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Theorem sseqval 27967
Description: Value of the strong sequence builder function. The set 
W represents here the words of length greater than or equal to the lenght of the initial sequence  M (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 )
Assertion
Ref Expression
sseqval  |-  ( 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 ) "> ) } ) ) ) ) )
Distinct variable groups:    x, y, F    x, M, y    ph, x, y
Allowed substitution hints:    S( x, y)    W( x, y)

Proof of Theorem sseqval
Dummy variables  f  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sseq 27963 . . 3  |- seqstr  =  ( m  e.  _V ,  f  e.  _V  |->  ( m  u.  ( lastS  o.  seq ( # `  m ) ( ( x  e. 
_V ,  y  e. 
_V  |->  ( x concat  <" (
f `  x ) "> ) ) ,  ( NN0  X.  {
( m concat  <" (
f `  m ) "> ) } ) ) ) ) )
21a1i 11 . 2  |-  ( ph  -> seqstr  =  ( m  e.  _V ,  f  e.  _V  |->  ( m  u.  ( lastS  o. 
seq ( # `  m
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( f `  x ) "> ) ) ,  ( NN0  X.  { ( m concat  <" ( f `
 m ) "> ) } ) ) ) ) ) )
3 simprl 755 . . 3  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  ->  m  =  M )
43fveq2d 5868 . . . . 5  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
( # `  m )  =  ( # `  M
) )
5 simp1rr 1062 . . . . . . . . 9  |-  ( ( ( ph  /\  (
m  =  M  /\  f  =  F )
)  /\  x  e.  _V  /\  y  e.  _V )  ->  f  =  F )
65fveq1d 5866 . . . . . . . 8  |-  ( ( ( ph  /\  (
m  =  M  /\  f  =  F )
)  /\  x  e.  _V  /\  y  e.  _V )  ->  ( f `  x )  =  ( F `  x ) )
76s1eqd 12572 . . . . . . 7  |-  ( ( ( ph  /\  (
m  =  M  /\  f  =  F )
)  /\  x  e.  _V  /\  y  e.  _V )  ->  <" ( f `
 x ) ">  =  <" ( F `  x ) "> )
87oveq2d 6298 . . . . . 6  |-  ( ( ( ph  /\  (
m  =  M  /\  f  =  F )
)  /\  x  e.  _V  /\  y  e.  _V )  ->  ( x concat  <" (
f `  x ) "> )  =  ( x concat  <" ( F `
 x ) "> ) )
98mpt2eq3dva 6343 . . . . 5  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" (
f `  x ) "> ) )  =  ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) )
10 simprr 756 . . . . . . . . . 10  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
f  =  F )
1110, 3fveq12d 5870 . . . . . . . . 9  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
( f `  m
)  =  ( F `
 M ) )
1211s1eqd 12572 . . . . . . . 8  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  ->  <" ( f `  m ) ">  =  <" ( F `
 M ) "> )
133, 12oveq12d 6300 . . . . . . 7  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
( m concat  <" (
f `  m ) "> )  =  ( M concat  <" ( F `
 M ) "> ) )
1413sneqd 4039 . . . . . 6  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  ->  { ( m concat  <" (
f `  m ) "> ) }  =  { ( M concat  <" ( F `  M ) "> ) } )
1514xpeq2d 5023 . . . . 5  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
( NN0  X.  { ( m concat  <" ( f `
 m ) "> ) } )  =  ( NN0  X.  { ( M concat  <" ( F `  M ) "> ) } ) )
164, 9, 15seqeq123d 12080 . . . 4  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  ->  seq ( # `  m
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( f `  x ) "> ) ) ,  ( NN0  X.  { ( m concat  <" ( f `
 m ) "> ) } ) )  =  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) )
1716coeq2d 5163 . . 3  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
( lastS  o.  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 ) "> ) } ) ) ) )
183, 17uneq12d 3659 . 2  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
( m  u.  ( lastS  o. 
