Theorem pfxpfx 11425

Description: A prefix of a prefix is a prefix. (Contributed by Alexander van der Vekens, 7-Apr-2018.) (Revised by AV, 8-May-2020.)

Assertion

Ref	Expression
pfxpfx	|- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> ( ( W prefix N ) prefix L ) = ( W prefix L ) )

Proof of Theorem pfxpfx

Step	Hyp	Ref	Expression
1		elfznn0 10470	. . . . . 6 |- ( N e. ( 0 ... (♯ ` W ) ) -> N e. NN0 )
2	1	anim2i 342	. . . . 5 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) -> ( W e. Word V /\ N e. NN0 ) )
3	2	3adant3 1044	. . . 4 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> ( W e. Word V /\ N e. NN0 ) )
4		pfxval 11391	. . . 4 |- ( ( W e. Word V /\ N e. NN0 ) -> ( W prefix N ) = ( W substr <. 0 , N >. ) )
5	3, 4	syl 14	. . 3 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> ( W prefix N ) = ( W substr <. 0 , N >. ) )
6	5	oveq1d 6073	. 2 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> ( ( W prefix N ) prefix L ) = ( ( W substr <. 0 , N >. ) prefix L ) )
7		simp1 1024	. . . 4 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> W e. Word V )
8		simp2 1025	. . . 4 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> N e. ( 0 ... (♯ ` W ) ) )
9		0elfz 10474	. . . . . 6 |- ( N e. NN0 -> 0 e. ( 0 ... N ) )
10	1, 9	syl 14	. . . . 5 |- ( N e. ( 0 ... (♯ ` W ) ) -> 0 e. ( 0 ... N ) )
11	10	3ad2ant2 1046	. . . 4 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> 0 e. ( 0 ... N ) )
12	7, 8, 11	3jca 1204	. . 3 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ 0 e. ( 0 ... N ) ) )
13	1	nn0cnd 9572	. . . . . . . . 9 |- ( N e. ( 0 ... (♯ ` W ) ) -> N e. CC )
14	13	subid1d 8589	. . . . . . . 8 |- ( N e. ( 0 ... (♯ ` W ) ) -> ( N - 0 ) = N )
15	14	eqcomd 2240	. . . . . . 7 |- ( N e. ( 0 ... (♯ ` W ) ) -> N = ( N - 0 ) )
16	15	adantl 277	. . . . . 6 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) -> N = ( N - 0 ) )
17	16	oveq2d 6074	. . . . 5 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) -> ( 0 ... N ) = ( 0 ... ( N - 0 ) ) )
18	17	eleq2d 2304	. . . 4 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) -> ( L e. ( 0 ... N ) <-> L e. ( 0 ... ( N - 0 ) ) ) )
19	18	biimp3a 1382	. . 3 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> L e. ( 0 ... ( N - 0 ) ) )
20		pfxswrd 11423	. . 3 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ 0 e. ( 0 ... N ) ) -> ( L e. ( 0 ... ( N - 0 ) ) -> ( ( W substr <. 0 , N >. ) prefix L ) = ( W substr <. 0 , ( 0 + L ) >. ) ) )
21	12, 19, 20	sylc 62	. 2 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> ( ( W substr <. 0 , N >. ) prefix L ) = ( W substr <. 0 , ( 0 + L ) >. ) )
22		elfznn0 10470	. . . . . . . 8 |- ( L e. ( 0 ... N ) -> L e. NN0 )
23	22	nn0cnd 9572	. . . . . . 7 |- ( L e. ( 0 ... N ) -> L e. CC )
24	23	addlidd 8439	. . . . . 6 |- ( L e. ( 0 ... N ) -> ( 0 + L ) = L )
25	24	opeq2d 3895	. . . . 5 |- ( L e. ( 0 ... N ) -> <. 0 , ( 0 + L ) >. = <. 0 , L >. )
26	25	oveq2d 6074	. . . 4 |- ( L e. ( 0 ... N ) -> ( W substr <. 0 , ( 0 + L ) >. ) = ( W substr <. 0 , L >. ) )
27	26	3ad2ant3 1047	. . 3 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> ( W substr <. 0 , ( 0 + L ) >. ) = ( W substr <. 0 , L >. ) )
28	22	anim2i 342	. . . . 5 |- ( ( W e. Word V /\ L e. ( 0 ... N ) ) -> ( W e. Word V /\ L e. NN0 ) )
29	28	3adant2 1043	. . . 4 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> ( W e. Word V /\ L e. NN0 ) )
30		pfxval 11391	. . . 4 |- ( ( W e. Word V /\ L e. NN0 ) -> ( W prefix L ) = ( W substr <. 0 , L >. ) )
31	29, 30	syl 14	. . 3 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> ( W prefix L ) = ( W substr <. 0 , L >. ) )
32	27, 31	eqtr4d 2270	. 2 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> ( W substr <. 0 , ( 0 + L ) >. ) = ( W prefix L ) )
33	6, 21, 32	3eqtrd 2271	1 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> ( ( W prefix N ) prefix L ) = ( W prefix L ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   <.cop 3697   ` cfv 5357  (class class class)co 6058   0cc0 8143   + caddc 8146   - cmin 8460   NN0cn0 9513   ...cfz 10361  ♯chash 11163  Word cword 11249   substr csubstr 11362   prefix cpfx 11389

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-substr 11363  df-pfx 11390

This theorem is referenced by:  pfxpfxid  11426