Theorem swrdwrdsymbg 11436

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 11423	. . . 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 1235	. . 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 11310	. . . . . . . . . 10 |- ( S e. Word A -> S : ( 0..^ (♯ ` S ) ) --> A )
4	3	ffund 5537	. . . . . . . . 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 11312	. . . . . . . . 9 |- ( S e. Word A -> dom S = ( 0..^ (♯ ` S ) ) )
8		elfzodifsumelfzo 10619	. . . . . . . . . . . . 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 2302	. . . . . . . . . . . 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 10428	. . . . . . . . . . 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 10428	. . . . . . . . . 10 |- ( M e. ( 0 ... N ) -> M e. ZZ )
23	22	ad2antrl 494	. . . . . . . . 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 10608	. . . . . . . 8 |- ( ( x e. ( 0..^ ( N - M ) ) /\ N e. ZZ /\ M e. ZZ ) -> ( x + M ) e. ( M..^ N ) )
26	17, 21, 24, 25	syl3anc 1278	. . . . . . 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 5950	. . . . . . . 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 1277	. . . . . 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 5863	. . . . 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 531	. . . . . . . . . . . 12 |- ( ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( N - M ) ) ) -> S e. Word A )
32		elfzoelz 10554	. . . . . . . . . . . . . 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 9772	. . . . . . . . . . . 12 |- ( ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ ( N - M ) ) ) -> ( x + M ) e. ZZ )
35		fvexg 5714	. . . . . . . . . . . 12 |- ( ( S e. Word A /\ ( x + M ) e. ZZ ) -> ( S ` ( x + M ) ) e. _V )
36	31, 34, 35	syl2anc 415	. . . . . . . . . . 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 2623	. . . . . . . . . 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 2238	. . . . . . . . . . 11 |- ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) ) = ( x e. ( 0..^ ( N - M ) ) |-> ( S ` ( x + M ) ) )
39	38	fnmpt 5510	. . . . . . . . . 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 9655	. . . . . . . . . . 11 |- 0 e. ZZ
42		elfzel2 10426	. . . . . . . . . . . 12 |- ( M e. ( 0 ... N ) -> N e. ZZ )
43	42, 22	zsubcld 9773	. . . . . . . . . . 11 |- ( M e. ( 0 ... N ) -> ( N - M ) e. ZZ )
44		fzofig 10869	. . . . . . . . . . 11 |- ( ( 0 e. ZZ /\ ( N - M ) e. ZZ ) -> ( 0..^ ( N - M ) ) e. Fin )
45	41, 43, 44	sylancr 418	. . . . . . . . . 10 |- ( M e. ( 0 ... N ) -> ( 0..^ ( N - M ) ) e. Fin )
46	45	ad2antrl 494	. . . . . . . . 9 |- ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) -> ( 0..^ ( N - M ) ) e. Fin )
47		fihashfn 11240	. . . . . . . . 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 415	. . . . . . . 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 10463	. . . . . . . . . 10 |- ( M e. ( 0 ... N ) -> ( N - M ) e. NN0 )
50		hashfzo0 11264	. . . . . . . . . 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 494	. . . . . . . 8 |- ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) -> (♯ ` ( 0..^ ( N - M ) ) ) = ( N - M ) )
53	48, 52	eqtrd 2271	. . . . . . 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 6101	. . . . . 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 5521	. . . . 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 494	. . . . 5 |- ( ( S e. Word A /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) ) -> ( N - M ) e. NN0 )
58	53, 57	eqeltrd 2315	. . . 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 11309	. . . 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 415	. . 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 2315	. 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 1230	1 |- ( ( S e. Word A /\ M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` S ) ) ) -> ( S substr <. M , N >. ) e. Word ( S " ( M..^ N ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   <.cop 3712   |-> cmpt 4192   dom cdm 4774   "cima 4777   Fun wfun 5371   Fn wfn 5372   -->wf 5373   ` cfv 5377  (class class class)co 6085   Fincfn 7022   0cc0 8179   + caddc 8182   - cmin 8497   NN0cn0 9563   ZZcz 9644   ...cfz 10411  ..^cfzo 10549  ♯chash 11214  Word cword 11304   substr csubstr 11417

This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-fzo 10550  df-ihash 11215  df-word 11305  df-substr 11418

This theorem is used by:  pfxwrdsymbg  11462