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Theorem sseqval 29294
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 ++ 
<" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M ++  <" ( 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 29290 . . 3  |- seqstr  =  ( m  e.  _V ,  f  e.  _V  |->  ( m  u.  ( lastS  o.  seq ( # `  m ) ( ( x  e. 
_V ,  y  e. 
_V  |->  ( x ++  <" ( f `  x
) "> )
) ,  ( NN0 
X.  { ( m ++ 
<" ( 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 ++ 
<" ( f `  x ) "> ) ) ,  ( NN0  X.  { ( m ++  <" ( f `
 m ) "> ) } ) ) ) ) ) )
3 simprl 772 . . 3  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  ->  m  =  M )
43fveq2d 5883 . . . . 5  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
( # `  m )  =  ( # `  M
) )
5 simp1rr 1096 . . . . . . . . 9  |-  ( ( ( ph  /\  (
m  =  M  /\  f  =  F )
)  /\  x  e.  _V  /\  y  e.  _V )  ->  f  =  F )
65fveq1d 5881 . . . . . . . 8  |-  ( ( ( ph  /\  (
m  =  M  /\  f  =  F )
)  /\  x  e.  _V  /\  y  e.  _V )  ->  ( f `  x )  =  ( F `  x ) )
76s1eqd 12793 . . . . . . 7  |-  ( ( ( ph  /\  (
m  =  M  /\  f  =  F )
)  /\  x  e.  _V  /\  y  e.  _V )  ->  <" ( f `
 x ) ">  =  <" ( F `  x ) "> )
87oveq2d 6324 . . . . . 6  |-  ( ( ( ph  /\  (
m  =  M  /\  f  =  F )
)  /\  x  e.  _V  /\  y  e.  _V )  ->  ( x ++  <" ( f `  x
) "> )  =  ( x ++  <" ( F `  x
) "> )
)
98mpt2eq3dva 6374 . . . . 5  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
( x  e.  _V ,  y  e.  _V  |->  ( x ++  <" (
f `  x ) "> ) )  =  ( x  e.  _V ,  y  e.  _V  |->  ( x ++  <" ( F `  x ) "> ) ) )
10 simprr 774 . . . . . . . . . 10  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
f  =  F )
1110, 3fveq12d 5885 . . . . . . . . 9  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
( f `  m
)  =  ( F `
 M ) )
1211s1eqd 12793 . . . . . . . 8  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  ->  <" ( f `  m ) ">  =  <" ( F `
 M ) "> )
133, 12oveq12d 6326 . . . . . . 7  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
( m ++  <" (
f `  m ) "> )  =  ( M ++  <" ( F `
 M ) "> ) )
1413sneqd 3971 . . . . . 6  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  ->  { ( m ++  <" ( f `  m
) "> ) }  =  { ( M ++  <" ( F `
 M ) "> ) } )
1514xpeq2d 4863 . . . . 5  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
( NN0  X.  { ( m ++  <" ( f `
 m ) "> ) } )  =  ( NN0  X.  { ( M ++  <" ( F `  M
) "> ) } ) )
164, 9, 15seqeq123d 12260 . . . 4  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  ->  seq ( # `  m
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++ 
<" ( f `  x ) "> ) ) ,  ( NN0  X.  { ( m ++  <" ( f `
 m ) "> ) } ) )  =  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) ) )
1716coeq2d 5002 . . 3  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
( lastS  o.  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 ) "> ) } ) ) ) )
183, 17uneq12d 3580 . 2  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
( m  u.  ( lastS  o. 
seq ( # `  m
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++ 
<" ( f `  x ) "> ) ) ,  ( NN0  X.  { ( m ++  <" ( f `
 m ) "> ) } ) ) ) )  =  ( M  u.  ( lastS  o. 
seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++ 
<" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M ++  <" ( F `
 M ) "> ) } ) ) ) ) )
19 sseqval.2 . . 3  |-  ( ph  ->  M  e. Word  S )
20 elex 3040 . . 3  |-  ( M  e. Word  S  ->  M  e.  _V )
2119, 20syl 17 . 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 12729 . . . . 5  |-  ( S  e.  _V  -> Word  S  e. 
_V )
26 inex1g 4539 . . . . 5  |-  (Word  S  e.  _V  ->  (Word  S  i^i  ( `' # " ( ZZ>=
`  ( # `  M
) ) ) )  e.  _V )
2724, 25, 263syl 18 . . . 4  |-  ( ph  ->  (Word  S  i^i  ( `' # " ( ZZ>= `  ( # `  M ) ) ) )  e. 
_V )
2823, 27syl5eqel 2553 . . 3  |-  ( ph  ->  W  e.  _V )
29 fex 6155 . . 3  |-  ( ( F : W --> S  /\  W  e.  _V )  ->  F  e.  _V )
3022, 28, 29syl2anc 673 . 2  |-  ( ph  ->  F  e.  _V )
31 df-lsw 12712 . . . . . 6  |- lastS  =  ( x  e.  _V  |->  ( x `  ( (
# `  x )  -  1 ) ) )
3231funmpt2 5626 . . . . 5  |-  Fun lastS
3332a1i 11 . . . 4  |-  ( ph  ->  Fun lastS  )
34 seqex 12253 . . . . 5  |-  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) )  e.  _V
3534a1i 11 . . . 4  |-  ( ph  ->  seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++ 
<" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M ++  <" ( F `
 M ) "> ) } ) )  e.  _V )
36 cofunexg 6776 . . . 4  |-  ( ( Fun lastS  /\  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x ++  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) )  e.  _V )  ->  ( lastS  o.  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) ) )  e.  _V )
3733, 35, 36syl2anc 673 . . 3  |-  ( ph  ->  ( lastS  o.  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) ) )  e.  _V )
38 unexg 6611 . . 3  |-  ( ( M  e.  _V  /\  ( lastS  o.  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x ++  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) ) )  e.  _V )  ->  ( M  u.  ( lastS  o.  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x ++  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) ) ) )  e. 
_V )
3921, 37, 38syl2anc 673 . 2  |-  ( ph  ->  ( M  u.  ( lastS  o. 
seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++ 
<" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M ++  <" ( F `
 M ) "> ) } ) ) ) )  e. 
_V )
402, 18, 21, 30, 39ovmpt2d 6443 1  |-  ( ph  ->  ( Mseqstr F )  =  ( M  u.  ( lastS  o.  seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++ 
<" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M ++  <" ( F `
 M ) "> ) } ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   _Vcvv 3031    u. cun 3388    i^i cin 3389   {csn 3959    X. cxp 4837   `'ccnv 4838   "cima 4842    o. ccom 4843   Fun wfun 5583   -->wf 5585   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   1c1 9558    - cmin 9880   NN0cn0 10893   ZZ>=cuz 11182    seqcseq 12251   #chash 12553  Word cword 12703   lastS clsw 12704   ++ cconcat 12705   <"cs1 12706  seqstrcsseq 29289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-map 7492  df-pm 7493  df-neg 9883  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-seq 12252  df-word 12711  df-lsw 12712  df-s1 12714  df-sseq 29290
This theorem is referenced by:  sseqfv1  29295  sseqfn  29296  sseqf  29298  sseqfv2  29300
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