pfxcl - Metamath Proof Explorer

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Theorem pfxcl 14629

Description: Closure of the prefix extractor. (Contributed by AV, 2-May-2020.)

**Assertion**
Ref	Expression
pfxcl	⊢ (𝑆 ∈ Word 𝐴 → (𝑆 prefix 𝐿) ∈ Word 𝐴)

Proof of Theorem pfxcl Dummy variables 𝑥 𝑠 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2813	. 2 ⊢ ((𝑆 prefix 𝐿) = ∅ → ((𝑆 prefix 𝐿) ∈ Word 𝐴 ↔ ∅ ∈ Word 𝐴))
2		n0 4339	. . . 4 ⊢ ((𝑆 prefix 𝐿) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑆 prefix 𝐿))
3		df-pfx 14623	. . . . . 6 ⊢ prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ₀ ↦ (𝑠 substr ⟨0, 𝑙⟩))
4	3	elmpocl2 7644	. . . . 5 ⊢ (𝑥 ∈ (𝑆 prefix 𝐿) → 𝐿 ∈ ℕ₀)
5	4	exlimiv 1925	. . . 4 ⊢ (∃𝑥 𝑥 ∈ (𝑆 prefix 𝐿) → 𝐿 ∈ ℕ₀)
6	2, 5	sylbi 216	. . 3 ⊢ ((𝑆 prefix 𝐿) ≠ ∅ → 𝐿 ∈ ℕ₀)
7		pfxval 14625	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ ℕ₀) → (𝑆 prefix 𝐿) = (𝑆 substr ⟨0, 𝐿⟩))
8		swrdcl 14597	. . . . 5 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨0, 𝐿⟩) ∈ Word 𝐴)
9	8	adantr 480	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ ℕ₀) → (𝑆 substr ⟨0, 𝐿⟩) ∈ Word 𝐴)
10	7, 9	eqeltrd 2825	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ ℕ₀) → (𝑆 prefix 𝐿) ∈ Word 𝐴)
11	6, 10	sylan2 592	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑆 prefix 𝐿) ≠ ∅) → (𝑆 prefix 𝐿) ∈ Word 𝐴)
12		wrd0 14491	. . 3 ⊢ ∅ ∈ Word 𝐴
13	12	a1i 11	. 2 ⊢ (𝑆 ∈ Word 𝐴 → ∅ ∈ Word 𝐴)
14	1, 11, 13	pm2.61ne 3019	1 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 prefix 𝐿) ∈ Word 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∃wex 1773 ∈ wcel 2098 ≠ wne 2932 Vcvv 3466 ∅c0 4315 ⟨cop 4627 (class class class)co 7402 0cc0 11107 ℕ₀cn0 12471 Word cword 14466 substr csubstr 14592 prefix cpfx 14622

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13486 df-fzo 13629 df-hash 14292 df-word 14467 df-substr 14593 df-pfx 14623

This theorem is referenced by: pfxfvlsw 14647 pfxeq 14648 ccatpfx 14653 lenrevpfxcctswrd 14664 wrdind 14674 wrd2ind 14675 pfxccatin12 14685 splcl 14704 spllen 14706 splfv1 14707 splfv2a 14708 splval2 14709 repswpfx 14737 cshwcl 14750 cshwlen 14751 cshwidxmod 14755 pfx2 14900 gsumspl 18765 psgnunilem5 19410 efgsres 19654 efgredleme 19659 efgredlemc 19661 efgcpbllemb 19671 frgpuplem 19688 wwlksm1edg 29630 wwlksnred 29641 wwlksnextwrd 29646 clwlkclwwlk 29750 clwwlkinwwlk 29788 clwwlkf 29795 wwlksubclwwlk 29806 pfxlsw2ccat 32609 wrdt2ind 32610 splfv3 32615 cycpmco2f1 32777 cycpmco2rn 32778 cycpmco2lem2 32780 cycpmco2lem3 32781 cycpmco2lem4 32782 cycpmco2lem5 32783 cycpmco2lem6 32784 cycpmco2 32786 signsvtn0 34101 signstfveq0 34108 revpfxsfxrev 34624 swrdrevpfx 34625 pfxwlk 34632 swrdwlk 34635

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