Theorem swrd0 12844

Description: A subword of an empty set is always the empty set. (Contributed by AV, 31-Mar-2018.) (Revised by AV, 20-Oct-2018.) (Proof shortened by AV, 2-May-2020.)

Assertion

Ref	Expression
swrd0	|- ( (/) substr <. F , L >. ) = (/)

Proof of Theorem swrd0

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

Step	Hyp	Ref	Expression
1		opelxp 4869	. . . 4 |- ( <. (/) , <. F , L >. >. e. ( _V X. ( ZZ X. ZZ ) ) <-> ( (/) e. _V /\ <. F , L >. e. ( ZZ X. ZZ ) ) )
2		opelxp 4869	. . . . 5 |- ( <. F , L >. e. ( ZZ X. ZZ ) <-> ( F e. ZZ /\ L e. ZZ ) )
3		swrdval 12827	. . . . . . 7 |- ( ( (/) e. _V /\ F e. ZZ /\ L e. ZZ ) -> ( (/) substr <. F , L >. ) = if ( ( F..^ L ) C_ dom (/) , ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) , (/) ) )
4		fzonlt0 11968	. . . . . . . . . . . . . . 15 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( -. F < L <-> ( F..^ L ) = (/) ) )
5	4	biimprd 231	. . . . . . . . . . . . . 14 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( ( F..^ L ) = (/) -> -. F < L ) )
6	5	con2d 119	. . . . . . . . . . . . 13 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( F < L -> -. ( F..^ L ) = (/) ) )
7	6	impcom 437	. . . . . . . . . . . 12 |- ( ( F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> -. ( F..^ L ) = (/) )
8		ss0 3768	. . . . . . . . . . . 12 |- ( ( F..^ L ) C_ (/) -> ( F..^ L ) = (/) )
9	7, 8	nsyl 125	. . . . . . . . . . 11 |- ( ( F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> -. ( F..^ L ) C_ (/) )
10		dm0 5054	. . . . . . . . . . . . 13 |- dom (/) = (/)
11	10	a1i 11	. . . . . . . . . . . 12 |- ( ( F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> dom (/) = (/) )
12	11	sseq2d 3446	. . . . . . . . . . 11 |- ( ( F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( ( F..^ L ) C_ dom (/) <-> ( F..^ L ) C_ (/) ) )
13	9, 12	mtbird 308	. . . . . . . . . 10 |- ( ( F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> -. ( F..^ L ) C_ dom (/) )
14	13	iffalsed 3883	. . . . . . . . 9 |- ( ( F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> if ( ( F..^ L ) C_ dom (/) , ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) , (/) ) = (/) )
15		0ss 3766	. . . . . . . . . . . . 13 |- (/) C_ (/)
16	15	a1i 11	. . . . . . . . . . . 12 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> (/) C_ (/) )
17	4	biimpac 494	. . . . . . . . . . . 12 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( F..^ L ) = (/) )
18	10	a1i 11	. . . . . . . . . . . 12 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> dom (/) = (/) )
19	16, 17, 18	3sstr4d 3461	. . . . . . . . . . 11 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( F..^ L ) C_ dom (/) )
20	19	iftrued 3880	. . . . . . . . . 10 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> if ( ( F..^ L ) C_ dom (/) , ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) , (/) ) = ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) )
21		zre 10965	. . . . . . . . . . . . . . . . 17 |- ( L e. ZZ -> L e. RR )
22		zre 10965	. . . . . . . . . . . . . . . . 17 |- ( F e. ZZ -> F e. RR )
23		lenlt 9730	. . . . . . . . . . . . . . . . . 18 |- ( ( L e. RR /\ F e. RR ) -> ( L <_ F <-> -. F < L ) )
24	23	bicomd 206	. . . . . . . . . . . . . . . . 17 |- ( ( L e. RR /\ F e. RR ) -> ( -. F < L <-> L <_ F ) )
25	21, 22, 24	syl2anr 486	. . . . . . . . . . . . . . . 16 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( -. F < L <-> L <_ F ) )
26		fzo0n 11967	. . . . . . . . . . . . . . . 16 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( L <_ F <-> ( 0..^ ( L - F ) ) = (/) ) )
27	25, 26	bitrd 261	. . . . . . . . . . . . . . 15 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( -. F < L <-> ( 0..^ ( L - F ) ) = (/) ) )
28	27	biimpac 494	. . . . . . . . . . . . . 14 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( 0..^ ( L - F ) ) = (/) )
29	28	mpteq1d 4477	. . . . . . . . . . . . 13 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) = ( x e. (/) |-> ( (/) ` ( x + F ) ) ) )
