Theorem swrdval 11175

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 11173	. . 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 5626	. . . . 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 6294	. . . . 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 5626	. . . . 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 6295	. . . . 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 6018	. . . . 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 4924	. . . . 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 6018	. . . . . 6 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( ( 2nd ` b ) - ( 1st ` b ) ) = ( L - F ) )
21	20	oveq2d 6016	. . . . 5 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( 0..^ ( ( 2nd ` b ) - ( 1st ` b ) ) ) = ( 0..^ ( L - F ) ) )
22	14	oveq2d 6016	. . . . . 6 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( x + ( 1st ` b ) ) = ( x + F ) )
23	17, 22	fveq12d 5633	. . . . 5 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( s ` ( x + ( 1st ` b ) ) ) = ( S ` ( x + F ) ) )
24	21, 23	mpteq12dv 4165	. . . 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 4750	. . 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 9454	. . . . 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 9570	. . . . 5 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> ( L - F ) e. ZZ )
35		fzofig 10649	. . . . 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 5865	. . 3 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> ( x e. ( 0..^ ( L - F ) ) |-> ( S ` ( x + F ) ) ) e. _V )
38		0ex 4210	. . . 4 |- (/) e. _V
39	38	a1i 9	. . 3 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> (/) e. _V )
40	37, 39	ifexd 4574	. 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 6131	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 4144   X. cxp 4716   dom cdm 4718   ` cfv 5317  (class class class)co 6000   e. cmpo 6002   1stc1st 6282   2ndc2nd 6283   Fincfn 6885   0cc0 7995   + caddc 7998   - cmin 8313   ZZcz 9442  ..^cfzo 10334   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-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-substr 11173

This theorem is referenced by:  swrd00g  11176  swrdclg  11177  swrdval2  11178  swrdlend  11185  swrdnd  11186  swrd0g  11187  pfxval  11201  fnpfx  11204