( lastS  o.  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) ) )  Fn  ( ZZ>=
`  ( # `  M
) ) )
16 fzouzdisj 11952 . . . 4  |-  ( ( 0..^ ( # `  M
) )  i^i  ( ZZ>=
`  ( # `  M
) ) )  =  (/)
1716a1i 11 . . 3  |-  ( ph  ->  ( ( 0..^ (
# `  M )
)  i^i  ( ZZ>= `  ( # `  M ) ) )  =  (/) )
18 fnun 5700 . . 3  |-  ( ( ( M  Fn  (
0..^ ( # `  M
) )  /\  ( lastS  o. 
seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++ 
<" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M ++  <" ( F `
 M ) "> ) } ) ) )  Fn  ( ZZ>=
`  ( # `  M
) ) )  /\  ( ( 0..^ (
# `  M )
)  i^i  ( ZZ>= `  ( # `  M ) ) )  =  (/) )  ->  ( M  u.  ( lastS  o.  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x ++  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) ) ) )  Fn  ( ( 0..^ (
# `  M )
)  u.  ( ZZ>= `  ( # `  M ) ) ) )
193, 15, 17, 18syl21anc 1263 . 2  |-  ( ph  ->  ( M  u.  ( lastS  o. 
seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++ 
<" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M ++  <" ( 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 29047 . . 3  |-  ( ph  ->  ( Mseqstr F )  =  ( M  u.  ( lastS  o.  seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++ 
<" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M ++  <" ( F `
 M ) "> ) } ) ) ) ) )
24 nn0uz 11193 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
25 elnn0uz 11196 . . . . . 6  |-  ( (
# `  M )  e.  NN0  <->  ( # `  M
)  e.  ( ZZ>= ` 
0 ) )
26 fzouzsplit 11951 . . . . . 6  |-  ( (
# `  M )  e.  ( ZZ>= `  0 )  ->  ( ZZ>= `  0 )  =  ( ( 0..^ ( # `  M
) )  u.  ( ZZ>=
`  ( # `  M
) ) ) )
2725, 26sylbi 198 . . . . 5  |-  ( (
# `  M )  e.  NN0  ->  ( ZZ>= ` 
0 )  =  ( ( 0..^ ( # `  M ) )  u.  ( ZZ>= `  ( # `  M
) ) ) )
281, 8, 273syl 18 . . . 4  |-  ( ph  ->  ( ZZ>= `  0 )  =  ( ( 0..^ ( # `  M
) )  u.  ( ZZ>=
`  ( # `  M
) ) ) )
2924, 28syl5eq 2482 . . 3  |-  ( ph  ->  NN0  =  ( ( 0..^ ( # `  M
) )  u.  ( ZZ>=
`  ( # `  M
) ) ) )
3023, 29fneq12d 5686 . 2  |-  ( ph  ->  ( ( Mseqstr F )  Fn  NN0  <->  ( M  u.  ( lastS  o.  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x ++  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) ) ) )  Fn  ( ( 0..^ (
# `  M )
)  u.  ( ZZ>= `  ( # `  M ) ) ) ) )
3119, 30mpbird 235 1  |-  ( ph  ->  ( Mseqstr F )  Fn  NN0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870   _Vcvv 3087    u. cun 3440    i^i cin 3441    C_ wss 3442   (/)c0 3767   {csn 4002    X. cxp 4852   `'ccnv 4853   ran crn 4855   "cima 4857    o. ccom 4858    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   0cc0 9538   1c1 9539    - cmin 9859   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159  ..^cfzo 11913    seqcseq 12210   #chash 12512  Word cword 12643   lastS clsw 12644   ++ cconcat 12645   <"cs1 12646  seqstrcsseq 29042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-fzo 11914  df-seq 12211  df-hash 12513  df-word 12651  df-lsw 12652  df-s1 12654  df-sseq 29043
This theorem is referenced by:  sseqfres  29052
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