Theorem repswpfx 41436

Description: A prefix of a repeated symbol word is a repeated symbol word. (Contributed by AV, 11-May-2020.)

Assertion

Ref	Expression
repswpfx	|- ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) -> ( ( S repeatS N ) prefix L ) = ( S repeatS L ) )

Proof of Theorem repswpfx

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		repsw 13522	. . . . 5 |- ( ( S e. V /\ N e. NN0 ) -> ( S repeatS N ) e. Word V )
2	1	3adant3 1081	. . . 4 |- ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) -> ( S repeatS N ) e. Word V )
3		repswlen 13523	. . . . . . . 8 |- ( ( S e. V /\ N e. NN0 ) -> ( # ` ( S repeatS N ) ) = N )
4	3	eqcomd 2628	. . . . . . 7 |- ( ( S e. V /\ N e. NN0 ) -> N = ( # ` ( S repeatS N ) ) )
5	4	oveq2d 6666	. . . . . 6 |- ( ( S e. V /\ N e. NN0 ) -> ( 0 ... N ) = ( 0 ... ( # ` ( S repeatS N ) ) ) )
6	5	eleq2d 2687	. . . . 5 |- ( ( S e. V /\ N e. NN0 ) -> ( L e. ( 0 ... N ) <-> L e. ( 0 ... ( # ` ( S repeatS N ) ) ) ) )
7	6	biimp3a 1432	. . . 4 |- ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) -> L e. ( 0 ... ( # ` ( S repeatS N ) ) ) )
8		pfxlen 41391	. . . 4 |- ( ( ( S repeatS N ) e. Word V /\ L e. ( 0 ... ( # ` ( S repeatS N ) ) ) ) -> ( # ` ( ( S repeatS N ) prefix L ) ) = L )
9	2, 7, 8	syl2anc 693	. . 3 |- ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) -> ( # ` ( ( S repeatS N ) prefix L ) ) = L )
10		elfznn0 12433	. . . . . 6 |- ( L e. ( 0 ... N ) -> L e. NN0 )
11	10	anim2i 593	. . . . 5 |- ( ( S e. V /\ L e. ( 0 ... N ) ) -> ( S e. V /\ L e. NN0 ) )
12	11	3adant2 1080	. . . 4 |- ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) -> ( S e. V /\ L e. NN0 ) )
13		repswlen 13523	. . . 4 |- ( ( S e. V /\ L e. NN0 ) -> ( # ` ( S repeatS L ) ) = L )
14	12, 13	syl 17	. . 3 |- ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) -> ( # ` ( S repeatS L ) ) = L )
15	9, 14	eqtr4d 2659	. 2 |- ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) -> ( # ` ( ( S repeatS N ) prefix L ) ) = ( # ` ( S repeatS L ) ) )
16		simpl1 1064	. . . . 5 |- ( ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) /\ i e. ( 0..^ ( # ` ( ( S repeatS N ) prefix L ) ) ) ) -> S e. V )
17		simpl2 1065	. . . . 5 |- ( ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) /\ i e. ( 0..^ ( # ` ( ( S repeatS N ) prefix L ) ) ) ) -> N e. NN0 )
18		elfzuz3 12339	. . . . . . . . 9 |- ( L e. ( 0 ... N ) -> N e. ( ZZ>= ` L ) )
19	18	3ad2ant3 1084	. . . . . . . 8 |- ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) -> N e. ( ZZ>= ` L ) )
20	9	fveq2d 6195	. . . . . . . 8 |- ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) -> ( ZZ>= ` ( # ` ( ( S repeatS N ) prefix L ) ) ) = ( ZZ>= ` L ) )
21	19, 20	eleqtrrd 2704	. . . . . . 7 |- ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) -> N e. ( ZZ>= ` ( # ` ( ( S repeatS N ) prefix L ) ) ) )
22		fzoss2 12496	. . . . . . 7 |- ( N e. ( ZZ>= ` ( # ` ( ( S repeatS N ) prefix L ) ) ) -> ( 0..^ ( # ` ( ( S repeatS N ) prefix L ) ) ) C_ ( 0..^ N ) )
23	21, 22	syl 17	. . . . . 6 |- ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) -> ( 0..^ ( # ` ( ( S repeatS N ) prefix L ) ) ) C_ ( 0..^ N ) )
24	23	sselda 3603	. . . . 5 |- ( ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) /\ i e. ( 0..^ ( # ` ( ( S repeatS N ) prefix L ) ) ) ) -> i e. ( 0..^ N ) )
25		repswsymb 13521	. . . . 5 |- ( ( S e. V /\ N e. NN0 /\ i e. ( 0..^ N ) ) -> ( ( S repeatS N ) ` i ) = S )
26	16, 17, 24, 25	syl3anc 1326	. . . 4 |- ( ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) /\ i e. ( 0..^ ( # ` ( ( S repeatS N ) prefix L ) ) ) ) -> ( ( S repeatS N ) ` i ) = S )
27	2	adantr 481	. . . . 5 |- ( ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) /\ i e. ( 0..^ ( # ` ( ( S repeatS N ) prefix L ) ) ) ) -> ( S repeatS N ) e. Word V )
