Theorem swrd0 12780

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 4879	. . . 4 |- ( <. (/) , <. F , L >. >. e. ( _V X. ( ZZ X. ZZ ) ) <-> ( (/) e. _V /\ <. F , L >. e. ( ZZ X. ZZ ) ) )
2		opelxp 4879	. . . . 5 |- ( <. F , L >. e. ( ZZ X. ZZ ) <-> ( F e. ZZ /\ L e. ZZ ) )
3		swrdval 12763	. . . . . . 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 11941	. . . . . . . . . . . . . . 15 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( -. F < L <-> ( F..^ L ) = (/) ) )
5	4	biimprd 226	. . . . . . . . . . . . . 14 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( ( F..^ L ) = (/) -> -. F < L ) )
6	5	con2d 118	. . . . . . . . . . . . 13 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( F < L -> -. ( F..^ L ) = (/) ) )
7	6	impcom 431	. . . . . . . . . . . 12 |- ( ( F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> -. ( F..^ L ) = (/) )
8		ss0 3793	. . . . . . . . . . . 12 |- ( ( F..^ L ) C_ (/) -> ( F..^ L ) = (/) )
9	7, 8	nsyl 124	. . . . . . . . . . 11 |- ( ( F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> -. ( F..^ L ) C_ (/) )
10		dm0 5063	. . . . . . . . . . . . 13 |- dom (/) = (/)
11	10	a1i 11	. . . . . . . . . . . 12 |- ( ( F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> dom (/) = (/) )
12	11	sseq2d 3492	. . . . . . . . . . 11 |- ( ( F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( ( F..^ L ) C_ dom (/) <-> ( F..^ L ) C_ (/) ) )
13	9, 12	mtbird 302	. . . . . . . . . 10 |- ( ( F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> -. ( F..^ L ) C_ dom (/) )
14	13	iffalsed 3920	. . . . . . . . 9 |- ( ( F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> if ( ( F..^ L ) C_ dom (/) , ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) , (/) ) = (/) )
15		0ss 3791	. . . . . . . . . . . . 13 |- (/) C_ (/)
16	15	a1i 11	. . . . . . . . . . . 12 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> (/) C_ (/) )
17	4	biimpac 488	. . . . . . . . . . . 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 3507	. . . . . . . . . . 11 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( F..^ L ) C_ dom (/) )
20	19	iftrued 3917	. . . . . . . . . 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 10941	. . . . . . . . . . . . . . . . 17 |- ( L e. ZZ -> L e. RR )
22		zre 10941	. . . . . . . . . . . . . . . . 17 |- ( F e. ZZ -> F e. RR )
23		lenlt 9712	. . . . . . . . . . . . . . . . . 18 |- ( ( L e. RR /\ F e. RR ) -> ( L <_ F <-> -. F < L ) )
24	23	bicomd 204	. . . . . . . . . . . . . . . . 17 |- ( ( L e. RR /\ F e. RR ) -> ( -. F < L <-> L <_ F ) )
25	21, 22, 24	syl2anr 480	. . . . . . . . . . . . . . . 16 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( -. F < L <-> L <_ F ) )
26		fzo0n 11940	. . . . . . . . . . . . . . . 16 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( L <_ F <-> ( 0..^ ( L - F ) ) = (/) ) )
27	25, 26	bitrd 256	. . . . . . . . . . . . . . 15 |- ( ( F e. ZZ /\ L e. ZZ ) -> ( -. F < L <-> ( 0..^ ( L - F ) ) = (/) ) )
28	27	biimpac 488	. . . . . . . . . . . . . 14 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( 0..^ ( L - F ) ) = (/) )
29	28	mpteq1d 4502	. . . . . . . . . . . . 13 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) = ( x e. (/) |-> ( (/) ` ( x + F ) ) ) )