seq ( # `  m
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( f `  x ) "> ) ) ,  ( NN0  X.  { ( m concat  <" ( f `
 m ) "> ) } ) ) ) )  =  ( M  u.  ( lastS  o. 
seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M concat  <" ( F `
 M ) "> ) } ) ) ) ) )
19 sseqval.2 . . 3  |-  ( ph  ->  M  e. Word  S )
20 elex 3122 . . 3  |-  ( M  e. Word  S  ->  M  e.  _V )
2119, 20syl 16 . 2  |-  ( ph  ->  M  e.  _V )
22 sseqval.4 . . 3  |-  ( ph  ->  F : W --> S )
23 sseqval.3 . . . 4  |-  W  =  (Word  S  i^i  ( `' # " ( ZZ>= `  ( # `  M ) ) ) )
24 sseqval.1 . . . . 5  |-  ( ph  ->  S  e.  _V )
25 wrdexg 12519 . . . . 5  |-  ( S  e.  _V  -> Word  S  e. 
_V )
26 inex1g 4590 . . . . 5  |-  (Word  S  e.  _V  ->  (Word  S  i^i  ( `' # " ( ZZ>=
`  ( # `  M
) ) ) )  e.  _V )
2724, 25, 263syl 20 . . . 4  |-  ( ph  ->  (Word  S  i^i  ( `' # " ( ZZ>= `  ( # `  M ) ) ) )  e. 
_V )
2823, 27syl5eqel 2559 . . 3  |-  ( ph  ->  W  e.  _V )
29 fex 6131 . . 3  |-  ( ( F : W --> S  /\  W  e.  _V )  ->  F  e.  _V )
3022, 28, 29syl2anc 661 . 2  |-  ( ph  ->  F  e.  _V )
31 df-lsw 12505 . . . . . 6  |- lastS  =  ( x  e.  _V  |->  ( x `  ( (
# `  x )  -  1 ) ) )
3231funmpt2 5623 . . . . 5  |-  Fun lastS
3332a1i 11 . . . 4  |-  ( ph  ->  Fun lastS  )
34 seqex 12073 . . . . 5  |-  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) )  e.  _V
3534a1i 11 . . . 4  |-  ( ph  ->  seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M concat  <" ( F `
 M ) "> ) } ) )  e.  _V )
36 cofunexg 6745 . . . 4  |-  ( ( Fun lastS  /\  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x concat  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) )  e.  _V )  ->  ( lastS  o.  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) )  e.  _V )
3733, 35, 36syl2anc 661 . . 3  |-  ( ph  ->  ( lastS  o.  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) )  e.  _V )
38 unexg 6583 . . 3  |-  ( ( M  e.  _V  /\  ( lastS  o.  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x concat  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) )  e.  _V )  ->  ( M  u.  ( lastS  o.  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x concat  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) ) )  e. 
_V )
3921, 37, 38syl2anc 661 . 2  |-  ( ph  ->  ( M  u.  ( lastS  o. 
seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M concat  <" ( F `
 M ) "> ) } ) ) ) )  e. 
_V )
402, 18, 21, 30, 39ovmpt2d 6412 1  |-  ( 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 ) "> ) } ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3113    u. cun 3474    i^i cin 3475   {csn 4027    X. cxp 4997   `'ccnv 4998   "cima 5002    o. ccom 5003   Fun wfun 5580   -->wf 5582   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   1c1 9489    - cmin 9801   NN0cn0 10791   ZZ>=cuz 11078    seqcseq 12071   #chash 12369  Word cword 12496   lastS clsw 12497   concat cconcat 12498   <"cs1 12499  seqstrcsseq 27962
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 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545
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-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-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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-map 7419  df-pm 7420  df-neg 9804  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-seq 12072  df-word 12504  df-lsw 12505  df-s1 12507  df-sseq 27963
This theorem is referenced by:  sseqfv1  27968  sseqfn  27969  sseqf  27971  sseqfv2  27973
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