Theorem swrd0 12624

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 5029	. . . 4 |- ( <. (/) , <. F , L >. >. e. ( _V X. ( ZZ X. ZZ ) ) <-> ( (/) e. _V /\ <. F , L >. e. ( ZZ X. ZZ ) ) )
2		opelxp 5029	. . . . 5 |- ( <. F , L >. e. ( ZZ X. ZZ ) <-> ( F e. ZZ /\ L e. ZZ ) )
3		swrdval 12610	. . . . . . 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 11817	. . . . . . . . . . . . . . 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 3816	. . . . . . . . . . . 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 5216	. . . . . . . . . . . . 13 |- dom (/) = (/)
11	10	a1i 11	. . . . . . . . . . . 12 |- ( ( F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> dom (/) = (/) )
12	11	sseq2d 3532	. . . . . . . . . . 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 3948	. . . . . . . . . 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 3814	. . . . . . . . . . . . 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 3547	. . . . . . . . . . 11 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( F..^ L ) C_ dom (/) )
21		iftrue 3945	. . . . . . . . . . 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 4507	. . . . . . . . . . . 12 |- ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) = { <. x , y >. | ( x e. ( 0..^ ( L - F ) ) /\ y = ( (/) ` ( x + F ) ) ) }
24		df-opab 4506	. . . . . . . . . . . 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 2496	. . . . . . . . . . 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 3789	. . . . . . . . . . . . . . . . . . . 20 |- -. x e. (/)
27	26	a1i 11	. . . . . . . . . . . . . . . . . . 19 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> -. x e. (/) )
28		zre 10869	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( F e. ZZ -> F e. RR )
29		zre 10869	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( L e. ZZ -> L e. RR )
30		posdif 10046	. . . . . . . . . . . . . . . . . . . . . . . 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 10906	. . . . . . . . . . . . . . . . . . . . . . . 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 10876	. . . . . . . . . . . . . . . . . . . . . . 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 11817	. . . . . . . . . . . . . . . . . . . . 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 2580	. . . . . . . . . . . . . . . . . 18 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> -. x e. ( 0..^ ( L - F ) ) )
44	43	intnanrd 915	. . . . . . . . . . . . . . . . 17 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> -. ( x e. ( 0..^ ( L - F ) ) /\ y = ( (/) ` ( x + F ) ) ) )
45	44	intnand 914	. . . . . . . . . . . . . . . 16 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> -. ( p = <. x , y >. /\ ( x e. ( 0..^ ( L - F ) ) /\ y = ( (/) ` ( x + F ) ) ) ) )
46	45	alrimivv 1696	. . . . . . . . . . . . . . 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 1631	. . . . . . . . . . . . . . 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 3574	. . . . . . . . . . . 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 3816	. . . . . . . . . . . 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 2520	. . . . . . . . . 10 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) = (/) )
54	22, 53	eqtrd 2508	. . . . . . . . 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 1014	. . . . . . 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 2508	. . . . . 6 |- ( ( (/) e. _V /\ F e. ZZ /\ L e. ZZ ) -> ( (/) substr <. F , L >. ) = (/) )
58	57	3expb 1197	. . . . 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 12513	. . . 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 6310	. . . . . 6 |- ( 0..^ ( ( 2nd ` b ) - ( 1st ` b ) ) ) e. _V
63	62	mptex 6132	. . . . 5 |- ( z e. ( 0..^ ( ( 2nd ` b ) - ( 1st ` b ) ) ) |-> ( s ` ( z + ( 1st ` b ) ) ) ) e. _V
64		0ex 4577	. . . . 5 |- (/) e. _V
65	63, 64	ifex 4008	. . . 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 6855	. . 3 |- dom substr = ( _V X. ( ZZ X. ZZ ) )
67	60, 66	eleq2s 2575	. 2 |- ( <. (/) , <. F , L >. >. e. dom substr -> ( (/) substr <. F , L >. ) = (/) )
68		df-ov 6288	. . 3 |- ( (/) substr <. F , L >. ) = ( substr ` <. (/) , <. F , L >. >. )
69		ndmfv 5890	. . 3 |- ( -. <. (/) , <. F , L >. >. e. dom substr -> ( substr ` <. (/) , <. F , L >. >. ) = (/) )
70	68, 69	syl5eq 2520	. 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 973   A.wal 1377   = wceq 1379   E.wex 1596   e. wcel 1767   {cab 2452   _Vcvv 3113   C_ wss 3476   (/)c0 3785   ifcif 3939   <.cop 4033   class class class wbr 4447   {copab 4504   |-> cmpt 4505   X. cxp 4997   dom cdm 4999   ` cfv 5588  (class class class)co 6285   1stc1st 6783   2ndc2nd 6784   RRcr 9492   0cc0 9493   + caddc 9496   < clt 9629   - cmin 9806   ZZcz 10865  ..^cfzo 11793   substr csubstr 12505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674  df-fzo 11794  df-substr 12513

This theorem is referenced by:  cshword  12728