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Theorem signshf 28182
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 "> concat  F )  oF  -  ( ( F concat  <" 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 9879 . . . 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 0re 9592 . . . . . . . 8  |-  0  e.  RR
43a1i 11 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
0  e.  RR )
54s1cld 12572 . . . . . 6  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  ->  <" 0 ">  e. Word  RR )
6 simpl 457 . . . . . 6  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  ->  F  e. Word  RR )
7 ccatcl 12552 . . . . . 6  |-  ( (
<" 0 ">  e. Word  RR  /\  F  e. Word  RR )  ->  ( <" 0 "> concat  F )  e. Word  RR )
85, 6, 7syl2anc 661 . . . . 5  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( <" 0 "> concat  F )  e. Word  RR )
9 wrdf 12513 . . . . 5  |-  ( (
<" 0 "> concat  F )  e. Word  RR  ->  (
<" 0 "> concat  F ) : ( 0..^ ( # `  ( <" 0 "> concat  F ) ) ) --> RR )
108, 9syl 16 . . . 4  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( <" 0 "> concat  F ) : ( 0..^ ( # `  ( <" 0 "> concat  F ) ) ) --> RR )
11 ccatlen 12553 . . . . . . . . 9  |-  ( (
<" 0 ">  e. Word  RR  /\  F  e. Word  RR )  ->  ( # `  ( <" 0 "> concat  F ) )  =  ( ( # `  <" 0 "> )  +  ( # `  F
) ) )
125, 6, 11syl2anc 661 . . . . . . . 8  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( # `  ( <" 0 "> concat  F ) )  =  ( ( # `  <" 0 "> )  +  ( # `  F
) ) )
13 s1len 12574 . . . . . . . . 9  |-  ( # `  <" 0 "> )  =  1
1413oveq1i 6292 . . . . . . . 8  |-  ( (
# `  <" 0 "> )  +  (
# `  F )
)  =  ( 1  +  ( # `  F
) )
1512, 14syl6eq 2524 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( # `  ( <" 0 "> concat  F ) )  =  ( 1  +  ( # `  F ) ) )
16 ax-1cn 9546 . . . . . . . . 9  |-  1  e.  CC
1716a1i 11 . . . . . . . 8  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
1  e.  CC )
18 wrdfin 12521 . . . . . . . . . 10  |-  ( F  e. Word  RR  ->  F  e. 
Fin )
19 hashcl 12390 . . . . . . . . . 10  |-  ( F  e.  Fin  ->  ( # `
 F )  e. 
NN0 )
206, 18, 193syl 20 . . . . . . . . 9  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( # `  F )  e.  NN0 )
2120nn0cnd 10850 . . . . . . . 8  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( # `  F )  e.  CC )
2217, 21addcomd 9777 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( 1  +  (
# `  F )
)  =  ( (
# `  F )  +  1 ) )
2315, 22eqtrd 2508 . . . . . 6  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( # `  ( <" 0 "> concat  F ) )  =  ( ( # `  F
)  +  1 ) )
2423oveq2d 6298 . . . . 5  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( 0..^ ( # `  ( <" 0 "> concat  F ) ) )  =  ( 0..^ ( ( # `  F
)  +  1 ) ) )
2524feq2d 5716 . . . 4  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( ( <" 0 "> concat  F ) : ( 0..^ ( # `  ( <" 0 "> concat  F ) ) ) --> RR  <->  (
<" 0 "> concat  F ) : ( 0..^ ( ( # `  F
)  +  1 ) ) --> RR ) )
2610, 25mpbid 210 . . 3  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( <" 0 "> concat  F ) : ( 0..^ ( ( # `  F )  +  1 ) ) --> RR )
27 remulcl 9573 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  x.  y
)  e.  RR )
2827adantl 466 . . . 4  |-  ( ( ( F  e. Word  RR  /\  C  e.  RR+ )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( x  x.  y )  e.  RR )
29 ccatcl 12552 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  <" 0 ">  e. Word  RR )  ->  ( F concat  <" 0 "> )  e. Word  RR )
