Theorem swrdpfx 11225

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 10298	. . . . . . 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 11192	. . . . 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 6009	. . 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 10302	. . . . . . . 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 10209	. . . . . . . . . 10 |- ( N e. ( 0 ... (♯ ` W ) ) -> N e. ZZ )
15		zcn 9439	. . . . . . . . . . . 12 |- ( N e. ZZ -> N e. CC )
16	15	subid1d 8434	. . . . . . . . . . 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 6010	. . . . . . 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 6010	. . . . . . 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 11223	. . . 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 10209	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> K e. ZZ )
29	28	zcnd 9558	. . . . . . . 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 8284	. . . . 5 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) /\ ( K e. ( 0 ... N ) /\ L e. ( K ... N ) ) ) -> ( 0 + K ) = K )
33		elfzelz 10209	. . . . . . . . 9 |- ( L e. ( K ... N ) -> L e. ZZ )
34	33	zcnd 9558	. . . . . . . 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 8284	. . . . 5 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) /\ ( K e. ( 0 ... N ) /\ L e. ( K ... N ) ) ) -> ( 0 + L ) = L )
38	32, 37	opeq12d 3864	. . . 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 6010	. . 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 5314  (class class class)co 5994   CCcc 7985   0cc0 7987   + caddc 7990   - cmin 8305   NN0cn0 9357   ZZcz 9434   ...cfz 10192  ♯chash 10984  Word cword 11058   substr csubstr 11163   prefix cpfx 11190

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 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-addcom 8087  ax-addass 8089  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-0id 8095  ax-rnegex 8096  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103

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 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-frec 6527  df-1o 6552  df-er 6670  df-en 6878  df-dom 6879  df-fin 6880  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-inn 9099  df-n0 9358  df-z 9435  df-uz 9711  df-fz 10193  df-fzo 10327  df-ihash 10985  df-word 11059  df-substr 11164  df-pfx 11191

This theorem is referenced by: (None)