Theorem swrd0 12448

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

Assertion

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

Proof of Theorem swrd0

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

Step	Hyp	Ref	Expression
1		opelxp 4980	. . . 4 |- ( <. (/) , <. F , L >. >. e. ( _V X. ( ZZ X. ZZ ) ) <-> ( (/) e. _V /\ <. F , L >. e. ( ZZ X. ZZ ) ) )
2		opelxp 4980	. . . . 5 |- ( <. F , L >. e. ( ZZ X. ZZ ) <-> ( F e. ZZ /\ L e. ZZ ) )
3		swrdval 12434	. . . . . . 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 11692	. . . . . . . . . . . . . . 15 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( -. F < L <-> ( F..^ L ) = (/) ) )
5	4	biimprd 223	. . . . . . . . . . . . . 14 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( ( F..^ L ) = (/) -> -. F < L ) )
6	5	con2d 115	. . . . . . . . . . . . 13 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( F < L -> -. ( F..^ L ) = (/) ) )
7	6	impcom 430	. . . . . . . . . . . 12 |- ( ( F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> -. ( F..^ L ) = (/) )
8		ss0 3779	. . . . . . . . . . . 12 |- ( ( F..^ L ) C_ (/) -> ( F..^ L ) = (/) )
9	7, 8	nsyl 121	. . . . . . . . . . 11 |- ( ( F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> -. ( F..^ L ) C_ (/) )
10		dm0 5164	. . . . . . . . . . . . 13 |- dom (/) = (/)
11	10	a1i 11	. . . . . . . . . . . 12 |- ( ( F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> dom (/) = (/) )
12	11	sseq2d 3495	. . . . . . . . . . 11 |- ( ( F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( ( F..^ L ) C_ dom (/) <-> ( F..^ L ) C_ (/) ) )
13	9, 12	mtbird 301	. . . . . . . . . 10 |- ( ( F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> -. ( F..^ L ) C_ dom (/) )
14		iffalse 3910	. . . . . . . . . 10 |- ( -. ( F..^ L ) C_ dom (/) -> if ( ( F..^ L ) C_ dom (/) , ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) , (/) ) = (/) )
15	13, 14	syl 16	. . . . . . . . 9 |- ( ( F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> if ( ( F..^ L ) C_ dom (/) , ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) , (/) ) = (/) )
16		0ss 3777	. . . . . . . . . . . . 13 |- (/) C_ (/)
17	16	a1i 11	. . . . . . . . . . . 12 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> (/) C_ (/) )
18	4	biimpac 486	. . . . . . . . . . . 12 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( F..^ L ) = (/) )
19	10	a1i 11	. . . . . . . . . . . 12 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> dom (/) = (/) )
20	17, 18, 19	3sstr4d 3510	. . . . . . . . . . 11 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( F..^ L ) C_ dom (/) )
21		iftrue 3908	. . . . . . . . . . 11 |- ( ( F..^ L ) C_ dom (/) -> if ( ( F..^ L ) C_ dom (/) , ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) , (/) ) = ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) )
22	20, 21	syl 16	. . . . . . . . . 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 ) ) ) )
23		df-mpt 4463	. . . . . . . . . . . 12 |- ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) = { <. x , y >. | ( x e. ( 0..^ ( L - F ) ) /\ y = ( (/) ` ( x + F ) ) ) }
24		df-opab 4462	. . . . . . . . . . . 12 |- { <. x , y >. | ( x e. ( 0..^ ( L - F ) ) /\ y = ( (/) ` ( x + F ) ) ) } = { p | E. x E. y ( p = <. x , y >. /\ ( x e. ( 0..^ ( L - F ) ) /\ y = ( (/) ` ( x + F ) ) ) ) }
