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Theorem sseqfv2 28084
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 28078 . . 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 5868 . 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 12527 . . . 4  |-  ( M  e. Word  S  ->  M  Fn  ( 0..^ ( # `  M ) ) )
82, 7syl 16 . . 3  |-  ( ph  ->  M  Fn  ( 0..^ ( # `  M
) ) )
9 fvex 5876 . . . . . 6  |-  ( x `
 ( ( # `  x )  -  1 ) )  e.  _V
10 df-lsw 12510 . . . . . 6  |- lastS  =  ( x  e.  _V  |->  ( x `  ( (
# `  x )  -  1 ) ) )
119, 10fnmpti 5709 . . . . 5  |- lastS  Fn  _V
1211a1i 11 . . . 4  |-  ( ph  -> lastS 
Fn  _V )
13 lencl 12529 . . . . . . 7  |-  ( M  e. Word  S  ->  ( # `
 M )  e. 
NN0 )
142, 13syl 16 . . . . . 6  |-  ( ph  ->  ( # `  M
)  e.  NN0 )
1514nn0zd 10965 . . . . 5  |-  ( ph  ->  ( # `  M
)  e.  ZZ )
16 seqfn 12088 . . . . 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 3524 . . . . 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 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
) ) )
2112, 17, 19, 20syl3anc 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
) ) )
22 fzouzdisj 11830 . . . . 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 5940 . . 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 1228 . 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 5678 . . . 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 5876 . . . . . . 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 6310 . . . . . . . . . 10  |-  ( x concat  <" ( F `  x ) "> )  e.  _V
3332rgen2w 2826 . . . . . . . . 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 2467 . . . . . . . . 9  |-  ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) )  =  ( x  e.  _V , 
y  e.  _V  |->  ( x concat  <" ( F `
 x ) "> ) )
3635fmpt2 6852 . . . . . . . 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 6431 . . . . . 6  |-  ( (
ph  /\  ( a  e.  _V  /\  b  e. 
_V ) )  -> 
( a ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) b )  e.  _V )
39 eqid 2467 . . . . . 6  |-  ( ZZ>= `  ( # `  M ) )  =  ( ZZ>= `  ( # `  M ) )
40 fvex 5876 . . . . . . 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 12095 . . . . 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 5735 . . . . 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 2558 . . 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 5944 . . 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 2512 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 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   {csn 4027    X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002    o. ccom 5003   Fun wfun 5582    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6285    |-> cmpt2 6287   0cc0 9493   1c1 9494    + caddc 9496    - cmin 9806   NN0cn0 10796   ZZcz 10865   ZZ>=cuz 11083  ..^cfzo 11793    seqcseq 12076   #chash 12374  Word cword 12501   lastS clsw 12502   concat cconcat 12503   <"cs1 12504  seqstrcsseq 28073
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 6577  ax-inf2 8059  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
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 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-pm 7424  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-card 8321  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674  df-fzo 11794  df-seq 12077  df-hash 12375  df-word 12509  df-lsw 12510  df-s1 12512  df-sseq 28074
This theorem is referenced by:  sseqp1  28085
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