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Theorem signstfvp 28703
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 ++  <" 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 996 . . . . . . 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 12622 . . . . . . . 8  |-  ( K  e.  RR  ->  <" K ">  e. Word  RR )
433ad2ant2 1018 . . . . . . 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 998 . . . . . . . 8  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  N  e.  ( 0..^ ( # `  F
) ) )
7 fzssfzo 28665 . . . . . . . 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 3499 . . . . . 6  |-  ( ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  /\  i  e.  ( 0 ... N ) )  ->  i  e.  ( 0..^ ( # `  F
) ) )
10 ccatval1 12603 . . . . . 6  |-  ( ( F  e. Word  RR  /\  <" K ">  e. Word  RR  /\  i  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F ++ 
<" K "> ) `  i )  =  ( F `  i ) )
112, 5, 9, 10syl3anc 1228 . . . . 5  |-  ( ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  /\  i  e.  ( 0 ... N ) )  ->  ( ( F ++ 
<" K "> ) `  i )  =  ( F `  i ) )
1211fveq2d 5876 . . . 4  |-  ( ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F
) ) )  /\  i  e.  ( 0 ... N ) )  ->  (sgn `  (
( F ++  <" K "> ) `  i
) )  =  (sgn
`  ( F `  i ) ) )
1312mpteq2dva 4543 . . 3  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( i  e.  ( 0 ... N
)  |->  (sgn `  (
( F ++  <" K "> ) `  i
) ) )  =  ( i  e.  ( 0 ... N ) 
|->  (sgn `  ( F `  i ) ) ) )
1413oveq2d 6312 . 2  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( W  gsumg  ( i  e.  ( 0 ... N )  |->  (sgn `  ( ( F ++  <" K "> ) `  i ) ) ) )  =  ( W 
gsumg  ( i  e.  ( 0 ... N ) 
|->  (sgn `  ( F `  i ) ) ) ) )
15 ccatcl 12601 . . . 4  |-  ( ( F  e. Word  RR  /\  <" K ">  e. Word  RR )  ->  ( F ++  <" K "> )  e. Word  RR )
161, 4, 15syl2anc 661 . . 3  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( F ++  <" K "> )  e. Word  RR )
17 lencl 12568 . . . . . . . 8  |-  ( F  e. Word  RR  ->  ( # `  F )  e.  NN0 )
1817nn0zd 10988 . . . . . . 7  |-  ( F  e. Word  RR  ->  ( # `  F )  e.  ZZ )
19 uzid 11120 . . . . . . 7  |-  ( (
# `  F )  e.  ZZ  ->  ( # `  F
)  e.  ( ZZ>= `  ( # `  F ) ) )
20 peano2uz 11159 . . . . . . 7  |-  ( (
# `  F )  e.  ( ZZ>= `  ( # `  F
) )  ->  (
( # `  F )  +  1 )  e.  ( ZZ>= `  ( # `  F
) ) )
21 fzoss2 11851 . . . . . . 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 1017 . . . . 5  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( 0..^ (
# `  F )
)  C_  ( 0..^ ( ( # `  F
)  +  1 ) ) )
2423, 6sseldd 3500 . . . 4  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  N  e.  ( 0..^ ( ( # `  F )  +  1 ) ) )
25 ccatlen 12602 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  <" K ">  e. Word  RR )  ->  ( # `
 ( F ++  <" K "> )
)  =  ( (
# `  F )  +  ( # `  <" K "> )
) )
261, 4, 25syl2anc 661 . . . . . 6  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( # `  ( F ++  <" K "> ) )  =  ( ( # `  F
)  +  ( # `  <" K "> ) ) )
27 s1len 12625 . . . . . . 7  |-  ( # `  <" K "> )  =  1
2827oveq2i 6307 . . . . . 6  |-  ( (
# `  F )  +  ( # `  <" K "> )
)  =  ( (
# `  F )  +  1 )
2926, 28syl6eq 2514 . . . . 5  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( # `  ( F ++  <" K "> ) )  =  ( ( # `  F
)  +  1 ) )
3029oveq2d 6312 . . . 4  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( 0..^ (
# `  ( F ++  <" K "> ) ) )  =  ( 0..^ ( (
# `  F )  +  1 ) ) )
3124, 30eleqtrrd 2548 . . 3  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  N  e.  ( 0..^ ( # `  ( F ++  <" 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 28696 . . 3  |-  ( ( ( F ++  <" K "> )  e. Word  RR  /\  N  e.  ( 0..^ ( # `  ( F ++  <" K "> ) ) ) )  ->  ( ( T `
 ( F ++  <" K "> )
) `  N )  =  ( W  gsumg  ( i  e.  ( 0 ... N )  |->  (sgn `  ( ( F ++  <" K "> ) `  i ) ) ) ) )
3716, 31, 36syl2anc 661 . 2  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F ++  <" K "> )
) `  N )  =  ( W  gsumg  ( i  e.  ( 0 ... N )  |->  (sgn `  ( ( F ++  <" K "> ) `  i ) ) ) ) )
3832, 33, 34, 35signstfval 28696 . . 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 2508 1  |-  ( ( F  e. Word  RR  /\  K  e.  RR  /\  N  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( T `
 ( F ++  <" K "> )
) `  N )  =  ( ( T `
 F ) `  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652    C_ wss 3471   ifcif 3944   {cpr 4034   {ctp 4036   <.cop 4038    |-> cmpt 4515   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    - cmin 9824   -ucneg 9825   ZZcz 10885   ZZ>=cuz 11106   ...cfz 11697  ..^cfzo 11820   #chash 12407  Word cword 12537   ++ cconcat 12539   <"cs1 12540  sgncsgn 12930   sum_csu 13519   ndxcnx 14640   Basecbs 14643   +g cplusg 14711    gsumg cgsu 14857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11821  df-hash 12408  df-word 12545  df-concat 12547  df-s1 12548
This theorem is referenced by:  signstfvneq0  28704  signstfvc  28706  signstfveq0  28709  signsvfn  28714
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