Theorem swrdval 11101

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 11099	. . 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 5576	. . . . 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 6236	. . . . 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 2260	. . 3 |- ( ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) /\ ( s = S /\ b = <. F , L >. ) ) -> ( 1st ` b ) = F )
9		fveq2 5576	. . . . 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 6237	. . . . 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 2260	. . 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 5962	. . . . 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 4880	. . . . 5 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> dom s = dom S )
19	16, 18	sseq12d 3224	. . . 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 5962	. . . . . 6 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( ( 2nd ` b ) - ( 1st ` b ) ) = ( L - F ) )
21	20	oveq2d 5960	. . . . 5 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( 0..^ ( ( 2nd ` b ) - ( 1st ` b ) ) ) = ( 0..^ ( L - F ) ) )
22	14	oveq2d 5960	. . . . . 6 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( x + ( 1st ` b ) ) = ( x + F ) )
23	17, 22	fveq12d 5583	. . . . 5 |- ( ( s = S /\ ( 1st ` b ) = F /\ ( 2nd ` b ) = L ) -> ( s ` ( x + ( 1st ` b ) ) ) = ( S ` ( x + F ) ) )
24	21, 23	mpteq12dv 4126	. . . 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 3593	. . 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 2783	. . 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 4707	. . 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 9384	. . . . 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 9500	. . . . 5 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> ( L - F ) e. ZZ )
35		fzofig 10577	. . . . 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 5811	. . 3 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> ( x e. ( 0..^ ( L - F ) ) |-> ( S ` ( x + F ) ) ) e. _V )
38		0ex 4171	. . . 4 |- (/) e. _V
39	38	a1i 9	. . 3 |- ( ( S e. V /\ F e. ZZ /\ L e. ZZ ) -> (/) e. _V )
40	37, 39	ifexd 4531	. 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 6073	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 2176   _Vcvv 2772   C_ wss 3166   (/)c0 3460   ifcif 3571   <.cop 3636   |-> cmpt 4105   X. cxp 4673   dom cdm 4675   ` cfv 5271  (class class class)co 5944   e. cmpo 5946   1stc1st 6224   2ndc2nd 6225   Fincfn 6827   0cc0 7925   + caddc 7928   - cmin 8243   ZZcz 9372  ..^cfzo 10264   substr csubstr 11098

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041

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 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-1o 6502  df-er 6620  df-en 6828  df-fin 6830  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-inn 9037  df-n0 9296  df-z 9373  df-uz 9649  df-fz 10131  df-fzo 10265  df-substr 11099

This theorem is referenced by:  swrd00g  11102  swrdclg  11103  swrdval2  11104  swrdlend  11111  swrdnd  11112  swrd0g  11113