Theorem signstres 30652

Description: Restriction of a zero skipping sign to a subword. (Contributed by Thierry Arnoux, 11-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
signstres	|- ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) -> ( ( T ` F ) |` ( 0..^ N ) ) = ( T ` ( F |` ( 0..^ N ) ) ) )

Distinct variable groups:   a, b, .+^   f, i, n, F   f, W, i, n   f, N, i, n

Allowed substitution hints:   .+^ ( f, i, j, n)   T( f, i, j, n, a, b)   F( j, a, b)   N( j, a, b)   V( f, i, j, n, a, b)   W( j, a, b)

Proof of Theorem signstres

Dummy variables g m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		signsv.p	. . . . . . . 8 |- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )
2		signsv.w	. . . . . . . 8 |- W = { <. ( Base ` ndx ) , { -u 1 , 0 , 1 } >. , <. ( +g ` ndx ) , .+^ >. }
3		signsv.t	. . . . . . . 8 |- T = ( f e. Word RR |-> ( n e. ( 0..^ ( # ` f ) ) |-> ( W gsumg ( i e. ( 0 ... n ) |-> (sgn ` ( f ` i ) ) ) ) ) )
4		signsv.v	. . . . . . . 8 |- V = ( f e. Word RR |-> sum_ j e. ( 1..^ ( # ` f ) ) if ( ( ( T ` f ) ` j ) =/= ( ( T ` f ) ` ( j - 1 ) ) , 1 , 0 ) )
5	1, 2, 3, 4	signstf 30643	. . . . . . 7 |- ( F e. Word RR -> ( T ` F ) e. Word RR )
6		wrdf 13310	. . . . . . 7 |- ( ( T ` F ) e. Word RR -> ( T ` F ) : ( 0..^ ( # ` ( T ` F ) ) ) --> RR )
7		ffn 6045	. . . . . . 7 |- ( ( T ` F ) : ( 0..^ ( # ` ( T ` F ) ) ) --> RR -> ( T ` F ) Fn ( 0..^ ( # ` ( T ` F ) ) ) )
8	5, 6, 7	3syl 18	. . . . . 6 |- ( F e. Word RR -> ( T ` F ) Fn ( 0..^ ( # ` ( T ` F ) ) ) )
9	1, 2, 3, 4	signstlen 30644	. . . . . . . 8 |- ( F e. Word RR -> ( # ` ( T ` F ) ) = ( # ` F ) )
10	9	oveq2d 6666	. . . . . . 7 |- ( F e. Word RR -> ( 0..^ ( # ` ( T ` F ) ) ) = ( 0..^ ( # ` F ) ) )
11	10	fneq2d 5982	. . . . . 6 |- ( F e. Word RR -> ( ( T ` F ) Fn ( 0..^ ( # ` ( T ` F ) ) ) <-> ( T ` F ) Fn ( 0..^ ( # ` F ) ) ) )
12	8, 11	mpbid 222	. . . . 5 |- ( F e. Word RR -> ( T ` F ) Fn ( 0..^ ( # ` F ) ) )
13		fnresin 29430	. . . . 5 |- ( ( T ` F ) Fn ( 0..^ ( # ` F ) ) -> ( ( T ` F ) |` ( 0..^ N ) ) Fn ( ( 0..^ ( # ` F ) ) i^i ( 0..^ N ) ) )
14	12, 13	syl 17	. . . 4 |- ( F e. Word RR -> ( ( T ` F ) |` ( 0..^ N ) ) Fn ( ( 0..^ ( # ` F ) ) i^i ( 0..^ N ) ) )
15	14	adantr 481	. . 3 |- ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) -> ( ( T ` F ) |` ( 0..^ N ) ) Fn ( ( 0..^ ( # ` F ) ) i^i ( 0..^ N ) ) )
16		elfzuz3 12339	. . . . . 6 |- ( N e. ( 0 ... ( # ` F ) ) -> ( # ` F ) e. ( ZZ>= ` N ) )
17		fzoss2 12496	. . . . . 6 |- ( ( # ` F ) e. ( ZZ>= ` N ) -> ( 0..^ N ) C_ ( 0..^ ( # ` F ) ) )
18	16, 17	syl 17	. . . . 5 |- ( N e. ( 0 ... ( # ` F ) ) -> ( 0..^ N ) C_ ( 0..^ ( # ` F ) ) )
19	18	adantl 482	. . . 4 |- ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) -> ( 0..^ N ) C_ ( 0..^ ( # ` F ) ) )
