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Theorem sseqval 26786
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 26782 . . 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 5710 . . . . 5  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
( # `  m )  =  ( # `  M
) )
5 simp1rr 1054 . . . . . . . . 9  |-  ( ( ( ph  /\  (
m  =  M  /\  f  =  F )
)  /\  x  e.  _V  /\  y  e.  _V )  ->  f  =  F )
65fveq1d 5708 . . . . . . . 8  |-  ( ( ( ph  /\  (
m  =  M  /\  f  =  F )
)  /\  x  e.  _V  /\  y  e.  _V )  ->  ( f `  x )  =  ( F `  x ) )
76s1eqd 12307 . . . . . . 7  |-  ( ( ( ph  /\  (
m  =  M  /\  f  =  F )
)  /\  x  e.  _V  /\  y  e.  _V )  ->  <" ( f `
 x ) ">  =  <" ( F `  x ) "> )
87oveq2d 6122 . . . . . 6  |-  ( ( ( ph  /\  (
m  =  M  /\  f  =  F )
)  /\  x  e.  _V  /\  y  e.  _V )  ->  ( x concat  <" (
f `  x ) "> )  =  ( x concat  <" ( F `
 x ) "> ) )
98mpt2eq3dva 6165 . . . . 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 5712 . . . . . . . . 9  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
( f `  m
)  =  ( F `
 M ) )
1211s1eqd 12307 . . . . . . . 8  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  ->  <" ( f `  m ) ">  =  <" ( F `
 M ) "> )
133, 12oveq12d 6124 . . . . . . 7  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
( m concat  <" (
f `  m ) "> )  =  ( M concat  <" ( F `
 M ) "> ) )
1413sneqd 3904 . . . . . 6  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  ->  { ( m concat  <" (
f `  m ) "> ) }  =  { ( M concat  <" ( F `  M ) "> ) } )
1514xpeq2d 4879 . . . . 5  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
( NN0  X.  { ( m concat  <" ( f `
 m ) "> ) } )  =  ( NN0  X.  { ( M concat  <" ( F `  M ) "> ) } ) )
164, 9, 15seqeq123d 11830 . . . 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 5017 . . 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 3526 . 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 2996 . . 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 12259 . . . . 5  |-  ( S  e.  _V  -> Word  S  e. 
_V )
26 inex1g 4450 . . . . 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 2527 . . 3  |-  ( ph  ->  W  e.  _V )
29 fex 5965 . . 3  |-  ( ( F : W --> S  /\  W  e.  _V )  ->  F  e.  _V )
3022, 28, 29syl2anc 661 . 2  |-  ( ph  ->  F  e.  _V )
31 df-lsw 12245 . . . . . 6  |- lastS  =  ( x  e.  _V  |->  ( x `  ( (
# `  x )  -  1 ) ) )
3231funmpt2 5470 . . . . 5  |-  Fun lastS
3332a1i 11 . . . 4  |-  ( ph  ->  Fun lastS  )
34 seqex 11823 . . . . 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 6556 . . . 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 6396 . . 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 6233 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 965    = wceq 1369    e. wcel 1756   _Vcvv 2987    u. cun 3341    i^i cin 3342   {csn 3892    X. cxp 4853   `'ccnv 4854   "cima 4858    o. ccom 4859   Fun wfun 5427   -->wf 5429   ` cfv 5433  (class class class)co 6106    e. cmpt2 6108   1c1 9298    - cmin 9610   NN0cn0 10594   ZZ>=cuz 10876    seqcseq 11821   #chash 12118  Word cword 12236   lastS clsw 12237   concat cconcat 12238   <"cs1 12239  seqstrcsseq 26781
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 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-inf2 7862  ax-cnex 9353  ax-resscn 9354
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-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-map 7231  df-pm 7232  df-neg 9613  df-z 10662  df-uz 10877  df-fz 11453  df-fzo 11564  df-seq 11822  df-word 12244  df-lsw 12245  df-s1 12247  df-sseq 26782
This theorem is referenced by:  sseqfv1  26787  sseqfn  26788  sseqf  26790  sseqfv2  26792
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