Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sseqfv1 Structured version   Unicode version

Theorem sseqfv1 26936
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 26935 . . 3  |-  ( 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 ) "> ) } ) ) ) ) )
65fveq1d 5804 . 2  |-  ( ph  ->  ( ( Mseqstr F ) `
 N )  =  ( ( M  u.  ( lastS  o.  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x concat  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) ) ) `  N ) )
7 wrdfn 12368 . . . 4  |-  ( M  e. Word  S  ->  M  Fn  ( 0..^ ( # `  M ) ) )
82, 7syl 16 . . 3  |-  ( ph  ->  M  Fn  ( 0..^ ( # `  M
) ) )
9 fvex 5812 . . . . . 6  |-  ( x `
 ( ( # `  x )  -  1 ) )  e.  _V
10 df-lsw 12351 . . . . . 6  |- lastS  =  ( x  e.  _V  |->  ( x `  ( (
# `  x )  -  1 ) ) )
119, 10fnmpti 5650 . . . . 5  |- lastS  Fn  _V
1211a1i 11 . . . 4  |-  ( ph  -> lastS 
Fn  _V )
13 lencl 12370 . . . . . . 7  |-  ( M  e. Word  S  ->  ( # `
 M )  e. 
NN0 )
142, 13syl 16 . . . . . 6  |-  ( ph  ->  ( # `  M
)  e.  NN0 )
1514nn0zd 10859 . . . . 5  |-  ( ph  ->  ( # `  M
)  e.  ZZ )
16 seqfn 11938 . . . . 5  |-  ( (
# `  M )  e.  ZZ  ->  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) )  Fn  ( ZZ>= `  ( # `  M ) ) )
1715, 16syl 16 . . . 4  |-  ( ph  ->  seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M concat  <" ( F `
 M ) "> ) } ) )  Fn  ( ZZ>= `  ( # `  M ) ) )
18 ssv 3487 . . . . 5  |-  ran  seq ( # `  M ) ( ( x  e. 
_V ,  y  e. 
_V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) )  C_  _V
1918a1i 11 . . . 4  |-  ( ph  ->  ran  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x concat  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) )  C_  _V )
20 fnco 5630 . . . 4  |-  ( ( lastS 
Fn  _V  /\  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) )  Fn  ( ZZ>= `  ( # `  M ) )  /\  ran  seq ( # `  M ) ( ( x  e. 
_V ,  y  e. 
_V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) )  C_  _V )  ->  ( lastS  o.  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) )  Fn  ( ZZ>=
`  ( # `  M
) ) )
2112, 17, 19, 20syl3anc 1219 . . 3  |-  ( ph  ->  ( lastS  o.  seq ( # `
 M ) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) )  Fn  ( ZZ>=
`  ( # `  M
) ) )
22 fzouzdisj 11705 . . . . 5  |-  ( ( 0..^ ( # `  M
) )  i^i  ( ZZ>=
`  ( # `  M
) ) )  =  (/)
2322a1i 11 . . . 4  |-  ( ph  ->  ( ( 0..^ (
# `  M )
)  i^i  ( ZZ>= `  ( # `  M ) ) )  =  (/) )
24 sseqfv1.4 . . . 4  |-  ( ph  ->  N  e.  ( 0..^ ( # `  M
) ) )
2523, 24jca 532 . . 3  |-  ( ph  ->  ( ( ( 0..^ ( # `  M
) )  i^i  ( ZZ>=
`  ( # `  M
) ) )  =  (/)  /\  N  e.  ( 0..^ ( # `  M
) ) ) )
26 fvun1 5874 . . 3  |-  ( ( M  Fn  ( 0..^ ( # `  M
) )  /\  ( lastS  o. 
seq ( # `  M
) ( ( x  e.  _V ,  y  e.  _V  |->  ( x concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  { ( M concat  <" ( 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 concat  <" ( F `  x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) ) ) `  N )  =  ( M `  N ) )
278, 21, 25, 26syl3anc 1219 . 2  |-  ( ph  ->  ( ( M  u.  ( lastS  o.  seq ( # `  M ) ( ( x  e.  _V , 
y  e.  _V  |->  ( x concat  <" ( F `
 x ) "> ) ) ,  ( NN0  X.  {
( M concat  <" ( F `  M ) "> ) } ) ) ) ) `  N )  =  ( M `  N ) )
286, 27eqtrd 2495 1  |-  ( ph  ->  ( ( Mseqstr F ) `
 N )  =  ( M `  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078    u. cun 3437    i^i cin 3438    C_ wss 3439   (/)c0 3748   {csn 3988    X. cxp 4949   `'ccnv 4950   ran crn 4952   "cima 4954    o. ccom 4955    Fn wfn 5524   -->wf 5525   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   0cc0 9396   1c1 9397    - cmin 9709   NN0cn0 10693   ZZcz 10760   ZZ>=cuz 10975  ..^cfzo 11668    seqcseq 11926   #chash 12223  Word cword 12342   lastS clsw 12343   concat cconcat 12344   <"cs1 12345  seqstrcsseq 26930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-fzo 11669  df-seq 11927  df-hash 12224  df-word 12350  df-lsw 12351  df-s1 12353  df-sseq 26931
This theorem is referenced by:  sseqfres  26940  fib0  26946  fib1  26947
  Copyright terms: Public domain W3C validator