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Theorem sseqfv2 29053
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 ++  <" ( F `  x
) "> )
) ,  ( NN0 
X.  { ( M ++ 
<" ( 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 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 ) "> ) } ) ) ) ) )
65fveq1d 5883 . 2  |-  ( ph  ->  ( ( Mseqstr F ) `
 N )  =  ( ( M  u.  ( lastS  o.  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x ++  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) ) ) ) `  N ) )
7 wrdfn 12672 . . . 4  |-  ( M  e. Word  S  ->  M  Fn  ( 0..^ ( # `  M ) ) )
82, 7syl 17 . . 3  |-  ( ph  ->  M  Fn  ( 0..^ ( # `  M
) ) )
9 fvex 5891 . . . . . 6  |-  ( x `
 ( ( # `  x )  -  1 ) )  e.  _V
10 df-lsw 12652 . . . . . 6  |- lastS  =  ( x  e.  _V  |->  ( x `  ( (
# `  x )  -  1 ) ) )
119, 10fnmpti 5724 . . . . 5  |- lastS  Fn  _V
1211a1i 11 . . . 4  |-  ( ph  -> lastS 
Fn  _V )
13 lencl 12674 . . . . . . 7  |-  ( M  e. Word  S  ->  ( # `
 M )  e. 
NN0 )
142, 13syl 17 . . . . . 6  |-  ( ph  ->  ( # `  M
)  e.  NN0 )
1514nn0zd 11038 . . . . 5  |-  ( ph  ->  ( # `  M
)  e.  ZZ )
16 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 ) ) )
1715, 16syl 17 . . . 4  |-  ( ph  ->  seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++ 
<" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M ++  <" ( F `
 M ) "> ) } ) )  Fn  ( ZZ>= `  ( # `  M ) ) )
18 ssv 3490 . . . . 5  |-  ran  seq ( # `  M ) ( ( x  e. 
_V ,  y  e. 
_V  |->  ( x ++  <" ( F `  x
) "> )
) ,  ( NN0 
X.  { ( M ++ 
<" ( F `  M ) "> ) } ) )  C_  _V
1918a1i 11 . . . 4  |-  ( ph  ->  ran  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x ++  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) )  C_  _V )
20 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
) ) )
2112, 17, 19, 20syl3anc 1264 . . 3  |-  ( ph  ->  ( lastS  o.  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) ) )  Fn  ( ZZ>=
`  ( # `  M
) ) )
22 fzouzdisj 11952 . . . 4  |-  ( ( 0..^ ( # `  M
) )  i^i  ( ZZ>=
`  ( # `  M
) ) )  =  (/)
2322a1i 11 . . 3  |-  ( ph  ->  ( ( 0..^ (
# `  M )
)  i^i  ( ZZ>= `  ( # `  M ) ) )  =  (/) )
24 sseqfv2.4 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  ( # `  M ) ) )
25 fvun2 5953 . . 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 ) ) )  =  (/)  /\  N  e.  ( ZZ>= `  ( # `  M ) ) ) )  -> 
( ( M  u.  ( lastS  o.  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x ++  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) ) ) ) `  N )  =  ( ( lastS  o.  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) ) ) `  N
) )
268, 21, 23, 24, 25syl112anc 1268 . 2  |-  ( ph  ->  ( ( M  u.  ( lastS  o.  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x ++  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) ) ) ) `  N )  =  ( ( lastS  o.  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) ) ) `  N
) )
27 fnfun 5691 . . . 4  |-  (  seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++ 
<" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M ++  <" ( F `
 M ) "> ) } ) )  Fn  ( ZZ>= `  ( # `  M ) )  ->  Fun  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) ) )
2817, 27syl 17 . . 3  |-  ( ph  ->  Fun  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x ++  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) ) )
29 fvex 5891 . . . . . . 7  |-  ( ( NN0  X.  { ( M ++  <" ( F `
 M ) "> ) } ) `
 ( # `  M
) )  e.  _V
3029a1i 11 . . . . . 6  |-  ( ph  ->  ( ( NN0  X.  { ( M ++  <" ( F `  M
) "> ) } ) `  ( # `
 M ) )  e.  _V )
31 ovex 6333 . . . . . . . . . 10  |-  ( x ++ 
<" ( F `  x ) "> )  e.  _V
3231rgen2w 2794 . . . . . . . . 9  |-  A. x  e.  _V  A. y  e. 
_V  ( x ++  <" ( F `  x
) "> )  e.  _V
3332a1i 11 . . . . . . . 8  |-  ( ph  ->  A. x  e.  _V  A. y  e.  _V  (
x ++  <" ( F `
 x ) "> )  e.  _V )
34 eqid 2429 . . . . . . . . 9  |-  ( x  e.  _V ,  y  e.  _V  |->  ( x ++ 
<" ( F `  x ) "> ) )  =  ( x  e.  _V , 
y  e.  _V  |->  ( x ++  <" ( F `
 x ) "> ) )
3534fmpt2 6874 . . . . . . . 8  |-  ( A. x  e.  _V  A. y  e.  _V  ( x ++  <" ( F `  x
) "> )  e.  _V  <->  ( x  e. 