20		incom 3805	. . . . . 6 |- ( ( 0..^ N ) i^i ( 0..^ ( # ` F ) ) ) = ( ( 0..^ ( # ` F ) ) i^i ( 0..^ N ) )
21		df-ss 3588	. . . . . . 7 |- ( ( 0..^ N ) C_ ( 0..^ ( # ` F ) ) <-> ( ( 0..^ N ) i^i ( 0..^ ( # ` F ) ) ) = ( 0..^ N ) )
22	21	biimpi 206	. . . . . 6 |- ( ( 0..^ N ) C_ ( 0..^ ( # ` F ) ) -> ( ( 0..^ N ) i^i ( 0..^ ( # ` F ) ) ) = ( 0..^ N ) )
23	20, 22	syl5eqr 2670	. . . . 5 |- ( ( 0..^ N ) C_ ( 0..^ ( # ` F ) ) -> ( ( 0..^ ( # ` F ) ) i^i ( 0..^ N ) ) = ( 0..^ N ) )
24	23	fneq2d 5982	. . . 4 |- ( ( 0..^ N ) C_ ( 0..^ ( # ` F ) ) -> ( ( ( T ` F ) |` ( 0..^ N ) ) Fn ( ( 0..^ ( # ` F ) ) i^i ( 0..^ N ) ) <-> ( ( T ` F ) |` ( 0..^ N ) ) Fn ( 0..^ N ) ) )
25	19, 24	syl 17	. . 3 |- ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) -> ( ( ( T ` F ) |` ( 0..^ N ) ) Fn ( ( 0..^ ( # ` F ) ) i^i ( 0..^ N ) ) <-> ( ( T ` F ) |` ( 0..^ N ) ) Fn ( 0..^ N ) ) )
26	15, 25	mpbid 222	. 2 |- ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) -> ( ( T ` F ) |` ( 0..^ N ) ) Fn ( 0..^ N ) )
27		wrdres 30617	. . . 4 |- ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) -> ( F |` ( 0..^ N ) ) e. Word RR )
28	1, 2, 3, 4	signstf 30643	. . . 4 |- ( ( F |` ( 0..^ N ) ) e. Word RR -> ( T ` ( F |` ( 0..^ N ) ) ) e. Word RR )
29		wrdf 13310	. . . 4 |- ( ( T ` ( F |` ( 0..^ N ) ) ) e. Word RR -> ( T ` ( F |` ( 0..^ N ) ) ) : ( 0..^ ( # ` ( T ` ( F |` ( 0..^ N ) ) ) ) ) --> RR )
30		ffn 6045	. . . 4 |- ( ( T ` ( F |` ( 0..^ N ) ) ) : ( 0..^ ( # ` ( T ` ( F |` ( 0..^ N ) ) ) ) ) --> RR -> ( T ` ( F |` ( 0..^ N ) ) ) Fn ( 0..^ ( # ` ( T ` ( F |` ( 0..^ N ) ) ) ) ) )
31	27, 28, 29, 30	4syl 19	. . 3 |- ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) -> ( T ` ( F |` ( 0..^ N ) ) ) Fn ( 0..^ ( # ` ( T ` ( F |` ( 0..^ N ) ) ) ) ) )
32	1, 2, 3, 4	signstlen 30644	. . . . . . 7 |- ( ( F |` ( 0..^ N ) ) e. Word RR -> ( # ` ( T ` ( F |` ( 0..^ N ) ) ) ) = ( # ` ( F |` ( 0..^ N ) ) ) )
33	27, 32	syl 17	. . . . . 6 |- ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) -> ( # ` ( T ` ( F |` ( 0..^ N ) ) ) ) = ( # ` ( F |` ( 0..^ N ) ) ) )
34		wrdfn 13319	. . . . . . . 8 |- ( F e. Word RR -> F Fn ( 0..^ ( # ` F ) ) )
35		fnssres 6004	. . . . . . . 8 |- ( ( F Fn ( 0..^ ( # ` F ) ) /\ ( 0..^ N ) C_ ( 0..^ ( # ` F ) ) ) -> ( F |` ( 0..^ N ) ) Fn ( 0..^ N ) )
36	34, 18, 35	syl2an 494	. . . . . . 7 |- ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) -> ( F |` ( 0..^ N ) ) Fn ( 0..^ N ) )
37		hashfn 13164	. . . . . . 7 |- ( ( F |` ( 0..^ N ) ) Fn ( 0..^ N ) -> ( # ` ( F |` ( 0..^ N ) ) ) = ( # ` ( 0..^ N ) ) )
38	36, 37	syl 17	. . . . . 6 |- ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) -> ( # ` ( F |` ( 0..^ N ) ) ) = ( # ` ( 0..^ N ) ) )
