Theorem swrdval 11398

Description: Value of a subword. (Contributed by Stefan O'Rear, 15-Aug-2015.)

Assertion

Ref	Expression
swrdval	|- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> ( S substr <. F , L >. ) = if ( ( F..^ L ) C_ dom S , ( x e. ( 0..^ ( L - F ) ) |-> ( S ` ( x + F ) ) ) , (/) ) )

Distinct variable groups:   x, S   x, F   x, L   x, V

Proof of Theorem swrdval

Dummy variables s b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-substr 11396	. . 3 |- substr = ( s e. _V , b e. ( ZZ X. ZZ ) |-> if ( ( ( 1st ` b )..^ ( 2nd ` b ) ) C_ dom s , ( x e. ( 0..^ ( ( 2nd ` b ) - ( 1st ` b ) ) ) |-> ( s ` ( x + ( 1st ` b ) ) ) ) , (/) ) )
2	1	a1i 9	. 2 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> substr = ( s e. _V , b e. ( ZZ X. ZZ ) |-> if ( ( ( 1st ` b )..^ ( 2nd ` b ) ) C_ dom s , ( x e. ( 0..^ ( ( 2nd ` b ) - ( 1st ` b ) ) ) |-> ( s ` ( x + ( 1st ` b ) ) ) ) , (/) ) ) )
3		simprl 535	. . 3 |- ( ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) /\ ( s = S /\ b = <. F , L >. ) ) -> s = S )
4		fveq2 5690	. . . . 5 |- ( b = <. F , L >. -> ( 1st ` b ) = ( 1st ` <. F , L >. ) )
5	4	adantl 277	. . . 4 |- ( ( s = S /\ b = <. F , L >. ) -> ( 1st ` b ) = ( 1st ` <. F , L >. ) )
6		op1stg 6374	. . . . 5 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( 1st ` <. F , L >. ) = F )
7	6	3adant1 1046	. . . 4 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> ( 1st ` <. F , L >. ) = F )
8	5, 7	sylan9eqr 2293	. . 3 |- ( ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) /\ ( s = S /\ b = <. F , L >. ) ) -> ( 1st ` b ) = F )
9		fveq2 5690	. . . . 5 |- ( b = <. F , L >. -> ( 2nd ` b ) = ( 2nd ` <. F , L >. ) )
10	9	adantl 277	. . . 4 |- ( ( s = S /\ b = <. F , L >. ) -> ( 2nd ` b ) = ( 2nd ` <. F , L >. ) )
11		op2ndg 6375	. . . . 5 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( 2nd ` <. F , L >. ) = L )
12	11	3adant1 1046	. . . 4 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> ( 2nd ` <. F , L >. ) = L )
13	10, 12	sylan9eqr 2293	. . 3 |- ( ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) /\ ( s = S /\ b = <. F , L >. ) ) -> ( 2nd ` b ) = L )
14		simp2 1029	. . . . . 6 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( 1st ` b ) = F )
15		simp3 1030	. . . . . 6 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( 2nd ` b ) = L )
16	14, 15	oveq12d 6093	. . . . 5 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( ( 1st ` b )..^ ( 2nd ` b ) ) = ( F..^ L ) )
17		simp1 1028	. . . . . 6 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> s = S )
18	17	dmeqd 4978	. . . . 5 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> dom s = dom S )
19	16, 18	sseq12d 3279	. . . 4 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( ( ( 1st ` b )..^ ( 2nd ` b ) ) C_ dom s <-> ( F..^ L ) C_ dom S ) )
20	15, 14	oveq12d 6093	. . . . . 6 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( ( 2nd ` b ) - ( 1st ` b ) ) = ( L - F ) )
21	20	oveq2d 6091	. . . . 5 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( 0..^ ( ( 2nd ` b ) - ( 1st ` b ) ) ) = ( 0..^ ( L - F ) ) )
22	14	oveq2d 6091	. . . . . 6 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( x + ( 1st ` b ) ) = ( x + F ) )
