Theorem swrdwrdsymbg 11191

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 11178	. . . 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 1228	. . 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 11072	. . . . . . . . . 10 |- ( S e. Word A -> S : ( 0..^ (♯ ` S ) ) --> A )
4	3	ffund 5476	. . . . . . . . 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 11074	. . . . . . . . 9 |- ( S e. Word A -> dom S = ( 0..^ (♯ ` S ) ) )
8		elfzodifsumelfzo 10402	. . . . . . . . . . . . 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 2293	. . . . . . . . . . . 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 10217	. . . . . . . . . . 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 10217	. . . . . . . . . 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 10391	. . . . . . . 8 |- ( ( x e. ( 0..^ ( N - M ) ) /\ N e. ZZ /\ M e. ZZ ) -> ( x + M ) e. ( M..^ N ) )
26	17, 21, 24, 25	syl3anc 1271	. . . . . . 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 5870	. . . . . . . 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 1270	. . . . . 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 5789	. . . . 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 10339	. . . . . . . . . . . . . 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 9569	. . . . . . . . . . . 12 |- ( ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( N - M ) ) ) -> ( x + M ) e. ZZ )
35		fvexg 5645	. . . . . . . . . . . 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 2603	. . . . . . . . . 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 2229	. . . . . . . . . . 11 |- ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) = ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) )
39	38	fnmpt 5449	. . . . . . . . . 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 9453	. . . . . . . . . . 11 |- 0 e. ZZ
42		elfzel2 10215	. . . . . . . . . . . 12 |- ( M e. ( 0 ... N ) -> N e. ZZ )
43	42, 22	zsubcld 9570	. . . . . . . . . . 11 |- ( M e. ( 0 ... N ) -> ( N - M ) e. ZZ )
44		fzofig 10649	. . . . . . . . . . 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 11017	. . . . . . . . 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 10249	. . . . . . . . . 10 |- ( M e. ( 0 ... N ) -> ( N - M ) e. NN0 )
50		hashfzo0 11040	. . . . . . . . . 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 2262	. . . . . . 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 6016	. . . . . 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 5460	. . . . 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 2306	. . . 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 11071	. . . 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 2306	. 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 1223	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 1002   = wceq 1395   e. wcel 2200   A.wral 2508   _Vcvv 2799   <.cop 3669   |-> cmpt 4144   dom cdm 4718   "cima 4721   Fun wfun 5311   Fn wfn 5312   -->wf 5313   ` cfv 5317  (class class class)co 6000   Fincfn 6885   0cc0 7995   + caddc 7998   - cmin 8313   NN0cn0 9365   ZZcz 9442   ...cfz 10200  ..^cfzo 10334  ♯chash 10992  Word cword 11066   substr csubstr 11172

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 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-1o 6560  df-er 6678  df-en 6886  df-dom 6887  df-fin 6888  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719  df-fz 10201  df-fzo 10335  df-ihash 10993  df-word 11067  df-substr 11173

This theorem is referenced by:  pfxwrdsymbg  11217