305, 29syldan 470 . . . . . 6  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( F concat  <" 0 "> )  e. Word  RR )
31 wrdf 12513 . . . . . 6  |-  ( ( F concat  <" 0 "> )  e. Word  RR  ->  ( F concat  <" 0 "> ) : ( 0..^ ( # `  ( F concat  <" 0 "> ) ) ) --> RR )
3230, 31syl 16 . . . . 5  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( F concat  <" 0 "> ) : ( 0..^ ( # `  ( F concat  <" 0 "> ) ) ) --> RR )
33 ccatlen 12553 . . . . . . . . 9  |-  ( ( F  e. Word  RR  /\  <" 0 ">  e. Word  RR )  ->  ( # `
 ( F concat  <" 0 "> ) )  =  ( ( # `  F
)  +  ( # `  <" 0 "> ) ) )
345, 33syldan 470 . . . . . . . 8  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( # `  ( F concat  <" 0 "> ) )  =  ( ( # `  F
)  +  ( # `  <" 0 "> ) ) )
3513oveq2i 6293 . . . . . . . 8  |-  ( (
# `  F )  +  ( # `  <" 0 "> )
)  =  ( (
# `  F )  +  1 )
3634, 35syl6eq 2524 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( # `  ( F concat  <" 0 "> ) )  =  ( ( # `  F
)  +  1 ) )
3736oveq2d 6298 . . . . . 6  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( 0..^ ( # `  ( F concat  <" 0 "> ) ) )  =  ( 0..^ ( ( # `  F
)  +  1 ) ) )
3837feq2d 5716 . . . . 5  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( ( F concat  <" 0 "> ) : ( 0..^ ( # `  ( F concat  <" 0 "> ) ) ) --> RR  <->  ( F concat  <" 0 "> ) : ( 0..^ ( ( # `  F )  +  1 ) ) --> RR ) )
3932, 38mpbid 210 . . . 4  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( F concat  <" 0 "> ) : ( 0..^ ( ( # `  F )  +  1 ) ) --> RR )
40 ovex 6307 . . . . 5  |-  ( 0..^ ( ( # `  F
)  +  1 ) )  e.  _V
4140a1i 11 . . . 4  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( 0..^ ( (
# `  F )  +  1 ) )  e.  _V )
42 simpr 461 . . . . 5  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  ->  C  e.  RR+ )
4342rpred 11252 . . . 4  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  ->  C  e.  RR )
4428, 39, 41, 43ofcf 27739 . . 3  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( ( F concat  <" 0 "> )𝑓/𝑐  x.  C ) : ( 0..^ ( ( # `  F )  +  1 ) ) --> RR )
45 inidm 3707 . . 3  |-  ( ( 0..^ ( ( # `  F )  +  1 ) )  i^i  (
0..^ ( ( # `  F )  +  1 ) ) )  =  ( 0..^ ( (
# `  F )  +  1 ) )
462, 26, 44, 41, 41, 45off 6536 . 2  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( ( <" 0 "> concat  F )  oF  -  ( ( F concat  <" 0 "> )𝑓/𝑐  x.  C ) ) : ( 0..^ ( (
# `  F )  +  1 ) ) --> RR )
47 signs.h . . 3  |-  H  =  ( ( <" 0 "> concat  F )  oF  -  ( ( F concat  <" 0 "> )𝑓/𝑐  x.  C ) )
4847feq1i 5721 . 2  |-  ( H : ( 0..^ ( ( # `  F
)  +  1 ) ) --> RR  <->  ( ( <" 0 "> concat  F )  oF  -  ( ( F concat  <" 0 "> )𝑓/𝑐  x.  C ) ) : ( 0..^ ( (
# `  F )  +  1 ) ) --> RR )
4946, 48sylibr 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 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113   ifcif 3939   {cpr 4029   {ctp 4031   <.cop 4033    |-> cmpt 4505   -->wf 5582   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284    oFcof 6520   Fincfn 7513   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493    - cmin 9801   -ucneg 9802   NN0cn0 10791   RR+crp 11216   ...cfz 11668  ..^cfzo 11788   #chash 12367  Word cword 12494   concat cconcat 12496   <"cs1 12497  sgncsgn 12876   sum_csu 13464   ndxcnx 14480   Basecbs 14483   +g cplusg 14548    gsumg cgsu 14689  ∘𝑓/𝑐cofc 27731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-hash 12368  df-word 12502  df-concat 12504  df-s1 12505  df-ofc 27732
This theorem is referenced by:  signshwrd  28183  signshlen  28184
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