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Theorem signshf 28720
Description:  H, corresponding to the word  F multiplied by  ( x  -  C ), as a function. (Contributed by Thierry Arnoux, 29-Sep-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 ) )
signs.h  |-  H  =  ( ( <" 0 "> ++  F )  oF  -  ( ( F ++  <" 0 "> )𝑓/𝑐  x.  C ) )
Assertion
Ref Expression
signshf  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  ->  H : ( 0..^ ( ( # `  F
)  +  1 ) ) --> RR )
Distinct variable groups:    a, b,  .+^    f, i, n, F    f, W, i, n
Allowed substitution hints:    C( f, i, j, n, a, b)    .+^ ( f, i, j, n)    T( f, i, j, n, a, b)    F( j, a, b)    H( f, i, j, n, a, b)    V( f, i, j, n, a, b)    W( j, a, b)

Proof of Theorem signshf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resubcl 9902 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  -  y
)  e.  RR )
21adantl 466 . . 3  |-  ( ( ( F  e. Word  RR  /\  C  e.  RR+ )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( x  -  y )  e.  RR )
3 0red 9614 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
0  e.  RR )
43s1cld 12623 . . . . . 6  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  ->  <" 0 ">  e. Word  RR )
5 simpl 457 . . . . . 6  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  ->  F  e. Word  RR )
6 ccatcl 12601 . . . . . 6  |-  ( (
<" 0 ">  e. Word  RR  /\  F  e. Word  RR )  ->  ( <" 0 "> ++  F )  e. Word  RR )
74, 5, 6syl2anc 661 . . . . 5  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( <" 0 "> ++  F )  e. Word  RR )
8 wrdf 12557 . . . . 5  |-  ( (
<" 0 "> ++  F )  e. Word  RR  ->  (
<" 0 "> ++  F ) : ( 0..^ ( # `  ( <" 0 "> ++  F ) ) ) --> RR )
97, 8syl 16 . . . 4  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( <" 0 "> ++  F ) : ( 0..^ ( # `  ( <" 0 "> ++  F ) ) ) --> RR )
10 ccatlen 12602 . . . . . . . . 9  |-  ( (
<" 0 ">  e. Word  RR  /\  F  e. Word  RR )  ->  ( # `  ( <" 0 "> ++  F ) )  =  ( ( # `  <" 0 "> )  +  (
# `  F )
) )
114, 5, 10syl2anc 661 . . . . . . . 8  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( # `  ( <" 0 "> ++  F ) )  =  ( ( # `  <" 0 "> )  +  ( # `  F
) ) )
12 s1len 12625 . . . . . . . . 9  |-  ( # `  <" 0 "> )  =  1
1312oveq1i 6306 . . . . . . . 8  |-  ( (
# `  <" 0 "> )  +  (
# `  F )
)  =  ( 1  +  ( # `  F
) )
1411, 13syl6eq 2514 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( # `  ( <" 0 "> ++  F ) )  =  ( 1  +  ( # `  F ) ) )
15 1cnd 9629 . . . . . . . 8  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
1  e.  CC )
16 wrdfin 12567 . . . . . . . . . 10  |-  ( F  e. Word  RR  ->  F  e. 
Fin )
17 hashcl 12430 . . . . . . . . . 10  |-  ( F  e.  Fin  ->  ( # `
 F )  e. 