_V ,  y  e. 
_V  |->  ( x ++  <" ( F `  x
) "> )
) : ( _V 
X.  _V ) --> _V )
3633, 35sylib 199 . . . . . . 7  |-  ( ph  ->  ( x  e.  _V ,  y  e.  _V  |->  ( x ++  <" ( F `  x ) "> ) ) : ( _V  X.  _V )
--> _V )
3736fovrnda 6454 . . . . . 6  |-  ( (
ph  /\  ( a  e.  _V  /\  b  e. 
_V ) )  -> 
( a ( x  e.  _V ,  y  e.  _V  |->  ( x ++ 
<" ( F `  x ) "> ) ) b )  e.  _V )
38 eqid 2429 . . . . . 6  |-  ( ZZ>= `  ( # `  M ) )  =  ( ZZ>= `  ( # `  M ) )
39 fvex 5891 . . . . . . 7  |-  ( ( NN0  X.  { ( M ++  <" ( F `
 M ) "> ) } ) `
 a )  e. 
_V
4039a1i 11 . . . . . 6  |-  ( (
ph  /\  a  e.  ( ZZ>= `  ( ( # `
 M )  +  1 ) ) )  ->  ( ( NN0 
X.  { ( M ++ 
<" ( F `  M ) "> ) } ) `  a
)  e.  _V )
4130, 37, 38, 15, 40seqf2 12229 . . . . 5  |-  ( ph  ->  seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++ 
<" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M ++  <" ( F `
 M ) "> ) } ) ) : ( ZZ>= `  ( # `  M ) ) --> _V )
42 fdm 5750 . . . . 5  |-  (  seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++ 
<" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M ++  <" ( F `
 M ) "> ) } ) ) : ( ZZ>= `  ( # `  M ) ) --> _V  ->  dom  seq ( # `  M ) ( ( x  e. 
_V ,  y  e. 
_V  |->  ( x ++  <" ( F `  x
) "> )
) ,  ( NN0 
X.  { ( M ++ 
<" ( F `  M ) "> ) } ) )  =  ( ZZ>= `  ( # `  M
) ) )
4341, 42syl 17 . . . 4  |-  ( ph  ->  dom  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x ++  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) )  =  ( ZZ>= `  ( # `  M ) ) )
4424, 43eleqtrrd 2520 . . 3  |-  ( ph  ->  N  e.  dom  seq ( # `  M ) ( ( x  e. 
_V ,  y  e. 
_V  |->  ( x ++  <" ( F `  x
) "> )
) ,  ( NN0 
X.  { ( M ++ 
<" ( F `  M ) "> ) } ) ) )
45 fvco 5957 . . 3  |-  ( ( Fun  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x ++  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) )  /\  N  e. 
dom  seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++ 
<" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M ++  <" ( F `
 M ) "> ) } ) ) )  ->  (
( lastS  o.  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) ) ) `  N
)  =  ( lastS  `  (  seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++ 
<" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M ++  <" ( F `
 M ) "> ) } ) ) `  N ) ) )
4628, 44, 45syl2anc 665 . 2  |-  ( ph  ->  ( ( lastS  o.  seq ( # `  M ) ( ( x  e. 
_V ,  y  e. 
_V  |->  ( x ++  <" ( F `  x
) "> )
) ,  ( NN0 
X.  { ( M ++ 
<" ( F `  M ) "> ) } ) ) ) `
 N )  =  ( lastS  `  (  seq ( # `  M ) ( ( x  e. 
_V ,  y  e. 
_V  |->  ( x ++  <" ( F `  x
) "> )
) ,  ( NN0 
X.  { ( M ++ 
<" ( F `  M ) "> ) } ) ) `  N ) ) )
476, 26, 463eqtrd 2474 1  |-  ( ph  ->  ( ( Mseqstr F ) `
 N )  =  ( lastS  `  (  seq ( # `  M ) ( ( x  e. 
_V ,  y  e. 
_V  |->  ( x ++  <" ( F `  x
) "> )
) ,  ( NN0 
X.  { ( M ++ 
<" ( F `  M ) "> ) } ) ) `  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   _Vcvv 3087    u. cun 3440    i^i cin 3441    C_ wss 3442   (/)c0 3767   {csn 4002    X. cxp 4852   `'ccnv 4853   dom cdm 4854   ran crn 4855   "cima 4857    o. ccom 4858   Fun wfun 5595    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   0cc0 9538   1c1 9539    + caddc 9541    - 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:  sseqp1  29054
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