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Theorem sseqfv1 28715
Description: Value of the strong sequence builder function at one of its initial values. (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 )
sseqfv1.4  |-  ( ph  ->  N  e.  ( 0..^ ( # `  M
) ) )
Assertion
Ref Expression
sseqfv1  |-  ( ph  ->  ( ( Mseqstr F ) `
 N )  =  ( M `  N
) )

Proof of Theorem sseqfv1
Dummy variables  x  y 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 28714 . . 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 5805 . 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 12517 . . . 4  |-  ( M  e. Word  S  ->  M  Fn  ( 0..^ ( # `  M ) ) )
82, 7syl 17 . . 3  |-  ( ph  ->  M  Fn  ( 0..^ ( # `  M
) ) )
9 fvex 5813 . . . . . 6  |-  ( x `
 ( ( # `  x )  -  1 ) )  e.  _V
10 df-lsw 12497 . . . . . 6  |- lastS  =  ( x  e.  _V  |->  ( x `  ( (
# `  x )  -  1 ) ) )
119, 10fnmpti 5646 . . . . 5  |- lastS  Fn  _V
1211a1i 11 . . . 4  |-  ( ph  -> lastS 
Fn  _V )
13 lencl 12519 . . . . . . 7  |-  ( M  e. Word  S  ->  ( # `
 M )  e. 
NN0 )
142, 13syl 17 . . . . . 6  |-  ( ph  ->  ( # `  M
)  e.  NN0 )
1514nn0zd 10924 . . . . 5  |-  ( ph  ->  ( # `  M
)  e.  ZZ )
16 seqfn 12071 . . . . 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 3459 . . . . 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 5624 . . . 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 1228 . . 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 11804 . . . 4  |-  ( ( 0..^ ( # `  M
) )  i^i  ( ZZ>=
`  ( # `  M
) ) )  =  (/)
2322a1i 11 . . 3  |-  ( ph  ->  ( ( 0..^ (
# `  M )
)  i^i  ( ZZ>= `  ( # `  M ) ) )  =  (/) )
24 sseqfv1.4 . . 3  |-  ( ph  ->  N  e.  ( 0..^ ( # `  M
) ) )
25 fvun1 5874 . . 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.  ( 0..^ ( # `  M
) ) ) )  ->  ( ( M  u.  ( lastS  o.  seq ( # `  M ) ( ( x  e. 
_V ,  y  e. 
_V  |->  ( x ++  <" ( F `  x
) "> )
) ,  ( NN0 
X.  { ( M ++ 
<" ( F `  M ) "> ) } ) ) ) ) `  N )  =  ( M `  N ) )
268, 21, 23, 24, 25syl112anc 1232 . 2  |-  ( ph  ->  ( ( M  u.  ( lastS  o.  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x ++  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M ++  <" ( F `  M ) "> ) } ) ) ) ) `  N )  =  ( M `  N ) )
276, 26eqtrd 2441 1  |-  ( ph  ->  ( ( Mseqstr F ) `
 N )  =  ( M `  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1403    e. wcel 1840   _Vcvv 3056    u. cun 3409    i^i cin 3410    C_ wss 3411   (/)c0 3735   {csn 3969    X. cxp 4938   `'ccnv 4939   ran crn 4941   "cima 4943    o. ccom 4944    Fn wfn 5518   -->wf 5519   ` cfv 5523  (class class class)co 6232    |-> cmpt2 6234   0cc0 9440   1c1 9441    - cmin 9759   NN0cn0 10754   ZZcz 10823   ZZ>=cuz 11043  ..^cfzo 11765    seqcseq 12059   #chash 12357  Word cword 12488   lastS clsw 12489   ++ cconcat 12490   <"cs1 12491  seqstrcsseq 28709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-inf2 8009  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-1o 7085  df-oadd 7089  df-er 7266  df-map 7377  df-pm 7378  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-card 8270  df-cda 8498  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-nn 10495  df-2 10553  df-n0 10755  df-z 10824  df-uz 11044  df-fz 11642  df-fzo 11766  df-seq 12060  df-hash 12358  df-word 12496  df-lsw 12497  df-s1 12499  df-sseq 28710
This theorem is referenced by:  sseqfres  28719  fib0  28725  fib1  28726
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