Theorem swrdwrdsymbg 11356

Description: A subword is a word over the symbols it consists of. (Contributed by AV, 2-Dec-2022.)

Assertion

Ref	Expression
swrdwrdsymbg	|- ( ( S e. Word A /\ M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) -> ( S substr <. M , N >. ) e. Word ( S " ( M..^ N ) ) )

Proof of Theorem swrdwrdsymbg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		swrdval2 11343	. . . 4 |- ( ( S e. Word A /\ M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) -> ( S substr <. M , N >. ) = ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) )
2	1	3expb 1231	. . 3 |- ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) -> ( S substr <. M , N >. ) = ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) )
3		wrdf 11230	. . . . . . . . . 10 |- ( S e. Word A -> S : ( 0..^ (♯ ` S ) ) --> A )
4	3	ffund 5512	. . . . . . . . 9 |- ( S e. Word A -> Fun S )
5	4	adantr 276	. . . . . . . 8 |- ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) -> Fun S )
6	5	adantr 276	. . . . . . 7 |- ( ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( N - M ) ) ) -> Fun S )
7		wrddm 11232	. . . . . . . . 9 |- ( S e. Word A -> dom S = ( 0..^ (♯ ` S ) ) )
8		elfzodifsumelfzo 10546	. . . . . . . . . . . . 13 |- ( ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) -> ( x e. ( 0..^ ( N - M ) ) -> ( x + M ) e. ( 0..^ (♯ ` S ) ) ) )
9	8	imp 124	. . . . . . . . . . . 12 |- ( ( ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) /\ x e. ( 0..^ ( N - M ) ) ) -> ( x + M ) e. ( 0..^ (♯ ` S ) ) )
10	9	adantl 277	. . . . . . . . . . 11 |- ( ( dom S = ( 0..^ (♯ ` S ) ) /\ ( ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) /\ x e. ( 0..^ ( N - M ) ) ) ) -> ( x + M ) e. ( 0..^ (♯ ` S ) ) )
11		eleq2 2296	. . . . . . . . . . . 12 |- ( dom S = ( 0..^ (♯ ` S ) ) -> ( ( x + M ) e. dom S <-> ( x + M ) e. ( 0..^ (♯ ` S ) ) ) )
12	11	adantr 276	. . . . . . . . . . 11 |- ( ( dom S = ( 0..^ (♯ ` S ) ) /\ ( ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) /\ x e. ( 0..^ ( N - M ) ) ) ) -> ( ( x + M ) e. dom S <-> ( x + M ) e. ( 0..^ (♯ ` S ) ) ) )
13	10, 12	mpbird 167	. . . . . . . . . 10 |- ( ( dom S = ( 0..^ (♯ ` S ) ) /\ ( ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) /\ x e. ( 0..^ ( N - M ) ) ) ) -> ( x + M ) e. dom S )
14	13	exp32 365	. . . . . . . . 9 |- ( dom S = ( 0..^ (♯ ` S ) ) -> ( ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) -> ( x e. ( 0..^ ( N - M ) ) -> ( x + M ) e. dom S ) ) )
15	7, 14	syl 14	. . . . . . . 8 |- ( S e. Word A -> ( ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) -> ( x e. ( 0..^ ( N - M ) ) -> ( x + M ) e. dom S ) ) )
16	15	imp31 256	. . . . . . 7 |- ( ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( N - M ) ) ) -> ( x + M ) e. dom S )
17		simpr 110	. . . . . . . 8 |- ( ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( N - M ) ) ) -> x e. ( 0..^ ( N - M ) ) )
18		elfzelz 10359	. . . . . . . . . . 11 |- ( N e. ( 0 ... (♯ ` S ) ) -> N e. ZZ )
19	18	adantl 277	. . . . . . . . . 10 |- ( ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) -> N e. ZZ )
20	19	adantl 277	. . . . . . . . 9 |- ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) -> N e. ZZ )
21	20	adantr 276	. . . . . . . 8 |- ( ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( N - M ) ) ) -> N e. ZZ )