25	23, 24	eqtri 2483	. . . . . . . . . . 11 |- ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) = { p | E. x E. y ( p = <. x , y >. /\ ( x e. ( 0..^ ( L - F ) ) /\ y = ( (/) ` ( x + F ) ) ) ) }
26		noel 3752	. . . . . . . . . . . . . . . . . . . 20 |- -. x e. (/)
27	26	a1i 11	. . . . . . . . . . . . . . . . . . 19 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> -. x e. (/) )
28		zre 10764	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( F e. ZZ -> F e. RR )
29		zre 10764	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( L e. ZZ -> L e. RR )
30		posdif 9946	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( F e. RR /\ L e. RR ) -> ( F < L <-> 0 < ( L - F ) ) )
31	28, 29, 30	syl2an 477	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( F < L <-> 0 < ( L - F ) ) )
32	31	biimprd 223	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( 0 < ( L - F ) -> F < L ) )
33	32	con3d 133	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( -. F < L -> -. 0 < ( L - F ) ) )
34	33	impcom 430	. . . . . . . . . . . . . . . . . . . 20 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> -. 0 < ( L - F ) )
35		zsubcl 10801	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( L e. ZZ /\ F e. ZZ ) -> ( L - F ) e. ZZ )
36	35	ancoms 453	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( L - F ) e. ZZ )
37		0z 10771	. . . . . . . . . . . . . . . . . . . . . . 23 |- 0 e. ZZ
38	36, 37	jctil 537	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( 0 e. ZZ /\ ( L - F ) e. ZZ ) )
39	38	adantl 466	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( 0 e. ZZ /\ ( L - F ) e. ZZ ) )
40		fzonlt0 11692	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( 0 e. ZZ /\ ( L - F ) e. ZZ ) -> ( -. 0 < ( L - F ) <-> ( 0..^ ( L - F ) ) = (/) ) )
41	39, 40	syl 16	. . . . . . . . . . . . . . . . . . . 20 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( -. 0 < ( L - F ) <-> ( 0..^ ( L - F ) ) = (/) ) )
42	34, 41	mpbid 210	. . . . . . . . . . . . . . . . . . 19 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( 0..^ ( L - F ) ) = (/) )
43	27, 42	neleqtrrd 2567	. . . . . . . . . . . . . . . . . 18 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> -. x e. ( 0..^ ( L - F ) ) )
44	43	intnanrd 908	. . . . . . . . . . . . . . . . 17 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> -. ( x e. ( 0..^ ( L - F ) ) /\ y = ( (/) ` ( x + F ) ) ) )
45	44	intnand 907	. . . . . . . . . . . . . . . 16 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> -. ( p = <. x , y >. /\ ( x e. ( 0..^ ( L - F ) ) /\ y = ( (/) ` ( x + F ) ) ) ) )
46	45	alrimivv 1687	. . . . . . . . . . . . . . 15 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> A. x A. y -. ( p = <. x , y >. /\ ( x e. ( 0..^ ( L - F ) ) /\ y = ( (/) ` ( x + F ) ) ) ) )
47		2nexaln 1622	. . . . . . . . . . . . . . 15 |- ( -. E. x E. y ( p = <. x , y >. /\ ( x e. ( 0..^ ( L - F ) ) /\ y = ( (/) ` ( x + F ) ) ) ) <-> A. x A. y -. ( p = <. x , y >. /\ ( x e. ( 0..^ ( L - F ) ) /\ y = ( (/) ` ( x + F ) ) ) ) )
48	46, 47	sylibr 212	. . . . . . . . . . . . . 14 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> -. E. x E. y ( p = <. x , y >. /\ ( x e. ( 0..^ ( L - F ) ) /\ y = ( (/) ` ( x + F ) ) ) ) )
49	48	pm2.21d 106	. . . . . . . . . . . . 13 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( E. x E. y ( p = <. x , y >. /\ ( x e. ( 0..^ ( L - F ) ) /\ y = ( (/) ` ( x + F ) ) ) ) -> p e. (/) ) )
