Theorem swrdval 11134

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 11132	. . 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 5594	. . . . 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 6254	. . . . 5 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( 1st ` <. F , L >. ) = F )
7	6	3adant1 1018	. . . 4 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> ( 1st ` <. F , L >. ) = F )
8	5, 7	sylan9eqr 2261	. . 3 |- ( ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) /\ ( s = S /\ b = <. F , L >. ) ) -> ( 1st ` b ) = F )
9		fveq2 5594	. . . . 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 6255	. . . . 5 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( 2nd ` <. F , L >. ) = L )
12	11	3adant1 1018	. . . 4 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> ( 2nd ` <. F , L >. ) = L )
13	10, 12	sylan9eqr 2261	. . 3 |- ( ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) /\ ( s = S /\ b = <. F , L >. ) ) -> ( 2nd ` b ) = L )
14		simp2 1001	. . . . . 6 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( 1st ` b ) = F )
15		simp3 1002	. . . . . 6 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( 2nd ` b ) = L )
16	14, 15	oveq12d 5980	. . . . 5 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( ( 1st ` b )..^ ( 2nd ` b ) ) = ( F..^ L ) )
17		simp1 1000	. . . . . 6 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> s = S )
18	17	dmeqd 4894	. . . . 5 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> dom s = dom S )
19	16, 18	sseq12d 3228	. . . 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 5980	. . . . . 6 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( ( 2nd ` b ) - ( 1st ` b ) ) = ( L - F ) )
21	20	oveq2d 5978	. . . . 5 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( 0..^ ( ( 2nd ` b ) - ( 1st ` b ) ) ) = ( 0..^ ( L - F ) ) )
22	14	oveq2d 5978	. . . . . 6 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( x + ( 1st ` b ) ) = ( x + F ) )
23	17, 22	fveq12d 5601	. . . . 5 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( s ` ( x + ( 1st ` b ) ) ) = ( S ` ( x + F ) ) )
24	21, 23	mpteq12dv 4137	. . . 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 3598	. . 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 1250	. 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 2785	. . 3 |- ( S e. V -> S e. _V )
28	27	3ad2ant1 1021	. 2 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> S e. _V )
29		opelxpi 4720	. . 3 |- ( ( F e. ZZ /\ L e. ZZ ) -> <. F , L >. e. ( ZZ X. ZZ ) )
30	29	3adant1 1018	. 2 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> <. F , L >. e. ( ZZ X. ZZ ) )
31		0zd 9414	. . . . 5 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> 0 e. ZZ )
32		simp3 1002	. . . . . 6 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> L e. ZZ )
33		simp2 1001	. . . . . 6 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> F e. ZZ )
34	32, 33	zsubcld 9530	. . . . 5 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> ( L - F ) e. ZZ )
35		fzofig 10609	. . . . 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 5829	. . 3 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> ( x e. ( 0..^ ( L - F ) ) |-> ( S ` ( x + F ) ) ) e. _V )
38		0ex 4182	. . . 4 |- (/) e. _V
39	38	a1i 9	. . 3 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> (/) e. _V )
40	37, 39	ifexd 4544	. 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 6091	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 981   = wceq 1373   e. wcel 2177   _Vcvv 2773   C_ wss 3170   (/)c0 3464   ifcif 3575   <.cop 3641   |-> cmpt 4116   X. cxp 4686   dom cdm 4688   ` cfv 5285  (class class class)co 5962   e. cmpo 5964   1stc1st 6242   2ndc2nd 6243   Fincfn 6845   0cc0 7955   + caddc 7958   - cmin 8273   ZZcz 9402  ..^cfzo 10294   substr csubstr 11131

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-addcom 8055  ax-addass 8057  ax-distr 8059  ax-i2m1 8060  ax-0lt1 8061  ax-0id 8063  ax-rnegex 8064  ax-cnre 8066  ax-pre-ltirr 8067  ax-pre-ltwlin 8068  ax-pre-lttrn 8069  ax-pre-apti 8070  ax-pre-ltadd 8071

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-id 4353  df-iord 4426  df-on 4428  df-ilim 4429  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-recs 6409  df-frec 6495  df-1o 6520  df-er 6638  df-en 6846  df-fin 6848  df-pnf 8139  df-mnf 8140  df-xr 8141  df-ltxr 8142  df-le 8143  df-sub 8275  df-neg 8276  df-inn 9067  df-n0 9326  df-z 9403  df-uz 9679  df-fz 10161  df-fzo 10295  df-substr 11132

This theorem is referenced by:  swrd00g  11135  swrdclg  11136  swrdval2  11137  swrdlend  11144  swrdnd  11145  swrd0g  11146  pfxval  11160  fnpfx  11163