Theorem pfxpfx 11463

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 10504	. . . . . 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 1048	. . . 4 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> ( W e. Word V /\ N e. NN0 ) )
4		pfxval 11429	. . . 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 6094	. 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 1028	. . . 4 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> W e. Word V )
8		simp2 1029	. . . 4 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> N e. ( 0 ... (♯ ` W ) ) )
9		0elfz 10508	. . . . . 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 1050	. . . 4 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> 0 e. ( 0 ... N ) )
12	7, 8, 11	3jca 1208	. . 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 9605	. . . . . . . . 9 |- ( N e. ( 0 ... (♯ ` W ) ) -> N e. CC )
14	13	subid1d 8620	. . . . . . . 8 |- ( N e. ( 0 ... (♯ ` W ) ) -> ( N - 0 ) = N )
15	14	eqcomd 2244	. . . . . . 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 6095	. . . . 5 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) -> ( 0 ... N ) = ( 0 ... ( N - 0 ) ) )
18	17	eleq2d 2308	. . . 4 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) -> ( L e. ( 0 ... N ) <-> L e. ( 0 ... ( N - 0 ) ) ) )
19	18	biimp3a 1386	. . 3 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> L e. ( 0 ... ( N - 0 ) ) )
20		pfxswrd 11461	. . 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 10504	. . . . . . . 8 |- ( L e. ( 0 ... N ) -> L e. NN0 )
23	22	nn0cnd 9605	. . . . . . 7 |- ( L e. ( 0 ... N ) -> L e. CC )
24	23	addlidd 8470	. . . . . 6 |- ( L e. ( 0 ... N ) -> ( 0 + L ) = L )
25	24	opeq2d 3909	. . . . 5 |- ( L e. ( 0 ... N ) -> <. 0 , ( 0 + L ) >. = <. 0 , L >. )
26	25	oveq2d 6095	. . . 4 |- ( L e. ( 0 ... N ) -> ( W substr <. 0 , ( 0 + L ) >. ) = ( W substr <. 0 , L >. ) )
27	26	3ad2ant3 1051	. . 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 1047	. . . 4 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) /\ L e. ( 0 ... N ) ) -> ( W e. Word V /\ L e. NN0 ) )
30		pfxval 11429	. . . 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 2274	. 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 2275	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 1009   = wceq 1402   e. wcel 2209   <.cop 3711   ` cfv 5375  (class class class)co 6079   0cc0 8173   + caddc 8176   - cmin 8491   NN0cn0 9546   ...cfz 10394  ♯chash 11197  Word cword 11287   substr csubstr 11400   prefix cpfx 11427

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-1o 6681  df-er 6801  df-en 7017  df-dom 7018  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-inn 9288  df-n0 9547  df-z 9628  df-uz 9905  df-fz 10395  df-fzo 10533  df-ihash 11198  df-word 11288  df-substr 11401  df-pfx 11428

This theorem is referenced by:  pfxpfxid  11464