22		elfzelz 10359	. . . . . . . . . 10 |- ( M e. ( 0 ... N ) -> M e. ZZ )
23	22	ad2antrl 490	. . . . . . . . 9 |- ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) -> M e. ZZ )
24	23	adantr 276	. . . . . . . 8 |- ( ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( N - M ) ) ) -> M e. ZZ )
25		fzoaddel2 10535	. . . . . . . 8 |- ( ( x e. ( 0..^ ( N - M ) ) /\ N e. ZZ /\ M e. ZZ ) -> ( x + M ) e. ( M..^ N ) )
26	17, 21, 24, 25	syl3anc 1274	. . . . . . 7 |- ( ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( N - M ) ) ) -> ( x + M ) e. ( M..^ N ) )
27		funfvima 5918	. . . . . . . 8 |- ( ( Fun S /\ ( x + M ) e. dom S ) -> ( ( x + M ) e. ( M..^ N ) -> ( S ` ( x + M ) ) e. ( S " ( M..^ N ) ) ) )
28	27	imp 124	. . . . . . 7 |- ( ( ( Fun S /\ ( x + M ) e. dom S ) /\ ( x + M ) e. ( M..^ N ) ) -> ( S ` ( x + M ) ) e. ( S " ( M..^ N ) ) )
29	6, 16, 26, 28	syl21anc 1273	. . . . . 6 |- ( ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( N - M ) ) ) -> ( S ` ( x + M ) ) e. ( S " ( M..^ N ) ) )
30	29	fmpttd 5832	. . . . 5 |- ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) -> ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) : ( 0..^ ( N - M ) ) --> ( S " ( M..^ N ) ) )
31		simpll 527	. . . . . . . . . . . 12 |- ( ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( N - M ) ) ) -> S e. Word A )
32		elfzoelz 10481	. . . . . . . . . . . . . 14 |- ( x e. ( 0..^ ( N - M ) ) -> x e. ZZ )
33	32	adantl 277	. . . . . . . . . . . . 13 |- ( ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( N - M ) ) ) -> x e. ZZ )
34	33, 24	zaddcld 9704	. . . . . . . . . . . 12 |- ( ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( N - M ) ) ) -> ( x + M ) e. ZZ )
35		fvexg 5689	. . . . . . . . . . . 12 |- ( ( S e. Word A /\ ( x + M ) e. ZZ ) -> ( S ` ( x + M ) ) e. _V )
36	31, 34, 35	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( N - M ) ) ) -> ( S ` ( x + M ) ) e. _V )
37	36	ralrimiva 2615	. . . . . . . . . 10 |- ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) -> A. x e. ( 0..^ ( N - M ) ) ( S ` ( x + M ) ) e. _V )
38		eqid 2232	. . . . . . . . . . 11 |- ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) = ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) )
39	38	fnmpt 5485	. . . . . . . . . 10 |- ( A. x e. ( 0..^ ( N - M ) ) ( S ` ( x + M ) ) e. _V -> ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) Fn ( 0..^ ( N - M ) ) )
40	37, 39	syl 14	. . . . . . . . 9 |- ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) -> ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) Fn ( 0..^ ( N - M ) ) )
41		0z 9588	. . . . . . . . . . 11 |- 0 e. ZZ
42		elfzel2 10357	. . . . . . . . . . . 12 |- ( M e. ( 0 ... N ) -> N e. ZZ )
43	42, 22	zsubcld 9705	. . . . . . . . . . 11 |- ( M e. ( 0 ... N ) -> ( N - M ) e. ZZ )
44		fzofig 10794	. . . . . . . . . . 11 |- ( ( 0 e. ZZ /\ ( N - M ) e. ZZ ) -> ( 0..^ ( N - M ) ) e. Fin )
45	41, 43, 44	sylancr 414	. . . . . . . . . 10 |- ( M e. ( 0 ... N ) -> ( 0..^ ( N - M ) ) e. Fin )
46	45	ad2antrl 490	. . . . . . . . 9 |- ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) -> ( 0..^ ( N - M ) ) e. Fin )
47		fihashfn 11164	. . . . . . . . 9 |- ( ( ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) Fn ( 0..^ ( N - M ) ) /\ ( 0..^ ( N - M ) ) e. Fin ) -> (♯ ` ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) ) = (♯ ` ( 0..^ ( N - M ) ) ) )
