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Theorem sseqfv2 26796
Description: Value of the strong sequence builder function. (Contributed by Thierry Arnoux, 21-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 )
sseqfv2.4  |-  ( ph  ->  N  e.  ( ZZ>= `  ( # `  M ) ) )
Assertion
Ref Expression
sseqfv2  |-  ( ph  ->  ( ( Mseqstr F ) `
 N )  =  ( lastS  `  (  seq ( # `  M ) ( ( x  e. 
_V ,  y  e. 
_V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) `  N ) ) )
Distinct variable groups:    x, y, F    x, M, y    ph, x, y    x, W, y
Allowed substitution hints:    S( x, y)    N( x, y)

Proof of Theorem sseqfv2
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sseqval.1 . . . 4  |-  ( ph  ->  S  e.  _V )
2 sseqval.2 . . . 4  |-  ( ph  ->  M  e. Word  S )
3 sseqval.3 . . . 4  |-  W  =  (Word  S  i^i  ( `' # " ( ZZ>= `  ( # `  M ) ) ) )
4 sseqval.4 . . . 4  |-  ( ph  ->  F : W --> S )
51, 2, 3, 4sseqval 26790 . . 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 ) "> ) } ) ) ) ) )
65fveq1d 5712 . 2  |-  ( ph  ->  ( ( Mseqstr F ) `
 N )  =  ( ( M  u.  ( lastS  o.  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x concat  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) ) ) `  N ) )
7 wrdfn 12266 . . . 4  |-  ( M  e. Word  S  ->  M  Fn  ( 0..^ ( # `  M ) ) )
82, 7syl 16 . . 3  |-  ( ph  ->  M  Fn  ( 0..^ ( # `  M
) ) )
9 fvex 5720 . . . . . 6  |-  ( x `
 ( ( # `  x )  -  1 ) )  e.  _V
10 df-lsw 12249 . . . . . 6  |- lastS  =  ( x  e.  _V  |->  ( x `  ( (
# `  x )  -  1 ) ) )
119, 10fnmpti 5558 . . . . 5  |- lastS  Fn  _V
1211a1i 11 . . . 4  |-  ( ph  -> lastS 
Fn  _V )
13 lencl 12268 . . . . . . 7  |-  ( M  e. Word  S  ->  ( # `
 M )  e. 
NN0 )
142, 13syl 16 . . . . . 6  |-  ( ph  ->  ( # `  M
)  e.  NN0 )
1514nn0zd 10764 . . . . 5  |-  ( ph  ->  ( # `  M
)  e.  ZZ )
16 seqfn 11837 . . . . 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 ) ) )
1715, 16syl 16 . . . 4  |-  ( ph  ->  seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M concat  <" ( F `
 M ) "> ) } ) )  Fn  ( ZZ>= `  ( # `  M ) ) )
18 ssv 3395 . . . . 5  |-  ran  seq ( # `  M ) ( ( x  e. 
_V ,  y  e. 
_V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) )  C_  _V
1918a1i 11 . . . 4  |-  ( ph  ->  ran  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x concat  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) )  C_  _V )
20 fnco 5538 . . . 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
) ) )
2112, 17, 19, 20syl3anc 1218 . . 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
) ) )
22 fzouzdisj 11604 . . . . 5  |-  ( ( 0..^ ( # `  M
) )  i^i  ( ZZ>=
`  ( # `  M
) ) )  =  (/)
2322a1i 11 . . . 4  |-  ( ph  ->  ( ( 0..^ (
# `  M )
)  i^i  ( ZZ>= `  ( # `  M ) ) )  =  (/) )
24 sseqfv2.4 . . . 4  |-  ( ph  ->  N  e.  ( ZZ>= `  ( # `  M ) ) )
2523, 24jca 532 . . 3  |-  ( ph  ->  ( ( ( 0..^ ( # `  M
) )  i^i  ( ZZ>=
`  ( # `  M
) ) )  =  (/)  /\  N  e.  (
ZZ>= `  ( # `  M
) ) ) )
26 fvun2 5782 . . 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 ) ) )  =  (/)  /\  N  e.  ( ZZ>= `  ( # `  M ) ) ) )  -> 
( ( M  u.  ( lastS  o.  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x concat  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) ) ) `  N )  =  ( ( lastS  o.  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) ) `  N
) )
278, 21, 25, 26syl3anc 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 ) "> ) } ) ) ) ) `  N )  =  ( ( lastS  o.  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) ) `  N
) )
28 fnfun 5527 . . . 4  |-  (  seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M concat  <" ( F `
 M ) "> ) } ) )  Fn  ( ZZ>= `  ( # `  M ) )  ->  Fun  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) )
2917, 28syl 16 . . 3  |-  ( ph  ->  Fun  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x concat  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) )
30 fvex 5720 . . . . . . 7  |-  ( ( NN0  X.  { ( M concat  <" ( F `
 M ) "> ) } ) `
 ( # `  M
) )  e.  _V
3130a1i 11 . . . . . 6  |-  ( ph  ->  ( ( NN0  X.  { ( M concat  <" ( F `  M ) "> ) } ) `
 ( # `  M
) )  e.  _V )
32 ovex 6135 . . . . . . . . . 10  |-  ( x concat  <" ( F `  x ) "> )  e.  _V
3332rgen2w 2803 . . . . . . . . 9  |-  A. x  e.  _V  A. y  e. 