39		elfznn0 12433	. . . . . . . 8 |- ( N e. ( 0 ... ( # ` F ) ) -> N e. NN0 )
40		hashfzo0 13217	. . . . . . . 8 |- ( N e. NN0 -> ( # ` ( 0..^ N ) ) = N )
41	39, 40	syl 17	. . . . . . 7 |- ( N e. ( 0 ... ( # ` F ) ) -> ( # ` ( 0..^ N ) ) = N )
42	41	adantl 482	. . . . . 6 |- ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) -> ( # ` ( 0..^ N ) ) = N )
43	33, 38, 42	3eqtrd 2660	. . . . 5 |- ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) -> ( # ` ( T ` ( F |` ( 0..^ N ) ) ) ) = N )
44	43	oveq2d 6666	. . . 4 |- ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) -> ( 0..^ ( # ` ( T ` ( F |` ( 0..^ N ) ) ) ) ) = ( 0..^ N ) )
45	44	fneq2d 5982	. . 3 |- ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) -> ( ( T ` ( F |` ( 0..^ N ) ) ) Fn ( 0..^ ( # ` ( T ` ( F |` ( 0..^ N ) ) ) ) ) <-> ( T ` ( F |` ( 0..^ N ) ) ) Fn ( 0..^ N ) ) )
46	31, 45	mpbid 222	. 2 |- ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) -> ( T ` ( F |` ( 0..^ N ) ) ) Fn ( 0..^ N ) )
47		fvres 6207	. . . . 5 |- ( m e. ( 0..^ N ) -> ( ( ( T ` F ) |` ( 0..^ N ) ) ` m ) = ( ( T ` F ) ` m ) )
48	47	ad3antlr 767	. . . 4 |- ( ( ( ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) /\ m e. ( 0..^ N ) ) /\ g e. Word RR ) /\ F = ( ( F |` ( 0..^ N ) ) ++ g ) ) -> ( ( ( T ` F ) |` ( 0..^ N ) ) ` m ) = ( ( T ` F ) ` m ) )
49		simpr 477	. . . . . 6 |- ( ( ( ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) /\ m e. ( 0..^ N ) ) /\ g e. Word RR ) /\ F = ( ( F |` ( 0..^ N ) ) ++ g ) ) -> F = ( ( F |` ( 0..^ N ) ) ++ g ) )
50	49	fveq2d 6195	. . . . 5 |- ( ( ( ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) /\ m e. ( 0..^ N ) ) /\ g e. Word RR ) /\ F = ( ( F |` ( 0..^ N ) ) ++ g ) ) -> ( T ` F ) = ( T ` ( ( F |` ( 0..^ N ) ) ++ g ) ) )
51	50	fveq1d 6193	. . . 4 |- ( ( ( ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) /\ m e. ( 0..^ N ) ) /\ g e. Word RR ) /\ F = ( ( F |` ( 0..^ N ) ) ++ g ) ) -> ( ( T ` F ) ` m ) = ( ( T ` ( ( F |` ( 0..^ N ) ) ++ g ) ) ` m ) )
52	27	ad3antrrr 766	. . . . 5 |- ( ( ( ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) /\ m e. ( 0..^ N ) ) /\ g e. Word RR ) /\ F = ( ( F |` ( 0..^ N ) ) ++ g ) ) -> ( F |` ( 0..^ N ) ) e. Word RR )
53		simplr 792	. . . . 5 |- ( ( ( ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) /\ m e. ( 0..^ N ) ) /\ g e. Word RR ) /\ F = ( ( F |` ( 0..^ N ) ) ++ g ) ) -> g e. Word RR )
54	38, 42	eqtrd 2656	. . . . . . . . 9 |- ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) -> ( # ` ( F |` ( 0..^ N ) ) ) = N )
55	54	oveq2d 6666	. . . . . . . 8 |- ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) -> ( 0..^ ( # ` ( F |` ( 0..^ N ) ) ) ) = ( 0..^ N ) )
