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Theorem signshf 27134
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 9785 . . . 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 9498 . . . . . . . 8  |-  0  e.  RR
43a1i 11 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
0  e.  RR )
54s1cld 12413 . . . . . 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 12393 . . . . . 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 12359 . . . . 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 12394 . . . . . . . . 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 12415 . . . . . . . . 9  |-  ( # `  <" 0 "> )  =  1
1413oveq1i 6211 . . . . . . . 8  |-  ( (
# `  <" 0 "> )  +  (
# `  F )
)  =  ( 1  +  ( # `  F
) )
1512, 14syl6eq 2511 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( # `  ( <" 0 "> concat  F ) )  =  ( 1  +  ( # `  F ) ) )
16 ax-1cn 9452 . . . . . . . . 9  |-  1  e.  CC
1716a1i 11 . . . . . . . 8  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
1  e.  CC )
18 wrdfin 12367 . . . . . . . . . 10  |-  ( F  e. Word  RR  ->  F  e. 
Fin )
19 hashcl 12244 . . . . . . . . . 10  |-  ( F  e.  Fin  ->  ( # `
 F )  e. 
NN0 )
206, 18, 193syl 20 . . . . . . . . 9  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( # `  F )  e.  NN0 )
2120nn0cnd 10750 . . . . . . . 8  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( # `  F )  e.  CC )
2217, 21addcomd 9683 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( 1  +  (
# `  F )
)  =  ( (
# `  F )  +  1 ) )
2315, 22eqtrd 2495 . . . . . 6  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( # `  ( <" 0 "> concat  F ) )  =  ( ( # `  F
)  +  1 ) )
2423oveq2d 6217 . . . . 5  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( 0..^ ( # `  ( <" 0 "> concat  F ) ) )  =  ( 0..^ ( ( # `  F
)  +  1 ) ) )
2524feq2d 5656 . . . 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 9479 . . . . 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 12393 . . . . . . 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 12359 . . . . . 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 12394 . . . . . . . . 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 6212 . . . . . . . 8  |-  ( (
# `  F )  +  ( # `  <" 0 "> )
)  =  ( (
# `  F )  +  1 )
3634, 35syl6eq 2511 . . . . . . 7  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( # `  ( F concat  <" 0 "> ) )  =  ( ( # `  F
)  +  1 ) )
3736oveq2d 6217 . . . . . 6  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( 0..^ ( # `  ( F concat  <" 0 "> ) ) )  =  ( 0..^ ( ( # `  F
)  +  1 ) ) )
3837feq2d 5656 . . . . 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 6226 . . . . 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 11139 . . . 4  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  ->  C  e.  RR )
4428, 39, 41, 43ofcf 26691 . . 3  |-  ( ( F  e. Word  RR  /\  C  e.  RR+ )  -> 
( ( F concat  <" 0 "> )𝑓/𝑐  x.  C ) : ( 0..^ ( ( # `  F )  +  1 ) ) --> RR )
45 inidm 3668 . . 3  |-  ( ( 0..^ ( ( # `  F )  +  1 ) )  i^i  (
0..^ ( ( # `  F )  +  1 ) ) )  =  ( 0..^ ( (
# `  F )  +  1 ) )
462, 26, 44, 41, 41, 45off 6445 . 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 5660 . 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 1370    e. wcel 1758    =/= wne 2648   _Vcvv 3078   ifcif 3900   {cpr 3988   {ctp 3990   <.cop 3992    |-> cmpt 4459   -->wf 5523   ` cfv 5527  (class class class)co 6201    |-> cmpt2 6203    oFcof 6429   Fincfn 7421   CCcc 9392   RRcr 9393   0cc0 9394   1c1 9395    + caddc 9397    x. cmul 9399    - cmin 9707   -ucneg 9708   NN0cn0 10691   RR+crp 11103   ...cfz 11555  ..^cfzo 11666   #chash 12221  Word cword 12340   concat cconcat 12342   <"cs1 12343  sgncsgn 12694   sum_csu 13282   ndxcnx 14290   Basecbs 14293   +g cplusg 14358    gsumg cgsu 14499  ∘𝑓/𝑐cofc 26683
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 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
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 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-of 6431  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-n0 10692  df-z 10759  df-uz 10974  df-rp 11104  df-fz 11556  df-fzo 11667  df-hash 12222  df-word 12348  df-concat 12350  df-s1 12351  df-ofc 26684
This theorem is referenced by:  signshwrd  27135  signshlen  27136
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