_V  ( x concat  <" ( F `  x ) "> )  e.  _V
3433a1i 11 . . . . . . . 8  |-  ( ph  ->  A. x  e.  _V  A. y  e.  _V  (
x concat  <" ( F `
 x ) "> )  e.  _V )
35 eqid 2443 . . . . . . . . 9  |-  ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) )  =  ( x  e.  _V , 
y  e.  _V  |->  ( x concat  <" ( F `
 x ) "> ) )
3635fmpt2 6660 . . . . . . . 8  |-  ( A. x  e.  _V  A. y  e.  _V  ( x concat  <" ( F `  x ) "> )  e.  _V  <->  ( x  e.  _V , 
y  e.  _V  |->  ( x concat  <" ( F `
 x ) "> ) ) : ( _V  X.  _V )
--> _V )
3734, 36sylib 196 . . . . . . 7  |-  ( ph  ->  ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) : ( _V  X.  _V )
--> _V )
3837fovrnda 6253 . . . . . 6  |-  ( (
ph  /\  ( a  e.  _V  /\  b  e. 
_V ) )  -> 
( a ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) b )  e.  _V )
39 eqid 2443 . . . . . 6  |-  ( ZZ>= `  ( # `  M ) )  =  ( ZZ>= `  ( # `  M ) )
40 fvex 5720 . . . . . . 7  |-  ( ( NN0  X.  { ( M concat  <" ( F `
 M ) "> ) } ) `
 a )  e. 
_V
4140a1i 11 . . . . . 6  |-  ( (
ph  /\  a  e.  ( ZZ>= `  ( ( # `
 M )  +  1 ) ) )  ->  ( ( NN0 
X.  { ( M concat  <" ( F `  M ) "> ) } ) `  a
)  e.  _V )
4231, 38, 39, 15, 41seqf2 11844 . . . . 5  |-  ( ph  ->  seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M concat  <" ( F `
 M ) "> ) } ) ) : ( ZZ>= `  ( # `  M ) ) --> _V )
43 fdm 5582 . . . . 5  |-  (  seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M concat  <" ( F `
 M ) "> ) } ) ) : ( ZZ>= `  ( # `  M ) ) --> _V  ->  dom  seq ( # `  M ) ( ( x  e. 
_V ,  y  e. 
_V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) )  =  ( ZZ>= `  ( # `  M ) ) )
4442, 43syl 16 . . . 4  |-  ( ph  ->  dom  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x concat  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) )  =  ( ZZ>= `  ( # `  M ) ) )
4524, 44eleqtrrd 2520 . . 3  |-  ( ph  ->  N  e.  dom  seq ( # `  M ) ( ( x  e. 
_V ,  y  e. 
_V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) )
46 fvco 5786 . . 3  |-  ( ( Fun  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x concat  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) )  /\  N  e. 
dom  seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M concat  <" ( F `
 M ) "> ) } ) ) )  ->  (
( lastS  o.  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) ) `  N
)  =  ( lastS  `  (  seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M concat  <" ( F `
 M ) "> ) } ) ) `  N ) ) )
4729, 45, 46syl2anc 661 . 2  |-  ( ph  ->  ( ( lastS  o.  seq ( # `  M ) ( ( x  e. 
_V ,  y  e. 
_V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) ) `  N
)  =  ( lastS  `  (  seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M concat  <" ( F `
 M ) "> ) } ) ) `  N ) ) )
486, 27, 473eqtrd 2479 1  |-  ( ph  ->  ( ( Mseqstr F ) `
 N )  =  ( lastS  `  (  seq ( # `  M ) ( ( x  e. 
_V ,  y  e. 
_V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) `  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2734   _Vcvv 2991    u. cun 3345    i^i cin 3346    C_ wss 3347   (/)c0 3656   {csn 3896    X. cxp 4857   `'ccnv 4858   dom cdm 4859   ran crn 4860   "cima 4862    o. ccom 4863   Fun wfun 5431    Fn wfn 5432   -->wf 5433   ` cfv 5437  (class class class)co 6110    e. cmpt2 6112   0cc0 9301   1c1 9302    + caddc 9304    - cmin 9614   NN0cn0 10598   ZZcz 10665   ZZ>=cuz 10880  ..^cfzo 11567    seqcseq 11825   #chash 12122  Word cword 12240   lastS clsw 12241   concat cconcat 12242   <"cs1 12243  seqstrcsseq 26785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-inf2 7866  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-1st 6596  df-2nd 6597  df-recs 6851  df-rdg 6885  df-1o 6939  df-oadd 6943  df-er 7120  df-map 7235  df-pm 7236  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-card 8128  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-nn 10342  df-n0 10599  df-z 10666  df-uz 10881  df-fz 11457  df-fzo 11568  df-seq 11826  df-hash 12123  df-word 12248  df-lsw 12249  df-s1 12251  df-sseq 26786
This theorem is referenced by:  sseqp1  26797
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