30	29	dmeqd 5042	. . . . . . . . . . . 12 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> dom ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) = dom ( x e. (/) |-> ( (/) ` ( x + F ) ) ) )
31		ral0 3865	. . . . . . . . . . . . 13 |- A. x e. (/) ( (/) ` ( x + F ) ) e. _V
32		dmmptg 5339	. . . . . . . . . . . . 13 |- ( A. x e. (/) ( (/) ` ( x + F ) ) e. _V -> dom ( x e. (/) |-> ( (/) ` ( x + F ) ) ) = (/) )
33	31, 32	mp1i 13	. . . . . . . . . . . 12 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> dom ( x e. (/) |-> ( (/) ` ( x + F ) ) ) = (/) )
34	30, 33	eqtrd 2505	. . . . . . . . . . 11 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> dom ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) = (/) )
35		mptrel 4966	. . . . . . . . . . . 12 |- Rel ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) )
36		reldm0 5058	. . . . . . . . . . . 12 |- ( Rel ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) -> ( ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) = (/) <-> dom ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) = (/) ) )
37	35, 36	mp1i 13	. . . . . . . . . . 11 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) = (/) <-> dom ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) = (/) ) )
38	34, 37	mpbird 240	. . . . . . . . . 10 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) = (/) )
39	20, 38	eqtrd 2505	. . . . . . . . 9 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> if ( ( F..^ L ) C_ dom (/) , ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) , (/) ) = (/) )
40	14, 39	pm2.61ian 807	. . . . . . . 8 |- ( ( F e. ZZ /\ L e. ZZ ) -> if ( ( F..^ L ) C_ dom (/) , ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) , (/) ) = (/) )
41	40	3adant1 1048	. . . . . . 7 |- ( ( (/) e. _V /\ F e. ZZ /\ L e. ZZ ) -> if ( ( F..^ L ) C_ dom (/) , ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) , (/) ) = (/) )
42	3, 41	eqtrd 2505	. . . . . 6 |- ( ( (/) e. _V /\ F e. ZZ /\ L e. ZZ ) -> ( (/) substr <. F , L >. ) = (/) )
43	42	3expb 1232	. . . . 5 |- ( ( (/) e. _V /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( (/) substr <. F , L >. ) = (/) )
44	2, 43	sylan2b 483	. . . 4 |- ( ( (/) e. _V /\ <. F , L >. e. ( ZZ X. ZZ ) ) -> ( (/) substr <. F , L >. ) = (/) )
45	1, 44	sylbi 200	. . 3 |- ( <. (/) , <. F , L >. >. e. ( _V X. ( ZZ X. ZZ ) ) -> ( (/) substr <. F , L >. ) = (/) )
46		df-substr 12715	. . . 4 |- substr = ( s e. _V , b e. ( ZZ X. ZZ ) |-> if ( ( ( 1st ` b )..^ ( 2nd ` b ) ) C_ dom s , ( z e. ( 0..^ ( ( 2nd ` b ) - ( 1st ` b ) ) ) |-> ( s ` ( z + ( 1st ` b ) ) ) ) , (/) ) )
47		ovex 6336	. . . . . 6 |- ( 0..^ ( ( 2nd ` b ) - ( 1st ` b ) ) ) e. _V
48	47	mptex 6152	. . . . 5 |- ( z e. ( 0..^ ( ( 2nd ` b ) - ( 1st ` b ) ) ) |-> ( s ` ( z + ( 1st ` b ) ) ) ) e. _V
49		0ex 4528	. . . . 5 |- (/) e. _V
50	48, 49	ifex 3940	. . . 4 |- if ( ( ( 1st ` b )..^ ( 2nd ` b ) ) C_ dom s , ( z e. ( 0..^ ( ( 2nd ` b ) - ( 1st ` b ) ) ) |-> ( s ` ( z + ( 1st ` b ) ) ) ) , (/) ) e. _V
51	46, 50	dmmpt2 6882	. . 3 |- dom substr = ( _V X. ( ZZ X. ZZ ) )
52	45, 51	eleq2s 2567	. 2 |- ( <. (/) , <. F , L >. >. e. dom substr -> ( (/) substr <. F , L >. ) = (/) )
53		df-ov 6311	. . 3 |- ( (/) substr <. F , L >. ) = ( substr ` <. (/) , <. F , L >. >. )
54		ndmfv 5903	. . 3 |- ( -. <. (/) , <. F , L >. >. e. dom substr -> ( substr ` <. (/) , <. F , L >. >. ) = (/) )
55	53, 54	syl5eq 2517	. 2 |- ( -. <. (/) , <. F , L >. >. e. dom substr -> ( (/) substr <. F , L >. ) = (/) )
56	52, 55	pm2.61i 169	1 |- ( (/) substr <. F , L >. ) = (/)

Syntax hints:   -. wn 3   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   A.wral 2756   _Vcvv 3031   C_ wss 3390   (/)c0 3722   ifcif 3872   <.cop 3965   class class class wbr 4395   |-> cmpt 4454   X. cxp 4837   dom cdm 4839   Rel wrel 4844   ` cfv 5589  (class class class)co 6308   1stc1st 6810   2ndc2nd 6811   RRcr 9556   0cc0 9557   + caddc 9560   < clt 9693   <_ cle 9694   - cmin 9880   ZZcz 10961  ..^cfzo 11942   substr csubstr 12707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-substr 12715

This theorem is referenced by:  cshword  12947  pfx0  39073  cshword2  39125