Theorem swrdpfx 11260

Description: A subword of a prefix is a subword. (Contributed by Alexander van der Vekens, 6-Apr-2018.) (Revised by AV, 8-May-2020.)

Assertion

Ref	Expression
swrdpfx	|- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) -> ( ( K e. ( 0 ... N ) /\ L e. ( K ... N ) ) -> ( ( W prefix N ) substr <. K , L >. ) = ( W substr <. K , L >. ) ) )

Proof of Theorem swrdpfx

Step	Hyp	Ref	Expression
1		elfznn0 10327	. . . . . . 7 |- ( N e. ( 0 ... (♯ ` W ) ) -> N e. NN0 )
2	1	anim2i 342	. . . . . 6 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) -> ( W e. Word V /\ N e. NN0 ) )
3	2	adantr 276	. . . . 5 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) /\ ( K e. ( 0 ... N ) /\ L e. ( K ... N ) ) ) -> ( W e. Word V /\ N e. NN0 ) )
4		pfxval 11227	. . . . 5 |- ( ( W e. Word V /\ N e. NN0 ) -> ( W prefix N ) = ( W substr <. 0 , N >. ) )
5	3, 4	syl 14	. . . 4 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) /\ ( K e. ( 0 ... N ) /\ L e. ( K ... N ) ) ) -> ( W prefix N ) = ( W substr <. 0 , N >. ) )
6	5	oveq1d 6025	. . 3 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) /\ ( K e. ( 0 ... N ) /\ L e. ( K ... N ) ) ) -> ( ( W prefix N ) substr <. K , L >. ) = ( ( W substr <. 0 , N >. ) substr <. K , L >. ) )
7		simpl 109	. . . . . 6 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) -> W e. Word V )
8		simpr 110	. . . . . 6 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) -> N e. ( 0 ... (♯ ` W ) ) )
9		0elfz 10331	. . . . . . . 8 |- ( N e. NN0 -> 0 e. ( 0 ... N ) )
10	1, 9	syl 14	. . . . . . 7 |- ( N e. ( 0 ... (♯ ` W ) ) -> 0 e. ( 0 ... N ) )
11	10	adantl 277	. . . . . 6 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) -> 0 e. ( 0 ... N ) )
12	7, 8, 11	3jca 1201	. . . . 5 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) -> ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ 0 e. ( 0 ... N ) ) )
13	12	adantr 276	. . . 4 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) /\ ( K e. ( 0 ... N ) /\ L e. ( K ... N ) ) ) -> ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ 0 e. ( 0 ... N ) ) )
14		elfzelz 10238	. . . . . . . . . 10 |- ( N e. ( 0 ... (♯ ` W ) ) -> N e. ZZ )
15		zcn 9467	. . . . . . . . . . . 12 |- ( N e. ZZ -> N e. CC )
16	15	subid1d 8462	. . . . . . . . . . 11 |- ( N e. ZZ -> ( N - 0 ) = N )
17	16	eqcomd 2235	. . . . . . . . . 10 |- ( N e. ZZ -> N = ( N - 0 ) )
18	14, 17	syl 14	. . . . . . . . 9 |- ( N e. ( 0 ... (♯ ` W ) ) -> N = ( N - 0 ) )
19	18	adantl 277	. . . . . . . 8 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) -> N = ( N - 0 ) )
20	19	oveq2d 6026	. . . . . . 7 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) -> ( 0 ... N ) = ( 0 ... ( N - 0 ) ) )
21	20	eleq2d 2299	. . . . . 6 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) -> ( K e. ( 0 ... N ) <-> K e. ( 0 ... ( N - 0 ) ) ) )
22	19	oveq2d 6026	. . . . . . 7 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) -> ( K ... N ) = ( K ... ( N - 0 ) ) )
23	22	eleq2d 2299	. . . . . 6 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) -> ( L e. ( K ... N ) <-> L e. ( K ... ( N - 0 ) ) ) )
24	21, 23	anbi12d 473	. . . . 5 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) -> ( ( K e. ( 0 ... N ) /\ L e. ( K ... N ) ) <-> ( K e. ( 0 ... ( N - 0 ) ) /\ L e. ( K ... ( N - 0 ) ) ) ) )
