Theorem swrdpfx 11335

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 10392	. . . . . . 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 11302	. . . . 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 6043	. . 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 10396	. . . . . . . 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 1204	. . . . 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 10303	. . . . . . . . . 10 |- ( N e. ( 0 ... (♯ ` W ) ) -> N e. ZZ )
15		zcn 9527	. . . . . . . . . . . 12 |- ( N e. ZZ -> N e. CC )
16	15	subid1d 8522	. . . . . . . . . . 11 |- ( N e. ZZ -> ( N - 0 ) = N )
17	16	eqcomd 2237	. . . . . . . . . 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 6044	. . . . . . 7 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) -> ( 0 ... N ) = ( 0 ... ( N - 0 ) ) )
21	20	eleq2d 2301	. . . . . 6 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) -> ( K e. ( 0 ... N ) <-> K e. ( 0 ... ( N - 0 ) ) ) )
22	19	oveq2d 6044	. . . . . . 7 |- ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) -> ( K ... N ) = ( K ... ( N - 0 ) ) )
23	22	eleq2d 2301	. . . . . 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 11333	. . . 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 10303	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> K e. ZZ )
29	28	zcnd 9646	. . . . . . . 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 8372	. . . . 5 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) /\ ( K e. ( 0 ... N ) /\ L e. ( K ... N ) ) ) -> ( 0 + K ) = K )
33		elfzelz 10303	. . . . . . . . 9 |- ( L e. ( K ... N ) -> L e. ZZ )
34	33	zcnd 9646	. . . . . . . 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 8372	. . . . 5 |- ( ( ( W e. Word V /\ N e. ( 0 ... (♯ ` W ) ) ) /\ ( K e. ( 0 ... N ) /\ L e. ( K ... N ) ) ) -> ( 0 + L ) = L )
38	32, 37	opeq12d 3875	. . . 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 6044	. . 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 2268	. 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 1005   = wceq 1398   e. wcel 2202   <.cop 3676   ` cfv 5333  (class class class)co 6028   CCcc 8073   0cc0 8075   + caddc 8078   - cmin 8393   NN0cn0 9445   ZZcz 9522   ...cfz 10286  ♯chash 11081  Word cword 11160   substr csubstr 11273   prefix cpfx 11300

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-inn 9187  df-n0 9446  df-z 9523  df-uz 9799  df-fz 10287  df-fzo 10421  df-ihash 11082  df-word 11161  df-substr 11274  df-pfx 11301

This theorem is referenced by: (None)