Theorem swrdval 11201

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 11199	. . 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 529	. . 3 |- ( ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) /\ ( s = S /\ b = <. F , L >. ) ) -> s = S )
4		fveq2 5632	. . . . 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 6305	. . . . 5 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( 1st ` <. F , L >. ) = F )
7	6	3adant1 1039	. . . 4 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> ( 1st ` <. F , L >. ) = F )
8	5, 7	sylan9eqr 2284	. . 3 |- ( ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) /\ ( s = S /\ b = <. F , L >. ) ) -> ( 1st ` b ) = F )
9		fveq2 5632	. . . . 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 6306	. . . . 5 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( 2nd ` <. F , L >. ) = L )
12	11	3adant1 1039	. . . 4 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> ( 2nd ` <. F , L >. ) = L )
13	10, 12	sylan9eqr 2284	. . 3 |- ( ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) /\ ( s = S /\ b = <. F , L >. ) ) -> ( 2nd ` b ) = L )
14		simp2 1022	. . . . . 6 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( 1st ` b ) = F )
15		simp3 1023	. . . . . 6 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( 2nd ` b ) = L )
16	14, 15	oveq12d 6028	. . . . 5 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( ( 1st ` b )..^ ( 2nd ` b ) ) = ( F..^ L ) )
17		simp1 1021	. . . . . 6 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> s = S )
18	17	dmeqd 4928	. . . . 5 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> dom s = dom S )
19	16, 18	sseq12d 3255	. . . 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 6028	. . . . . 6 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( ( 2nd ` b ) - ( 1st ` b ) ) = ( L - F ) )
21	20	oveq2d 6026	. . . . 5 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( 0..^ ( ( 2nd ` b ) - ( 1st ` b ) ) ) = ( 0..^ ( L - F ) ) )
22	14	oveq2d 6026	. . . . . 6 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( x + ( 1st ` b ) ) = ( x + F ) )
23	17, 22	fveq12d 5639	. . . . 5 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( s ` ( x + ( 1st ` b ) ) ) = ( S ` ( x + F ) ) )
24	21, 23	mpteq12dv 4166	. . . 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 3625	. . 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 1271	. 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 2811	. . 3 |- ( S e. V -> S e. _V )
28	27	3ad2ant1 1042	. 2 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> S e. _V )
29		opelxpi 4752	. . 3 |- ( ( F e. ZZ /\ L e. ZZ ) -> <. F , L >. e. ( ZZ X. ZZ ) )
30	29	3adant1 1039	. 2 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> <. F , L >. e. ( ZZ X. ZZ ) )
31		0zd 9474	. . . . 5 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> 0 e. ZZ )
32		simp3 1023	. . . . . 6 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> L e. ZZ )
33		simp2 1022	. . . . . 6 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> F e. ZZ )
34	32, 33	zsubcld 9590	. . . . 5 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> ( L - F ) e. ZZ )
35		fzofig 10671	. . . . 5 |- ( ( 0 e. ZZ /\ ( L - F ) e. ZZ ) -> ( 0..^ ( L - F ) ) e. Fin )
36	31, 34, 35	syl2anc 411	. . . 4 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> ( 0..^ ( L - F ) ) e. Fin )
37	36	mptexd 5873	. . 3 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> ( x e. ( 0..^ ( L - F ) ) |-> ( S ` ( x + F ) ) ) e. _V )
38		0ex 4211	. . . 4 |- (/) e. _V
39	38	a1i 9	. . 3 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> (/) e. _V )
40	37, 39	ifexd 4576	. 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 6141	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 1002   = wceq 1395   e. wcel 2200   _Vcvv 2799   C_ wss 3197   (/)c0 3491   ifcif 3602   <.cop 3669   |-> cmpt 4145   X. cxp 4718   dom cdm 4720   ` cfv 5321  (class class class)co 6010   e. cmpo 6012   1stc1st 6293   2ndc2nd 6294   Fincfn 6900   0cc0 8015   + caddc 8018   - cmin 8333   ZZcz 9462  ..^cfzo 10355   substr csubstr 11198

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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-addass 8117  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-0id 8123  ax-rnegex 8124  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131

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 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-frec 6548  df-1o 6573  df-er 6693  df-en 6901  df-fin 6903  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-inn 9127  df-n0 9386  df-z 9463  df-uz 9739  df-fz 10222  df-fzo 10356  df-substr 11199

This theorem is referenced by:  swrd00g  11202  swrdclg  11203  swrdval2  11204  swrdlend  11211  swrdnd  11212  swrd0g  11213  pfxval  11227  fnpfx  11230