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Theorem signstfvp 27136
Description: Zero-skipping sign in a word compared to a shorter word. (Contributed by Thierry Arnoux, 8-Oct-2018.)
Hypotheses
Ref Expression
signsv.p  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
signsv.w  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
signsv.t  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
signsv.v  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
Assertion
Ref Expression
signstfvp  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F concat  <" K "> ) ) `  N )  =  ( ( T `  F
) `  N )
)
Distinct variable groups:    a, b,  .+^    f, i, n, F    f, K, i, n    f, W, i, n    i, N, n
Allowed substitution hints:    .+^ ( f, i,
j, n)    T( f,
i, j, n, a, b)    F( j, a, b)    K( j, a, b)    N( f, j, a, b)    V( f, i, j, n, a, b)    W( j, a, b)

Proof of Theorem signstfvp
StepHypRef Expression
1 simp1 988 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  F  e. Word  RR )
21adantr 465 . . . . . 6  |-  ( ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  /\  i  e.  ( 0 ... N ) )  ->  F  e. Word  RR )
3 s1cl 12414 . . . . . . . 8  |-  ( K  e.  RR  ->  <" K ">  e. Word  RR )
433ad2ant2 1010 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  <" K ">  e. Word  RR )
54adantr 465 . . . . . 6  |-  ( ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  /\  i  e.  ( 0 ... N ) )  ->  <" K ">  e. Word  RR )
6 simp3 990 . . . . . . . 8  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  N  e.  ( 0..^ ( # `  F
) ) )
7 fzssfzo 27098 . . . . . . . 8  |-  ( N  e.  ( 0..^ (
# `  F )
)  ->  ( 0 ... N )  C_  ( 0..^ ( # `  F
) ) )
86, 7syl 16 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( 0 ... N )  C_  (
0..^ ( # `  F
) ) )
98sselda 3467 . . . . . 6  |-  ( ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  /\  i  e.  ( 0 ... N ) )  ->  i  e.  ( 0..^ ( # `  F
) ) )
10 ccatval1 12397 . . . . . 6  |-  ( ( F  e. Word  RR  /\  <" K ">  e. Word  RR  /\  i  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F concat  <" K "> ) `  i )  =  ( F `  i ) )
112, 5, 9, 10syl3anc 1219 . . . . 5  |-  ( ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  /\  i  e.  ( 0 ... N ) )  ->  ( ( F concat  <" K "> ) `  i )  =  ( F `  i ) )
1211fveq2d 5806 . . . 4  |-  ( ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  /\  i  e.  ( 0 ... N ) )  ->  (sgn `  (
( F concat  <" K "> ) `  i
) )  =  (sgn
`  ( F `  i ) ) )
1312mpteq2dva 4489 . . 3  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( i  e.  ( 0 ... N
)  |->  (sgn `  (
( F concat  <" K "> ) `  i
) ) )  =  ( i  e.  ( 0 ... N ) 
|->  (sgn `  ( F `  i ) ) ) )
1413oveq2d 6219 . 2  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( W  gsumg  ( i  e.  ( 0 ... N )  |->  (sgn `  ( ( F concat  <" K "> ) `  i
) ) ) )  =  ( W  gsumg  ( i  e.  ( 0 ... N )  |->  (sgn `  ( F `  i ) ) ) ) )
15 ccatcl 12395 . . . 4  |-  ( ( F  e. Word  RR  /\  <" K ">  e. Word  RR )  ->  ( F concat  <" K "> )  e. Word  RR )
161, 4, 15syl2anc 661 . . 3  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( F concat  <" K "> )  e. Word  RR )
17 lencl 12370 . . . . . . . 8  |-  ( F  e. Word  RR  ->  ( # `  F )  e.  NN0 )
1817nn0zd 10859 . . . . . . 7  |-  ( F  e. Word  RR  ->  ( # `  F )  e.  ZZ )
19 uzid 10989 . . . . . . 7  |-  ( (
# `  F )  e.  ZZ  ->  ( # `  F
)  e.  ( ZZ>= `  ( # `  F ) ) )
20 peano2uz 11022 . . . . . . 7  |-  ( (
# `  F )  e.  ( ZZ>= `  ( # `  F
) )  ->  (
( # `  F )  +  1 )  e.  ( ZZ>= `  ( # `  F
) ) )
21 fzoss2 11697 . . . . . . 7  |-  ( ( ( # `  F
)  +  1 )  e.  ( ZZ>= `  ( # `
 F ) )  ->  ( 0..^ (
# `  F )
)  C_  ( 0..^ ( ( # `  F
