swrdcl - Metamath Proof Explorer

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Theorem swrdcl 14599

Description: Closure of the subword extractor. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

**Assertion**
Ref	Expression
swrdcl	⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝐹, 𝐿⟩) ∈ Word 𝐴)

Proof of Theorem swrdcl Dummy variables 𝑠 𝑏 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2825	. 2 ⊢ ((𝑆 substr ⟨𝐹, 𝐿⟩) = ∅ → ((𝑆 substr ⟨𝐹, 𝐿⟩) ∈ Word 𝐴 ↔ ∅ ∈ Word 𝐴))
2		n0 4294	. . . 4 ⊢ ((𝑆 substr ⟨𝐹, 𝐿⟩) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑆 substr ⟨𝐹, 𝐿⟩))
3		df-substr 14595	. . . . . . 7 ⊢ substr = (𝑠 ∈ V, 𝑏 ∈ (ℤ × ℤ) ↦ if(((1^st ‘𝑏)..^(2^nd ‘𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2^nd ‘𝑏) − (1^st ‘𝑏))) ↦ (𝑠‘(𝑥 + (1^st ‘𝑏)))), ∅))
4	3	elmpocl2 7603	. . . . . 6 ⊢ (𝑥 ∈ (𝑆 substr ⟨𝐹, 𝐿⟩) → ⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ))
5		opelxp 5660	. . . . . 6 ⊢ (⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ) ↔ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
6	4, 5	sylib 218	. . . . 5 ⊢ (𝑥 ∈ (𝑆 substr ⟨𝐹, 𝐿⟩) → (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
7	6	exlimiv 1932	. . . 4 ⊢ (∃𝑥 𝑥 ∈ (𝑆 substr ⟨𝐹, 𝐿⟩) → (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
8	2, 7	sylbi 217	. . 3 ⊢ ((𝑆 substr ⟨𝐹, 𝐿⟩) ≠ ∅ → (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
9		swrdval 14597	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 substr ⟨𝐹, 𝐿⟩) = if((𝐹..^𝐿) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))), ∅))
10		wrdf 14471	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Word 𝐴 → 𝑆:(0..^(♯‘𝑆))⟶𝐴)
11	10	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → 𝑆:(0..^(♯‘𝑆))⟶𝐴)
12	11	ad2antrr 727	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹..^𝐿) ⊆ dom 𝑆) ∧ 𝑥 ∈ (0..^(𝐿 − 𝐹))) → 𝑆:(0..^(♯‘𝑆))⟶𝐴)
13		simplr 769	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹..^𝐿) ⊆ dom 𝑆) ∧ 𝑥 ∈ (0..^(𝐿 − 𝐹))) → (𝐹..^𝐿) ⊆ dom 𝑆)
14		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹..^𝐿) ⊆ dom 𝑆) ∧ 𝑥 ∈ (0..^(𝐿 − 𝐹))) → 𝑥 ∈ (0..^(𝐿 − 𝐹)))
15		simpll3 1216	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹..^𝐿) ⊆ dom 𝑆) ∧ 𝑥 ∈ (0..^(𝐿 − 𝐹))) → 𝐿 ∈ ℤ)
16		simpll2 1215	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹..^𝐿) ⊆ dom 𝑆) ∧ 𝑥 ∈ (0..^(𝐿 − 𝐹))) → 𝐹 ∈ ℤ)
17		fzoaddel2 13666	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (0..^(𝐿 − 𝐹)) ∧ 𝐿 ∈ ℤ ∧ 𝐹 ∈ ℤ) → (𝑥 + 𝐹) ∈ (𝐹..^𝐿))
18	14, 15, 16, 17	syl3anc 1374	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹..^𝐿) ⊆ dom 𝑆) ∧ 𝑥 ∈ (0..^(𝐿 − 𝐹))) → (𝑥 + 𝐹) ∈ (𝐹..^𝐿))
19	13, 18	sseldd 3923	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹..^𝐿) ⊆ dom 𝑆) ∧ 𝑥 ∈ (0..^(𝐿 − 𝐹))) → (𝑥 + 𝐹) ∈ dom 𝑆)
20	12	fdmd 6672	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹..^𝐿) ⊆ dom 𝑆) ∧ 𝑥 ∈ (0..^(𝐿 − 𝐹))) → dom 𝑆 = (0..^(♯‘𝑆)))
