Theorem signshf 30665

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.

Step	Hyp	Ref	Expression
1		resubcl 10345	. . . 4 |- ( ( x e. RR /\ y e. RR ) -> ( x - y ) e. RR )
2	1	adantl 482	. . 3 |- ( ( ( F e. Word RR /\ C e. RR+ ) /\ ( x e. RR /\ y e. RR ) ) -> ( x - y ) e. RR )
3		0red 10041	. . . . . . 7 |- ( ( F e. Word RR /\ C e. RR+ ) -> 0 e. RR )
4	3	s1cld 13383	. . . . . 6 |- ( ( F e. Word RR /\ C e. RR+ ) -> <" 0 "> e. Word RR )
5		simpl 473	. . . . . 6 |- ( ( F e. Word RR /\ C e. RR+ ) -> F e. Word RR )
6		ccatcl 13359	. . . . . 6 |- ( ( <" 0 "> e. Word RR /\ F e. Word RR ) -> ( <" 0 "> ++ F ) e. Word RR )
7	4, 5, 6	syl2anc 693	. . . . 5 |- ( ( F e. Word RR /\ C e. RR+ ) -> ( <" 0 "> ++ F ) e. Word RR )
8		wrdf 13310	. . . . 5 |- ( ( <" 0 "> ++ F ) e. Word RR -> ( <" 0 "> ++ F ) : ( 0..^ ( # ` ( <" 0 "> ++ F ) ) ) --> RR )
9	7, 8	syl 17	. . . 4 |- ( ( F e. Word RR /\ C e. RR+ ) -> ( <" 0 "> ++ F ) : ( 0..^ ( # ` ( <" 0 "> ++ F ) ) ) --> RR )
10		ccatlen 13360	. . . . . . . . 9 |- ( ( <" 0 "> e. Word RR /\ F e. Word RR ) -> ( # ` ( <" 0 "> ++ F ) ) = ( ( # ` <" 0 "> ) + ( # ` F ) ) )
11	4, 5, 10	syl2anc 693	. . . . . . . 8 |- ( ( F e. Word RR /\ C e. RR+ ) -> ( # ` ( <" 0 "> ++ F ) ) = ( ( # ` <" 0 "> ) + ( # ` F ) ) )
12		s1len 13385	. . . . . . . . 9 |- ( # ` <" 0 "> ) = 1
13	12	oveq1i 6660	. . . . . . . 8 |- ( ( # ` <" 0 "> ) + ( # ` F ) ) = ( 1 + ( # ` F ) )
14	11, 13	syl6eq 2672	. . . . . . 7 |- ( ( F e. Word RR /\ C e. RR+ ) -> ( # ` ( <" 0 "> ++ F ) ) = ( 1 + ( # ` F ) ) )
15		1cnd 10056	. . . . . . . 8 |- ( ( F e. Word RR /\ C e. RR+ ) -> 1 e. CC )
16		wrdfin 13323	. . . . . . . . . 10 |- ( F e. Word RR -> F e. Fin )
17		hashcl 13147	. . . . . . . . . 10 |- ( F e. Fin -> ( # ` F ) e. NN0 )
18	5, 16, 17	3syl 18	. . . . . . . . 9 |- ( ( F e. Word RR /\ C e. RR+ ) -> ( # ` F ) e. NN0 )
19	18	nn0cnd 11353	. . . . . . . 8 |- ( ( F e. Word RR /\ C e. RR+ ) -> ( # ` F ) e. CC )
20	15, 19	addcomd 10238	. . . . . . 7 |- ( ( F e. Word RR /\ C e. RR+ ) -> ( 1 + ( # ` F ) ) = ( ( # ` F ) + 1 ) )
21	14, 20	eqtrd 2656	. . . . . 6 |- ( ( F e. Word RR /\ C e. RR+ ) -> ( # ` ( <" 0 "> ++ F ) ) = ( ( # ` F ) + 1 ) )
22	21	oveq2d 6666	. . . . 5 |- ( ( F e. Word RR /\ C e. RR+ ) -> ( 0..^ ( # ` ( <" 0 "> ++ F ) ) ) = ( 0..^ ( ( # ` F ) + 1 ) ) )
23	22	feq2d 6031	. . . 4 |- ( ( F e. Word RR /\ C e. RR+ ) -> ( ( <" 0 "> ++ F ) : ( 0..^ ( # ` ( <" 0 "> ++ F ) ) ) --> RR <-> ( <" 0 "> ++ F ) : ( 0..^ ( ( # ` F ) + 1 ) ) --> RR ) )
24	9, 23	mpbid 222	. . 3 |- ( ( F e. Word RR /\ C e. RR+ ) -> ( <" 0 "> ++ F ) : ( 0..^ ( ( # ` F ) + 1 ) ) --> RR )
25		remulcl 10021	. . . . 5 |- ( ( x e. RR /\ y e. RR ) -> ( x x. y ) e. RR )
26	25	adantl 482	. . . 4 |- ( ( ( F e. Word RR /\ C e. RR+ ) /\ ( x e. RR /\ y e. RR ) ) -> ( x x. y ) e. RR )
27		ccatcl 13359	. . . . . . 7 |- ( ( F e. Word RR /\ <" 0 "> e. Word RR ) -> ( F ++ <" 0 "> ) e. Word RR )
