Theorem swrdwrdsymbg 11381

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 11368	. . . 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 11255	. . . . . . . . . 10 |- ( S e. Word A -> S : ( 0..^ (♯ ` S ) ) --> A )
4	3	ffund 5517	. . . . . . . . 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 11257	. . . . . . . . 9 |- ( S e. Word A -> dom S = ( 0..^ (♯ ` S ) ) )
8		elfzodifsumelfzo 10568	. . . . . . . . . . . . 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 2298	. . . . . . . . . . . 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 10378	. . . . . . . . . . 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 10378	. . . . . . . . . 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 10557	. . . . . . . 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 5923	. . . . . . . 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 5837	. . . . 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 10503	. . . . . . . . . . . . . 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 9722	. . . . . . . . . . . 12 |- ( ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( N - M ) ) ) -> ( x + M ) e. ZZ )
35		fvexg 5694	. . . . . . . . . . . 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 2617	. . . . . . . . . 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 2234	. . . . . . . . . . 11 |- ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) = ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) )
39	38	fnmpt 5490	. . . . . . . . . 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 9605	. . . . . . . . . . 11 |- 0 e. ZZ
42		elfzel2 10376	. . . . . . . . . . . 12 |- ( M e. ( 0 ... N ) -> N e. ZZ )
43	42, 22	zsubcld 9723	. . . . . . . . . . 11 |- ( M e. ( 0 ... N ) -> ( N - M ) e. ZZ )
44		fzofig 10818	. . . . . . . . . . 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 11189	. . . . . . . . 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 10412	. . . . . . . . . 10 |- ( M e. ( 0 ... N ) -> ( N - M ) e. NN0 )
50		hashfzo0 11213	. . . . . . . . . 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 2267	. . . . . . 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 6074	. . . . . 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 5501	. . . . 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 2311	. . . 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 11254	. . . 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 2311	. 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 2205   A.wral 2522   _Vcvv 2815   <.cop 3697   |-> cmpt 4176   dom cdm 4754   "cima 4757   Fun wfun 5351   Fn wfn 5352   -->wf 5353   ` cfv 5357  (class class class)co 6058   Fincfn 6988   0cc0 8143   + caddc 8146   - cmin 8460   NN0cn0 9513   ZZcz 9594   ...cfz 10361  ..^cfzo 10498  ♯chash 11163  Word cword 11249   substr csubstr 11362

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

This theorem is referenced by:  pfxwrdsymbg  11407