NN0 )
185, 16, 173syl 20 . . . . . . . . 9  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( # `  F )  e.  NN0 )
1918nn0cnd 10875 . . . . . . . 8  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( # `  F )  e.  CC )
2015, 19addcomd 9799 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( 1  +  (
# `  F )
)  =  ( (
# `  F )  +  1 ) )
2114, 20eqtrd 2498 . . . . . 6  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( # `  ( <" 0 "> ++  F ) )  =  ( ( # `  F
)  +  1 ) )
2221oveq2d 6312 . . . . 5  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( 0..^ ( # `  ( <" 0 "> ++  F ) ) )  =  ( 0..^ ( ( # `  F
)  +  1 ) ) )
2322feq2d 5724 . . . 4  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( ( <" 0 "> ++  F ) : ( 0..^ ( # `  ( <" 0 "> ++  F ) ) ) --> RR  <->  ( <" 0 "> ++  F ) : ( 0..^ ( ( # `  F
)  +  1 ) ) --> RR ) )
249, 23mpbid 210 . . 3  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( <" 0 "> ++  F ) : ( 0..^ ( (
# `  F )  +  1 ) ) --> RR )
25 remulcl 9594 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  x.  y
)  e.  RR )
2625adantl 466 . . . 4  |-  ( ( ( F  e. Word  RR  /\  C  e.  RR+ )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( x  x.  y )  e.  RR )
27 ccatcl 12601 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  <" 0 ">  e. Word  RR )  ->  ( F ++  <" 0 "> )  e. Word  RR )
284, 27syldan 470 . . . . . 6  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( F ++  <" 0 "> )  e. Word  RR )
29 wrdf 12557 . . . . . 6  |-  ( ( F ++  <" 0 "> )  e. Word  RR  ->  ( F ++  <" 0 "> ) : ( 0..^ ( # `  ( F ++  <" 0 "> ) ) ) --> RR )
3028, 29syl 16 . . . . 5  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( F ++  <" 0 "> ) : ( 0..^ ( # `  ( F ++  <" 0 "> ) ) ) --> RR )
31 ccatlen 12602 . . . . . . . . 9  |-  ( ( F  e. Word  RR  /\  <" 0 ">  e. Word  RR )  ->  ( # `
 ( F ++  <" 0 "> )
)  =  ( (
# `  F )  +  ( # `  <" 0 "> )
) )
324, 31syldan 470 . . . . . . . 8  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( # `  ( F ++ 
<" 0 "> ) )  =  ( ( # `  F
)  +  ( # `  <" 0 "> ) ) )
3312oveq2i 6307 . . . . . . . 8  |-  ( (
# `  F )  +  ( # `  <" 0 "> )
)  =  ( (
# `  F )  +  1 )
3432, 33syl6eq 2514 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( # `  ( F ++ 
<" 0 "> ) )  =  ( ( # `  F
)  +  1 ) )
3534oveq2d 6312 . . . . . 6  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( 0..^ ( # `  ( F ++  <" 0 "> ) ) )  =  ( 0..^ ( ( # `  F
)  +  1 ) ) )
3635feq2d 5724 . . . . 5  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( ( F ++  <" 0 "> ) : ( 0..^ (
# `  ( F ++  <" 0 "> ) ) ) --> RR  <->  ( F ++  <" 0 "> ) : ( 0..^ ( ( # `  F )  +  1 ) ) --> RR ) )
3730, 36mpbid 210 . . . 4  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( F ++  <" 0 "> ) : ( 0..^ ( ( # `  F )  +  1 ) ) --> RR )
38 ovex 6324 . . . . 5  |-  ( 0..^ ( ( # `  F
)  +  1 ) )  e.  _V
3938a1i 11 . . . 4  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( 0..^ ( (
# `  F )  +  1 ) )  e.  _V )
40 simpr 461 . . . . 5  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  ->  C  e.  RR+ )
4140rpred 11281 . . . 4  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  ->  C  e.  RR )
4226, 37, 39, 41ofcf 28263 . . 3  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( ( F ++  <" 0 "> )𝑓/𝑐  x.  C
) : ( 0..^ ( ( # `  F
)  +  1 ) ) --> RR )
43 inidm 3703 . . 3  |-  ( ( 0..^ ( ( # `  F )  +  1 ) )  i^i  (
0..^ ( ( # `  F )  +  1 ) ) )  =  ( 0..^ ( (
# `  F )  +  1 ) )
442, 24, 42, 39, 39, 43off 6553 . 2  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( ( <" 0 "> ++  F )  oF  -  ( ( F ++  <" 0 "> )𝑓/𝑐  x.  C ) ) : ( 0..^ ( (
# `  F )  +  1 ) ) --> RR )
45 signs.h . . 3  |-  H  =  ( ( <" 0 "> ++  F )  oF  -  ( ( F ++  <" 0 "> )𝑓/𝑐  x.  C ) )
4645feq1i 5729 . 2  |-  ( H : ( 0..^ ( ( # `  F
)  +  1 ) ) --> RR  <->  ( ( <" 0 "> ++  F )  oF  -  ( ( F ++  <" 0 "> )𝑓/𝑐  x.  C
) ) : ( 0..^ ( ( # `  F )  +  1 ) ) --> RR )
4744, 46sylibr 212 1  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  ->  H : ( 0..^ ( ( # `  F
)  +  1 ) ) --> RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   _Vcvv 3109   ifcif 3944   {cpr 4034   {ctp 4036   <.cop 4038    |-> cmpt 4515   -->wf 5590   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298    oFcof 6537   Fincfn 7535   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514    - cmin 9824   -ucneg 9825   NN0cn0 10816   RR+crp 11245   ...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  ∘𝑓/𝑐cofc 28255
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-of 6539  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-rp 11246  df-fz 11698  df-fzo 11821  df-hash 12408  df-word 12545  df-concat 12547  df-s1 12548  df-ofc 28256
This theorem is referenced by:  signshwrd  28721  signshlen  28722
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