50	49	abssdv 3537	. . . . . . . . . . . 12 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> { p | E. x E. y ( p = <. x , y >. /\ ( x e. ( 0..^ ( L - F ) ) /\ y = ( (/) ` ( x + F ) ) ) ) } C_ (/) )
51		ss0 3779	. . . . . . . . . . . 12 |- ( { p | E. x E. y ( p = <. x , y >. /\ ( x e. ( 0..^ ( L - F ) ) /\ y = ( (/) ` ( x + F ) ) ) ) } C_ (/) -> { p | E. x E. y ( p = <. x , y >. /\ ( x e. ( 0..^ ( L - F ) ) /\ y = ( (/) ` ( x + F ) ) ) ) } = (/) )
52	50, 51	syl 16	. . . . . . . . . . 11 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> { p | E. x E. y ( p = <. x , y >. /\ ( x e. ( 0..^ ( L - F ) ) /\ y = ( (/) ` ( x + F ) ) ) ) } = (/) )
53	25, 52	syl5eq 2507	. . . . . . . . . 10 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) = (/) )
54	22, 53	eqtrd 2495	. . . . . . . . 9 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> if ( ( F..^ L ) C_ dom (/) , ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) , (/) ) = (/) )
55	15, 54	pm2.61ian 788	. . . . . . . 8 |- ( ( F e. ZZ /\ L e. ZZ ) -> if ( ( F..^ L ) C_ dom (/) , ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) , (/) ) = (/) )
56	55	3adant1 1006	. . . . . . 7 |- ( ( (/) e. _V /\ F e. ZZ /\ L e. ZZ ) -> if ( ( F..^ L ) C_ dom (/) , ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) , (/) ) = (/) )
57	3, 56	eqtrd 2495	. . . . . 6 |- ( ( (/) e. _V /\ F e. ZZ /\ L e. ZZ ) -> ( (/) substr <. F , L >. ) = (/) )
58	57	3expb 1189	. . . . 5 |- ( ( (/) e. _V /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( (/) substr <. F , L >. ) = (/) )
59	2, 58	sylan2b 475	. . . 4 |- ( ( (/) e. _V /\ <. F , L >. e. ( ZZ X. ZZ ) ) -> ( (/) substr <. F , L >. ) = (/) )
60	1, 59	sylbi 195	. . 3 |- ( <. (/) , <. F , L >. >. e. ( _V X. ( ZZ X. ZZ ) ) -> ( (/) substr <. F , L >. ) = (/) )
61		df-substr 12354	. . . 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 ) ) ) ) , (/) ) )
62		ovex 6228	. . . . . 6 |- ( 0..^ ( ( 2nd ` b ) - ( 1st ` b ) ) ) e. _V
63	62	mptex 6060	. . . . 5 |- ( z e. ( 0..^ ( ( 2nd ` b ) - ( 1st ` b ) ) ) |-> ( s ` ( z + ( 1st ` b ) ) ) ) e. _V
64		0ex 4533	. . . . 5 |- (/) e. _V
65	63, 64	ifex 3969	. . . 4 |- if ( ( ( 1st ` b )..^ ( 2nd ` b ) ) C_ dom s , ( z e. ( 0..^ ( ( 2nd ` b ) - ( 1st ` b ) ) ) |-> ( s ` ( z + ( 1st ` b ) ) ) ) , (/) ) e. _V
66	61, 65	dmmpt2 6757	. . 3 |- dom substr = ( _V X. ( ZZ X. ZZ ) )
67	60, 66	eleq2s 2562	. 2 |- ( <. (/) , <. F , L >. >. e. dom substr -> ( (/) substr <. F , L >. ) = (/) )
68		df-ov 6206	. . 3 |- ( (/) substr <. F , L >. ) = ( substr ` <. (/) , <. F , L >. >. )
69		ndmfv 5826	. . 3 |- ( -. <. (/) , <. F , L >. >. e. dom substr -> ( substr ` <. (/) , <. F , L >. >. ) = (/) )
70	68, 69	syl5eq 2507	. 2 |- ( -. <. (/) , <. F , L >. >. e. dom substr -> ( (/) substr <. F , L >. ) = (/) )
71	67, 70	pm2.61i 164	1 |- ( (/) substr <. F , L >. ) = (/)

Syntax hints:   -. wn 3   <-> wb 184   /\ wa 369   /\ w3a 965   A.wal 1368   = wceq 1370   E.wex 1587   e. wcel 1758   {cab 2439   _Vcvv 3078   C_ wss 3439   (/)c0 3748   ifcif 3902   <.cop 3994   class class class wbr 4403   {copab 4460   |-> cmpt 4461   X. cxp 4949   dom cdm 4951   ` cfv 5529  (class class class)co 6203   1stc1st 6688   2ndc2nd 6689   RRcr 9395   0cc0 9396   + caddc 9399   < clt 9532   - cmin 9709   ZZcz 10760  ..^cfzo 11668   substr csubstr 12346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-fzo 11669  df-substr 12354

This theorem is referenced by:  cshword  12549