56	55	eleq2d 2687	. . . . . . 7 |- ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) -> ( m e. ( 0..^ ( # ` ( F |` ( 0..^ N ) ) ) ) <-> m e. ( 0..^ N ) ) )
57	56	biimpar 502	. . . . . 6 |- ( ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) /\ m e. ( 0..^ N ) ) -> m e. ( 0..^ ( # ` ( F |` ( 0..^ N ) ) ) ) )
58	57	ad2antrr 762	. . . . 5 |- ( ( ( ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) /\ m e. ( 0..^ N ) ) /\ g e. Word RR ) /\ F = ( ( F |` ( 0..^ N ) ) ++ g ) ) -> m e. ( 0..^ ( # ` ( F |` ( 0..^ N ) ) ) ) )
59	1, 2, 3, 4	signstfvc 30651	. . . . 5 |- ( ( ( F |` ( 0..^ N ) ) e. Word RR /\ g e. Word RR /\ m e. ( 0..^ ( # ` ( F |` ( 0..^ N ) ) ) ) ) -> ( ( T ` ( ( F |` ( 0..^ N ) ) ++ g ) ) ` m ) = ( ( T ` ( F |` ( 0..^ N ) ) ) ` m ) )
60	52, 53, 58, 59	syl3anc 1326	. . . 4 |- ( ( ( ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) /\ m e. ( 0..^ N ) ) /\ g e. Word RR ) /\ F = ( ( F |` ( 0..^ N ) ) ++ g ) ) -> ( ( T ` ( ( F |` ( 0..^ N ) ) ++ g ) ) ` m ) = ( ( T ` ( F |` ( 0..^ N ) ) ) ` m ) )
61	48, 51, 60	3eqtrd 2660	. . 3 |- ( ( ( ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) /\ m e. ( 0..^ N ) ) /\ g e. Word RR ) /\ F = ( ( F |` ( 0..^ N ) ) ++ g ) ) -> ( ( ( T ` F ) |` ( 0..^ N ) ) ` m ) = ( ( T ` ( F |` ( 0..^ N ) ) ) ` m ) )
62		wrdsplex 30618	. . . 4 |- ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) -> E. g e. Word RR F = ( ( F |` ( 0..^ N ) ) ++ g ) )
63	62	adantr 481	. . 3 |- ( ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) /\ m e. ( 0..^ N ) ) -> E. g e. Word RR F = ( ( F |` ( 0..^ N ) ) ++ g ) )
64	61, 63	r19.29a 3078	. 2 |- ( ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) /\ m e. ( 0..^ N ) ) -> ( ( ( T ` F ) |` ( 0..^ N ) ) ` m ) = ( ( T ` ( F |` ( 0..^ N ) ) ) ` m ) )
65	26, 46, 64	eqfnfvd 6314	1 |- ( ( F e. Word RR /\ N e. ( 0 ... ( # ` F ) ) ) -> ( ( T ` F ) |` ( 0..^ N ) ) = ( T ` ( F |` ( 0..^ N ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   i^i cin 3573   C_ wss 3574   ifcif 4086   {cpr 4179   {ctp 4181   <.cop 4183   |-> cmpt 4729   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   RRcr 9935   0cc0 9936   1c1 9937   - cmin 10266   -ucneg 10267   NN0cn0 11292   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   ++ cconcat 13293  sgncsgn 13826   sum_csu 14416   ndxcnx 15854   Basecbs 15857   +g cplusg 15941   gsum_g cgsu 16101

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-rmo 2920  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-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-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-sgn 13827  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295

This theorem is referenced by: (None)