30	29	dmeqd 5052	. . . . . . . . . . . 12 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> dom ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) = dom ( x e. (/) |-> ( (/) ` ( x + F ) ) ) )
31		ral0 3902	. . . . . . . . . . . . 13 |- A. x e. (/) ( (/) ` ( x + F ) ) e. _V
32		dmmptg 5347	. . . . . . . . . . . . 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 2463	. . . . . . . . . . 11 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> dom ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) = (/) )
35		mptrel 4976	. . . . . . . . . . . 12 |- Rel ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) )
36		reldm0 5067	. . . . . . . . . . . 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 235	. . . . . . . . . 10 |- ( ( -. F < L /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) = (/) )
39	20, 38	eqtrd 2463	. . . . . . . . 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 797	. . . . . . . 8 |- ( ( F e. ZZ /\ L e. ZZ ) -> if ( ( F..^ L ) C_ dom (/) , ( x e. ( 0..^ ( L - F ) ) |-> ( (/) ` ( x + F ) ) ) , (/) ) = (/) )
41	40	3adant1 1023	. . . . . . 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 2463	. . . . . 6 |- ( ( (/) e. _V /\ F e. ZZ /\ L e. ZZ ) -> ( (/) substr <. F , L >. ) = (/) )
43	42	3expb 1206	. . . . 5 |- ( ( (/) e. _V /\ ( F e. ZZ /\ L e. ZZ ) ) -> ( (/) substr <. F , L >. ) = (/) )
44	2, 43	sylan2b 477	. . . 4 |- ( ( (/) e. _V /\ <. F , L >. e. ( ZZ X. ZZ ) ) -> ( (/) substr <. F , L >. ) = (/) )
45	1, 44	sylbi 198	. . 3 |- ( <. (/) , <. F , L >. >. e. ( _V X. ( ZZ X. ZZ ) ) -> ( (/) substr <. F , L >. ) = (/) )
46		df-substr 12660	. . . 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 6329	. . . . . 6 |- ( 0..^ ( ( 2nd ` b ) - ( 1st ` b ) ) ) e. _V
48	47	mptex 6147	. . . . 5 |- ( z e. ( 0..^ ( ( 2nd ` b ) - ( 1st ` b ) ) ) |-> ( s ` ( z + ( 1st ` b ) ) ) ) e. _V
49		0ex 4552	. . . . 5 |- (/) e. _V
50	48, 49	ifex 3977	. . . 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 6873	. . 3 |- dom substr = ( _V X. ( ZZ X. ZZ ) )
52	45, 51	eleq2s 2530	. 2 |- ( <. (/) , <. F , L >. >. e. dom substr -> ( (/) substr <. F , L >. ) = (/) )
53		df-ov 6304	. . 3 |- ( (/) substr <. F , L >. ) = ( substr ` <. (/) , <. F , L >. >. )
54		ndmfv 5901	. . 3 |- ( -. <. (/) , <. F , L >. >. e. dom substr -> ( substr ` <. (/) , <. F , L >. >. ) = (/) )
55	53, 54	syl5eq 2475	. 2 |- ( -. <. (/) , <. F , L >. >. e. dom substr -> ( (/) substr <. F , L >. ) = (/) )
56	52, 55	pm2.61i 167	1 |- ( (/) substr <. F , L >. ) = (/)

Syntax hints:   -. wn 3   <-> wb 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1868   A.wral 2775   _Vcvv 3081   C_ wss 3436   (/)c0 3761   ifcif 3909   <.cop 4002   class class class wbr 4420   |-> cmpt 4479   X. cxp 4847   dom cdm 4849   Rel wrel 4854   ` cfv 5597  (class class class)co 6301   1stc1st 6801   2ndc2nd 6802   RRcr 9538   0cc0 9539   + caddc 9542   < clt 9675   <_ cle 9676   - cmin 9860   ZZcz 10937  ..^cfzo 11915   substr csubstr 12652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-1st 6803  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-substr 12660

This theorem is referenced by:  cshword  12883  pfx0  38637  cshword2  38689