48	40, 46, 47	syl2anc 411	. . . . . . . 8 |- ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) -> (♯ ` ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) ) = (♯ ` ( 0..^ ( N - M ) ) ) )
49		fznn0sub 10391	. . . . . . . . . 10 |- ( M e. ( 0 ... N ) -> ( N - M ) e. NN0 )
50		hashfzo0 11188	. . . . . . . . . 10 |- ( ( N - M ) e. NN0 -> (♯ ` ( 0..^ ( N - M ) ) ) = ( N - M ) )
51	49, 50	syl 14	. . . . . . . . 9 |- ( M e. ( 0 ... N ) -> (♯ ` ( 0..^ ( N - M ) ) ) = ( N - M ) )
52	51	ad2antrl 490	. . . . . . . 8 |- ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) -> (♯ ` ( 0..^ ( N - M ) ) ) = ( N - M ) )
53	48, 52	eqtrd 2265	. . . . . . 7 |- ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) -> (♯ ` ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) ) = ( N - M ) )
54	53	oveq2d 6066	. . . . . 6 |- ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) -> ( 0..^ (♯ ` ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) ) ) = ( 0..^ ( N - M ) ) )
55	54	feq2d 5496	. . . . 5 |- ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) : ( 0..^ (♯ ` ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) ) ) --> ( S " ( M..^ N ) ) <-> ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) : ( 0..^ ( N - M ) ) --> ( S " ( M..^ N ) ) ) )
56	30, 55	mpbird 167	. . . 4 |- ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) -> ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) : ( 0..^ (♯ ` ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) ) ) --> ( S " ( M..^ N ) ) )
57	49	ad2antrl 490	. . . . 5 |- ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) -> ( N - M ) e. NN0 )
58	53, 57	eqeltrd 2309	. . . 4 |- ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) -> (♯ ` ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) ) e. NN0 )
59		iswrdinn0 11229	. . . 4 |- ( ( ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) : ( 0..^ (♯ ` ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) ) ) --> ( S " ( M..^ N ) ) /\ (♯ ` ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) ) e. NN0 ) -> ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) e. Word ( S " ( M..^ N ) ) )
60	56, 58, 59	syl2anc 411	. . 3 |- ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) -> ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) e. Word ( S " ( M..^ N ) ) )
61	2, 60	eqeltrd 2309	. 2 |- ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) -> ( S substr <. M , N >. ) e. Word ( S " ( M..^ N ) ) )
62	61	3impb 1226	1 |- ( ( S e. Word A /\ M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) -> ( S substr <. M , N >. ) e. Word ( S " ( M..^ N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2203   A.wral 2520   _Vcvv 2813   <.cop 3692   |-> cmpt 4171   dom cdm 4749   "cima 4752   Fun wfun 5346   Fn wfn 5347   -->wf 5348   ` cfv 5352  (class class class)co 6050   Fincfn 6975   0cc0 8127   + caddc 8130   - cmin 8444   NN0cn0 9496   ZZcz 9577   ...cfz 10342  ..^cfzo 10476  ♯chash 11138  Word cword 11224   substr csubstr 11337

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343  df-fzo 10477  df-ihash 11139  df-word 11225  df-substr 11338

This theorem is referenced by:  pfxwrdsymbg  11382