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Theorem sseqval 28833
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 28829 . . 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 756 . . 3  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  ->  m  =  M )
43fveq2d 5853 . . . . 5  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
( # `  m )  =  ( # `  M
) )
5 simp1rr 1063 . . . . . . . . 9  |-  ( ( ( ph  /\  (
m  =  M  /\  f  =  F )
)  /\  x  e.  _V  /\  y  e.  _V )  ->  f  =  F )
65fveq1d 5851 . . . . . . . 8  |-  ( ( ( ph  /\  (
m  =  M  /\  f  =  F )
)  /\  x  e.  _V  /\  y  e.  _V )  ->  ( f `  x )  =  ( F `  x ) )
76s1eqd 12667 . . . . . . 7  |-  ( ( ( ph  /\  (
m  =  M  /\  f  =  F )
)  /\  x  e.  _V  /\  y  e.  _V )  ->  <" ( f `
 x ) ">  =  <" ( F `  x ) "> )
87oveq2d 6294 . . . . . 6  |-  ( ( ( ph  /\  (
m  =  M  /\  f  =  F )
)  /\  x  e.  _V  /\  y  e.  _V )  ->  ( x ++  <" ( f `  x
) "> )  =  ( x ++  <" ( F `  x
) "> )
)
98mpt2eq3dva 6342 . . . . 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 758 . . . . . . . . . 10  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
f  =  F )
1110, 3fveq12d 5855 . . . . . . . . 9  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
( f `  m
)  =  ( F `
 M ) )
1211s1eqd 12667 . . . . . . . 8  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  ->  <" ( f `  m ) ">  =  <" ( F `
 M ) "> )
133, 12oveq12d 6296 . . . . . . 7  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
( m ++  <" (
f `  m ) "> )  =  ( M ++  <" ( F `
 M ) "> ) )
1413sneqd 3984 . . . . . 6  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  ->  { ( m ++  <" ( f `  m
) "> ) }  =  { ( M ++  <" ( F `
 M ) "> ) } )
1514xpeq2d 4847 . . . . 5  |-  ( (
ph  /\  ( m  =  M  /\  f  =  F ) )  -> 
( NN0  X.  { ( m ++  <" ( f `
 m ) "> ) } )  =  ( NN0  X.  { ( M ++  <" ( F `  M
) "> ) } ) )
164, 9, 15seqeq123d 12160 . . . 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 4986 . . 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 3598 . 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 3068 . . 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 12609 . . . . 5  |-  ( S  e.  _V  -> Word  S  e. 
_V )
26 inex1g 4537 . . . . 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 2494 . . 3  |-  ( ph  ->  W  e.  _V )
29 fex 6126 . . 3  |-  ( ( F : W --> S  /\  W  e.  _V )  ->  F  e.  _V )
3022, 28, 29syl2anc 659 . 2  |-  ( ph  ->  F  e.  _V )
31 df-lsw 12592 . . . . . 6  |- lastS  =  ( x  e.  _V  |->  ( x `  ( (
# `  x )  -  1 ) ) )
3231funmpt2 5606 . . . . 5  |-  Fun lastS
3332a1i 11 . . . 4  |-  ( ph  ->  Fun lastS  )
34 seqex 12153 . . . . 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 6748 . . . 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 659 . . 3  |-  ( ph  ->  ( lastS  o.  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x ++  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) ) )  e.  _V )
38 unexg 6583 . . 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 659 . 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 6411 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   _Vcvv 3059    u. cun 3412    i^i cin 3413   {csn 3972    X. cxp 4821   `'ccnv 4822   "cima 4826    o. ccom 4827   Fun wfun 5563   -->wf 5565   ` cfv 5569  (class class class)co 6278    |-> cmpt2 6280   1c1 9523    - cmin 9841   NN0cn0 10836   ZZ>=cuz 11127    seqcseq 12151   #chash 12452  Word cword 12583   lastS clsw 12584   ++ cconcat 12585   <"cs1 12586  seqstrcsseq 28828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-map 7459  df-pm 7460  df-neg 9844  df-z 10906  df-uz 11128  df-fz 11727  df-fzo 11855  df-seq 12152  df-word 12591  df-lsw 12592  df-s1 12594  df-sseq 28829
This theorem is referenced by:  sseqfv1  28834  sseqfn  28835  sseqf  28837  sseqfv2  28839
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