Theorem pfxpfx 11282

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 10342	. . . . . 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 1041	. . . 4 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> ( W e. Word V /\ N e. NN0 ) )
4		pfxval 11248	. . . 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 6028	. 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 1021	. . . 4 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> W e. Word V )
8		simp2 1022	. . . 4 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> N e. ( 0 ... (♯ ` W ) ) )
9		0elfz 10346	. . . . . 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 1043	. . . 4 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> 0 e. ( 0 ... N ) )
12	7, 8, 11	3jca 1201	. . 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 9450	. . . . . . . . 9 |- ( N e. ( 0 ... (♯ ` W ) ) -> N e. CC )
14	13	subid1d 8472	. . . . . . . 8 |- ( N e. ( 0 ... (♯ ` W ) ) -> ( N - 0 ) = N )
15	14	eqcomd 2235	. . . . . . 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 6029	. . . . 5 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) -> ( 0 ... N ) = ( 0 ... ( N - 0 ) ) )
18	17	eleq2d 2299	. . . 4 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) -> ( L e. ( 0 ... N ) <-> L e. ( 0 ... ( N - 0 ) ) ) )
19	18	biimp3a 1379	. . 3 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> L e. ( 0 ... ( N - 0 ) ) )
20		pfxswrd 11280	. . 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 10342	. . . . . . . 8 |- ( L e. ( 0 ... N ) -> L e. NN0 )
23	22	nn0cnd 9450	. . . . . . 7 |- ( L e. ( 0 ... N ) -> L e. CC )
24	23	addlidd 8322	. . . . . 6 |- ( L e. ( 0 ... N ) -> ( 0 + L ) = L )
25	24	opeq2d 3867	. . . . 5 |- ( L e. ( 0 ... N ) -> <. 0 , ( 0 + L ) >. = <. 0 , L >. )
26	25	oveq2d 6029	. . . 4 |- ( L e. ( 0 ... N ) -> ( W substr <. 0 , ( 0 + L ) >. ) = ( W substr <. 0 , L >. ) )
27	26	3ad2ant3 1044	. . 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 1040	. . . 4 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> ( W e. Word V /\ L e. NN0 ) )
30		pfxval 11248	. . . 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 2265	. 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 2266	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 1002   = wceq 1395   e. wcel 2200   <.cop 3670   ` cfv 5324  (class class class)co 6013   0cc0 8025   + caddc 8028   - cmin 8343   NN0cn0 9395   ...cfz 10236  ♯chash 11030  Word cword 11106   substr csubstr 11219   prefix cpfx 11246

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 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-addass 8127  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-0id 8133  ax-rnegex 8134  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-1o 6577  df-er 6697  df-en 6905  df-dom 6906  df-fin 6907  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-inn 9137  df-n0 9396  df-z 9473  df-uz 9749  df-fz 10237  df-fzo 10371  df-ihash 11031  df-word 11107  df-substr 11220  df-pfx 11247

This theorem is referenced by:  pfxpfxid  11283