21	19, 20	eleqtrd 2839	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹..^𝐿) ⊆ dom 𝑆) ∧ 𝑥 ∈ (0..^(𝐿 − 𝐹))) → (𝑥 + 𝐹) ∈ (0..^(♯‘𝑆)))
22	12, 21	ffvelcdmd 7031	. . . . . . . 8 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹..^𝐿) ⊆ dom 𝑆) ∧ 𝑥 ∈ (0..^(𝐿 − 𝐹))) → (𝑆‘(𝑥 + 𝐹)) ∈ 𝐴)
23	22	fmpttd 7061	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹..^𝐿) ⊆ dom 𝑆) → (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))):(0..^(𝐿 − 𝐹))⟶𝐴)
24		iswrdi 14470	. . . . . . 7 ⊢ ((𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))):(0..^(𝐿 − 𝐹))⟶𝐴 → (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))) ∈ Word 𝐴)
25	23, 24	syl 17	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹..^𝐿) ⊆ dom 𝑆) → (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))) ∈ Word 𝐴)
26		wrd0 14492	. . . . . . 7 ⊢ ∅ ∈ Word 𝐴
27	26	a1i 11	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ ¬ (𝐹..^𝐿) ⊆ dom 𝑆) → ∅ ∈ Word 𝐴)
28	25, 27	ifclda 4503	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → if((𝐹..^𝐿) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))), ∅) ∈ Word 𝐴)
29	9, 28	eqeltrd 2837	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 substr ⟨𝐹, 𝐿⟩) ∈ Word 𝐴)
30	29	3expb 1121	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ)) → (𝑆 substr ⟨𝐹, 𝐿⟩) ∈ Word 𝐴)
31	8, 30	sylan2 594	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝐹, 𝐿⟩) ≠ ∅) → (𝑆 substr ⟨𝐹, 𝐿⟩) ∈ Word 𝐴)
32	26	a1i 11	. 2 ⊢ (𝑆 ∈ Word 𝐴 → ∅ ∈ Word 𝐴)
33	1, 31, 32	pm2.61ne 3018	1 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝐹, 𝐿⟩) ∈ Word 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 ∃wex 1781 ∈ wcel 2114 ≠ wne 2933 Vcvv 3430 ⊆ wss 3890 ∅c0 4274 ifcif 4467 ⟨cop 4574 ↦ cmpt 5167 × cxp 5622 dom cdm 5624 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 1^st c1st 7933 2^nd c2nd 7934 0cc0 11029 + caddc 11032 − cmin 11368 ℤcz 12515 ..^cfzo 13599 ♯chash 14283 Word cword 14466 substr csubstr 14594

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-n0 12429 df-z 12516 df-uz 12780 df-fz 13453 df-fzo 13600 df-hash 14284 df-word 14467 df-substr 14595

This theorem is referenced by: swrdf 14604 swrdspsleq 14619 swrds1 14620 ccatswrd 14622 swrdccat2 14623 pfxcl 14631 ccatpfx 14654 swrdswrd 14658 lenrevpfxcctswrd 14665 pfxccatin12 14686 swrdccat 14688 swrdccat3blem 14692 splcl 14705 spllen 14707 splfv1 14708 splfv2a 14709 splval2 14710 cshwcl 14751 cshwlen 14752 cshwidxmod 14756 gsumspl 18803 psgnunilem2 19461 efgredleme 19709 efgredlemc 19711 efgcpbllemb 19721 frgpuplem 19738 wrdsplex 33011 splfv3 33033 gsumwrd2dccatlem 33153 gsumwrd2dccat 33154 cycpmco2f1 33200 cycpmco2rn 33201 cycpmco2lem2 33203 cycpmco2lem3 33204 cycpmco2lem4 33205 cycpmco2lem5 33206 cycpmco2lem6 33207 cycpmco2 33209 elrgspnlem2 33319 revpfxsfxrev 35314

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