28	4, 27	syldan 487	. . . . . 6 |- ( ( F e. Word RR /\ C e. RR+ ) -> ( F ++ <" 0 "> ) e. Word RR )
29		wrdf 13310	. . . . . 6 |- ( ( F ++ <" 0 "> ) e. Word RR -> ( F ++ <" 0 "> ) : ( 0..^ ( # ` ( F ++ <" 0 "> ) ) ) --> RR )
30	28, 29	syl 17	. . . . 5 |- ( ( F e. Word RR /\ C e. RR+ ) -> ( F ++ <" 0 "> ) : ( 0..^ ( # ` ( F ++ <" 0 "> ) ) ) --> RR )
31		ccatlen 13360	. . . . . . . . 9 |- ( ( F e. Word RR /\ <" 0 "> e. Word RR ) -> ( # ` ( F ++ <" 0 "> ) ) = ( ( # ` F ) + ( # ` <" 0 "> ) ) )
32	4, 31	syldan 487	. . . . . . . 8 |- ( ( F e. Word RR /\ C e. RR+ ) -> ( # ` ( F ++ <" 0 "> ) ) = ( ( # ` F ) + ( # ` <" 0 "> ) ) )
33	12	oveq2i 6661	. . . . . . . 8 |- ( ( # ` F ) + ( # ` <" 0 "> ) ) = ( ( # ` F ) + 1 )
34	32, 33	syl6eq 2672	. . . . . . 7 |- ( ( F e. Word RR /\ C e. RR+ ) -> ( # ` ( F ++ <" 0 "> ) ) = ( ( # ` F ) + 1 ) )
35	34	oveq2d 6666	. . . . . 6 |- ( ( F e. Word RR /\ C e. RR+ ) -> ( 0..^ ( # ` ( F ++ <" 0 "> ) ) ) = ( 0..^ ( ( # ` F ) + 1 ) ) )
36	35	feq2d 6031	. . . . 5 |- ( ( F e. Word RR /\ C e. RR+ ) -> ( ( F ++ <" 0 "> ) : ( 0..^ ( # ` ( F ++ <" 0 "> ) ) ) --> RR <-> ( F ++ <" 0 "> ) : ( 0..^ ( ( # ` F ) + 1 ) ) --> RR ) )
37	30, 36	mpbid 222	. . . 4 |- ( ( F e. Word RR /\ C e. RR+ ) -> ( F ++ <" 0 "> ) : ( 0..^ ( ( # ` F ) + 1 ) ) --> RR )
38		ovexd 6680	. . . 4 |- ( ( F e. Word RR /\ C e. RR+ ) -> ( 0..^ ( ( # ` F ) + 1 ) ) e. _V )
39		simpr 477	. . . . 5 |- ( ( F e. Word RR /\ C e. RR+ ) -> C e. RR+ )
40	39	rpred 11872	. . . 4 |- ( ( F e. Word RR /\ C e. RR+ ) -> C e. RR )
41	26, 37, 38, 40	ofcf 30165	. . 3 |- ( ( F e. Word RR /\ C e. RR+ ) -> ( ( F ++ <" 0 "> )∘𝑓/𝑐 x. C ) : ( 0..^ ( ( # ` F ) + 1 ) ) --> RR )
42		inidm 3822	. . 3 |- ( ( 0..^ ( ( # ` F ) + 1 ) ) i^i ( 0..^ ( ( # ` F ) + 1 ) ) ) = ( 0..^ ( ( # ` F ) + 1 ) )
43	2, 24, 41, 38, 38, 42	off 6912	. 2 |- ( ( F e. Word RR /\ C e. RR+ ) -> ( ( <" 0 "> ++ F ) oF - ( ( F ++ <" 0 "> )∘𝑓/𝑐 x. C ) ) : ( 0..^ ( ( # ` F ) + 1 ) ) --> RR )
44		signs.h	. . 3 |- H = ( ( <" 0 "> ++ F ) oF - ( ( F ++ <" 0 "> )∘𝑓/𝑐 x. C ) )
45	44	feq1i 6036	. 2 |- ( H : ( 0..^ ( ( # ` F ) + 1 ) ) --> RR <-> ( ( <" 0 "> ++ F ) oF - ( ( F ++ <" 0 "> )∘𝑓/𝑐 x. C ) ) : ( 0..^ ( ( # ` F ) + 1 ) ) --> RR )
46	43, 45	sylibr 224	1 |- ( ( F e. Word RR /\ C e. RR+ ) -> H : ( 0..^ ( ( # ` F ) + 1 ) ) --> RR )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   ifcif 4086   {cpr 4179   {ctp 4181   <.cop 4183   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   oFcof 6895   Fincfn 7955   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   NN0cn0 11292   RR+crp 11832   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   ++ cconcat 13293   <"cs1 13294  sgncsgn 13826   sum_csu 14416   ndxcnx 15854   Basecbs 15857   +g cplusg 15941   gsum_g cgsu 16101  ∘_𝑓/𝑐cofc 30157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-ofc 30158

This theorem is referenced by:  signshwrd  30666  signshlen  30667