25	24	biimpa 296	. . . 4 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) /\ ( K e. ( 0 ... N ) /\ L e. ( K ... N ) ) ) -> ( K e. ( 0 ... ( N - 0 ) ) /\ L e. ( K ... ( N - 0 ) ) ) )
26		swrdswrd 11258	. . . 4 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ 0 e. ( 0 ... N ) ) -> ( ( K e. ( 0 ... ( N - 0 ) ) /\ L e. ( K ... ( N - 0 ) ) ) -> ( ( W substr <. 0 , N >. ) substr <. K , L >. ) = ( W substr <. ( 0 + K ) , ( 0 + L ) >. ) ) )
27	13, 25, 26	sylc 62	. . 3 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) /\ ( K e. ( 0 ... N ) /\ L e. ( K ... N ) ) ) -> ( ( W substr <. 0 , N >. ) substr <. K , L >. ) = ( W substr <. ( 0 + K ) , ( 0 + L ) >. ) )
28		elfzelz 10238	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> K e. ZZ )
29	28	zcnd 9586	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> K e. CC )
30	29	adantr 276	. . . . . . 7 |- ( ( K e. ( 0 ... N ) /\ L e. ( K ... N ) ) -> K e. CC )
31	30	adantl 277	. . . . . 6 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) /\ ( K e. ( 0 ... N ) /\ L e. ( K ... N ) ) ) -> K e. CC )
32	31	addlidd 8312	. . . . 5 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) /\ ( K e. ( 0 ... N ) /\ L e. ( K ... N ) ) ) -> ( 0 + K ) = K )
33		elfzelz 10238	. . . . . . . . 9 |- ( L e. ( K ... N ) -> L e. ZZ )
34	33	zcnd 9586	. . . . . . . 8 |- ( L e. ( K ... N ) -> L e. CC )
35	34	adantl 277	. . . . . . 7 |- ( ( K e. ( 0 ... N ) /\ L e. ( K ... N ) ) -> L e. CC )
36	35	adantl 277	. . . . . 6 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) /\ ( K e. ( 0 ... N ) /\ L e. ( K ... N ) ) ) -> L e. CC )
37	36	addlidd 8312	. . . . 5 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) /\ ( K e. ( 0 ... N ) /\ L e. ( K ... N ) ) ) -> ( 0 + L ) = L )
38	32, 37	opeq12d 3865	. . . 4 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) /\ ( K e. ( 0 ... N ) /\ L e. ( K ... N ) ) ) -> <. ( 0 + K ) , ( 0 + L ) >. = <. K , L >. )
39	38	oveq2d 6026	. . 3 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) /\ ( K e. ( 0 ... N ) /\ L e. ( K ... N ) ) ) -> ( W substr <. ( 0 + K ) , ( 0 + L ) >. ) = ( W substr <. K , L >. ) )
40	6, 27, 39	3eqtrd 2266	. 2 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) /\ ( K e. ( 0 ... N ) /\ L e. ( K ... N ) ) ) -> ( ( W prefix N ) substr <. K , L >. ) = ( W substr <. K , L >. ) )
41	40	ex 115	1 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) -> ( ( K e. ( 0 ... N ) /\ L e. ( K ... N ) ) -> ( ( W prefix N ) substr <. K , L >. ) = ( W substr <. K , L >. ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   <.cop 3669   ` cfv 5321  (class class class)co 6010   CCcc 8013   0cc0 8015   + caddc 8018   - cmin 8333   NN0cn0 9385   ZZcz 9462   ...cfz 10221  ♯chash 11014  Word cword 11089   substr csubstr 11198   prefix cpfx 11225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-addass 8117  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-0id 8123  ax-rnegex 8124  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-frec 6548  df-1o 6573  df-er 6693  df-en 6901  df-dom 6902  df-fin 6903  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-inn 9127  df-n0 9386  df-z 9463  df-uz 9739  df-fz 10222  df-fzo 10356  df-ihash 11015  df-word 11090  df-substr 11199  df-pfx 11226

This theorem is referenced by: (None)