28	7	adantr 481	. . . . 5 |- ( ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) /\ i e. ( 0..^ ( # ` ( ( S repeatS N ) prefix L ) ) ) ) -> L e. ( 0 ... ( # ` ( S repeatS N ) ) ) )
29	9	oveq2d 6666	. . . . . . 7 |- ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) -> ( 0..^ ( # ` ( ( S repeatS N ) prefix L ) ) ) = ( 0..^ L ) )
30	29	eleq2d 2687	. . . . . 6 |- ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) -> ( i e. ( 0..^ ( # ` ( ( S repeatS N ) prefix L ) ) ) <-> i e. ( 0..^ L ) ) )
31	30	biimpa 501	. . . . 5 |- ( ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) /\ i e. ( 0..^ ( # ` ( ( S repeatS N ) prefix L ) ) ) ) -> i e. ( 0..^ L ) )
32		pfxfv 41399	. . . . 5 |- ( ( ( S repeatS N ) e. Word V /\ L e. ( 0 ... ( # ` ( S repeatS N ) ) ) /\ i e. ( 0..^ L ) ) -> ( ( ( S repeatS N ) prefix L ) ` i ) = ( ( S repeatS N ) ` i ) )
33	27, 28, 31, 32	syl3anc 1326	. . . 4 |- ( ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) /\ i e. ( 0..^ ( # ` ( ( S repeatS N ) prefix L ) ) ) ) -> ( ( ( S repeatS N ) prefix L ) ` i ) = ( ( S repeatS N ) ` i ) )
34	10	3ad2ant3 1084	. . . . . 6 |- ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) -> L e. NN0 )
35	34	adantr 481	. . . . 5 |- ( ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) /\ i e. ( 0..^ ( # ` ( ( S repeatS N ) prefix L ) ) ) ) -> L e. NN0 )
36		repswsymb 13521	. . . . 5 |- ( ( S e. V /\ L e. NN0 /\ i e. ( 0..^ L ) ) -> ( ( S repeatS L ) ` i ) = S )
37	16, 35, 31, 36	syl3anc 1326	. . . 4 |- ( ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) /\ i e. ( 0..^ ( # ` ( ( S repeatS N ) prefix L ) ) ) ) -> ( ( S repeatS L ) ` i ) = S )
38	26, 33, 37	3eqtr4d 2666	. . 3 |- ( ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) /\ i e. ( 0..^ ( # ` ( ( S repeatS N ) prefix L ) ) ) ) -> ( ( ( S repeatS N ) prefix L ) ` i ) = ( ( S repeatS L ) ` i ) )
39	38	ralrimiva 2966	. 2 |- ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) -> A. i e. ( 0..^ ( # ` ( ( S repeatS N ) prefix L ) ) ) ( ( ( S repeatS N ) prefix L ) ` i ) = ( ( S repeatS L ) ` i ) )
40		pfxcl 41386	. . . 4 |- ( ( S repeatS N ) e. Word V -> ( ( S repeatS N ) prefix L ) e. Word V )
41	2, 40	syl 17	. . 3 |- ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) -> ( ( S repeatS N ) prefix L ) e. Word V )
42		repsw 13522	. . . 4 |- ( ( S e. V /\ L e. NN0 ) -> ( S repeatS L ) e. Word V )
43	12, 42	syl 17	. . 3 |- ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) -> ( S repeatS L ) e. Word V )
44		eqwrd 13346	. . 3 |- ( ( ( ( S repeatS N ) prefix L ) e. Word V /\ ( S repeatS L ) e. Word V ) -> ( ( ( S repeatS N ) prefix L ) = ( S repeatS L ) <-> ( ( # ` ( ( S repeatS N ) prefix L ) ) = ( # ` ( S repeatS L ) ) /\ A. i e. ( 0..^ ( # ` ( ( S repeatS N ) prefix L ) ) ) ( ( ( S repeatS N ) prefix L ) ` i ) = ( ( S repeatS L ) ` i ) ) ) )
45	41, 43, 44	syl2anc 693	. 2 |- ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) -> ( ( ( S repeatS N ) prefix L ) = ( S repeatS L ) <-> ( ( # ` ( ( S repeatS N ) prefix L ) ) = ( # ` ( S repeatS L ) ) /\ A. i e. ( 0..^ ( # ` ( ( S repeatS N ) prefix L ) ) ) ( ( ( S repeatS N ) prefix L ) ` i ) = ( ( S repeatS L ) ` i ) ) ) )
46	15, 39, 45	mpbir2and 957	1 |- ( ( S e. V /\ N e. NN0 /\ L e. ( 0 ... N ) ) -> ( ( S repeatS N ) prefix L ) = ( S repeatS L ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   ` cfv 5888  (class class class)co 6650   0cc0 9936   NN0cn0 11292   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   repeatS creps 13298   prefix cpfx 41381

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-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-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-substr 13303  df-reps 13306  df-pfx 41382

This theorem is referenced by: (None)