23	17, 22	fveq12d 5697	. . . . 5 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( s ` ( x + ( 1st ` b ) ) ) = ( S ` ( x + F ) ) )
24	21, 23	mpteq12dv 4208	. . . 4 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( x e. ( 0..^ ( ( 2nd ` b ) - ( 1st ` b ) ) ) |-> ( s ` ( x + ( 1st ` b ) ) ) ) = ( x e. ( 0..^ ( L - F ) ) |-> ( S ` ( x + F ) ) ) )
25	19, 24	ifbieq1d 3660	. . 3 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> if ( ( ( 1st ` b )..^ ( 2nd ` b ) ) C_ dom s , ( x e. ( 0..^ ( ( 2nd ` b ) - ( 1st ` b ) ) ) |-> ( s ` ( x + ( 1st ` b ) ) ) ) , (/) ) = if ( ( F..^ L ) C_ dom S , ( x e. ( 0..^ ( L - F ) ) |-> ( S ` ( x + F ) ) ) , (/) ) )
26	3, 8, 13, 25	syl3anc 1278	. 2 |- ( ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) /\ ( s = S /\ b = <. F , L >. ) ) -> if ( ( ( 1st ` b )..^ ( 2nd ` b ) ) C_ dom s , ( x e. ( 0..^ ( ( 2nd ` b ) - ( 1st ` b ) ) ) |-> ( s ` ( x + ( 1st ` b ) ) ) ) , (/) ) = if ( ( F..^ L ) C_ dom S , ( x e. ( 0..^ ( L - F ) ) |-> ( S ` ( x + F ) ) ) , (/) ) )
27		elex 2833	. . 3 |- ( S e. V -> S e. _V )
28	27	3ad2ant1 1049	. 2 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> S e. _V )
29		opelxpi 4801	. . 3 |- ( ( F e. ZZ /\ L e. ZZ ) -> <. F , L >. e. ( ZZ X. ZZ ) )
30	29	3adant1 1046	. 2 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> <. F , L >. e. ( ZZ X. ZZ ) )
31		0zd 9635	. . . . 5 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> 0 e. ZZ )
32		simp3 1030	. . . . . 6 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> L e. ZZ )
33		simp2 1029	. . . . . 6 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> F e. ZZ )
34	32, 33	zsubcld 9752	. . . . 5 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> ( L - F ) e. ZZ )
35		fzofig 10847	. . . . 5 |- ( ( 0 e. ZZ /\ ( L - F ) e. ZZ ) -> ( 0..^ ( L - F ) ) e. Fin )
36	31, 34, 35	syl2anc 415	. . . 4 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> ( 0..^ ( L - F ) ) e. Fin )
37	36	mptexd 5935	. . 3 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> ( x e. ( 0..^ ( L - F ) ) |-> ( S ` ( x + F ) ) ) e. _V )
38		0ex 4255	. . . 4 |- (/) e. _V
39	38	a1i 9	. . 3 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> (/) e. _V )
40	37, 39	ifexd 4625	. 2 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> if ( ( F..^ L ) C_ dom S , ( x e. ( 0..^ ( L - F ) ) |-> ( S ` ( x + F ) ) ) , (/) ) e. _V )
41	2, 26, 28, 30, 40	ovmpod 6206	1 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> ( S substr <. F , L >. ) = if ( ( F..^ L ) C_ dom S , ( x e. ( 0..^ ( L - F ) ) |-> ( S ` ( x + F ) ) ) , (/) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   _Vcvv 2821   C_ wss 3220   (/)c0 3520   ifcif 3635   <.cop 3708   |-> cmpt 4187   X. cxp 4767   dom cdm 4769   ` cfv 5372  (class class class)co 6075   e. cmpo 6077   1stc1st 6362   2ndc2nd 6363   Fincfn 7012   0cc0 8169   + caddc 8172   - cmin 8487   ZZcz 9623  ..^cfzo 10527   substr csubstr 11395

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-en 7013  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528  df-substr 11396

This theorem is referenced by:  swrd00g  11399  swrdclg  11400  swrdval2  11401  swrdlend  11408  swrdnd  11409  swrd0g  11410  pfxval  11424  fnpfx  11427