)  +  1 ) ) )
2218, 19, 20, 214syl 21 . . . . . 6  |-  ( F  e. Word  RR  ->  ( 0..^ ( # `  F
) )  C_  (
0..^ ( ( # `  F )  +  1 ) ) )
23223ad2ant1 1009 . . . . 5  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( 0..^ (
# `  F )
)  C_  ( 0..^ ( ( # `  F
)  +  1 ) ) )
2423, 6sseldd 3468 . . . 4  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  N  e.  ( 0..^ ( ( # `  F )  +  1 ) ) )
25 ccatlen 12396 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  <" K ">  e. Word  RR )  ->  ( # `
 ( F concat  <" K "> ) )  =  ( ( # `  F
)  +  ( # `  <" K "> ) ) )
261, 4, 25syl2anc 661 . . . . . 6  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( # `  ( F concat  <" K "> ) )  =  ( ( # `  F
)  +  ( # `  <" K "> ) ) )
27 s1len 12417 . . . . . . 7  |-  ( # `  <" K "> )  =  1
2827oveq2i 6214 . . . . . 6  |-  ( (
# `  F )  +  ( # `  <" K "> )
)  =  ( (
# `  F )  +  1 )
2926, 28syl6eq 2511 . . . . 5  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( # `  ( F concat  <" K "> ) )  =  ( ( # `  F
)  +  1 ) )
3029oveq2d 6219 . . . 4  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( 0..^ (
# `  ( F concat  <" K "> ) ) )  =  ( 0..^ ( (
# `  F )  +  1 ) ) )
3124, 30eleqtrrd 2545 . . 3  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  N  e.  ( 0..^ ( # `  ( F concat  <" K "> ) ) ) )
32 signsv.p . . . 4  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
33 signsv.w . . . 4  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
34 signsv.t . . . 4  |-  T  =  ( f  e. Word  RR  |->  ( n  e.  (
0..^ ( # `  f
) )  |->  ( W 
gsumg  ( i  e.  ( 0 ... n ) 
|->  (sgn `  ( f `  i ) ) ) ) ) )
35 signsv.v . . . 4  |-  V  =  ( f  e. Word  RR  |->  sum_ j  e.  ( 1..^ ( # `  f
) ) if ( ( ( T `  f ) `  j
)  =/=  ( ( T `  f ) `
 ( j  - 
1 ) ) ,  1 ,  0 ) )
3632, 33, 34, 35signstfval 27129 . . 3  |-  ( ( ( F concat  <" K "> )  e. Word  RR  /\  N  e.  ( 0..^ ( # `  ( F concat  <" K "> ) ) ) )  ->  ( ( T `
 ( F concat  <" K "> ) ) `  N )  =  ( W  gsumg  ( i  e.  ( 0 ... N ) 
|->  (sgn `  ( ( F concat  <" K "> ) `  i ) ) ) ) )
3716, 31, 36syl2anc 661 . 2  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F concat  <" K "> ) ) `  N )  =  ( W  gsumg  ( i  e.  ( 0 ... N ) 
|->  (sgn `  ( ( F concat  <" K "> ) `  i ) ) ) ) )
3832, 33, 34, 35signstfval 27129 . . 3  |-  ( ( F  e. Word  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  -> 
( ( T `  F ) `  N
)  =  ( W 
gsumg  ( i  e.  ( 0 ... N ) 
|->  (sgn `  ( F `  i ) ) ) ) )
391, 6, 38syl2anc 661 . 2  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 F ) `  N )  =  ( W  gsumg  ( i  e.  ( 0 ... N ) 
|->  (sgn `  ( F `  i ) ) ) ) )
4014, 37, 393eqtr4d 2505 1  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F concat  <" K "> ) ) `  N )  =  ( ( T `  F
) `  N )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648    C_ wss 3439   ifcif 3902   {cpr 3990   {ctp 3992   <.cop 3994    |-> cmpt 4461   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   RRcr 9395   0cc0 9396   1c1 9397    + caddc 9399    - cmin 9709   -ucneg 9710   ZZcz 10760   ZZ>=cuz 10975   ...cfz 11557  ..^cfzo 11668   #chash 12223  Word cword 12342   concat cconcat 12344   <"cs1 12345  sgncsgn 12696   sum_csu 13284   ndxcnx 14292   Basecbs 14295   +g cplusg 14360    gsumg cgsu 14501
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-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-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-hash 12224  df-word 12350  df-concat 12352  df-s1 12353
This theorem is referenced by:  signstfvneq0  27137  signstfvc  27139  signstfveq0  27142  signsvfn  27147
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