pfxcl - Metamath Proof Explorer

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Theorem pfxcl 14654

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 2817	. 2 ⊢ ((𝑆 prefix 𝐿) = ∅ → ((𝑆 prefix 𝐿) ∈ Word 𝐴 ↔ ∅ ∈ Word 𝐴))
2		n0 4343	. . . 4 ⊢ ((𝑆 prefix 𝐿) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑆 prefix 𝐿))
3		df-pfx 14648	. . . . . 6 ⊢ prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ₀ ↦ (𝑠 substr ⟨0, 𝑙⟩))
4	3	elmpocl2 7659	. . . . 5 ⊢ (𝑥 ∈ (𝑆 prefix 𝐿) → 𝐿 ∈ ℕ₀)
5	4	exlimiv 1926	. . . 4 ⊢ (∃𝑥 𝑥 ∈ (𝑆 prefix 𝐿) → 𝐿 ∈ ℕ₀)
6	2, 5	sylbi 216	. . 3 ⊢ ((𝑆 prefix 𝐿) ≠ ∅ → 𝐿 ∈ ℕ₀)
7		pfxval 14650	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ ℕ₀) → (𝑆 prefix 𝐿) = (𝑆 substr ⟨0, 𝐿⟩))
8		swrdcl 14622	. . . . 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 2829	. . 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 14516	. . 3 ⊢ ∅ ∈ Word 𝐴
13	12	a1i 11	. 2 ⊢ (𝑆 ∈ Word 𝐴 → ∅ ∈ Word 𝐴)
14	1, 11, 13	pm2.61ne 3023	1 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 prefix 𝐿) ∈ Word 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∃wex 1774 ∈ wcel 2099 ≠ wne 2936 Vcvv 3470 ∅c0 4319 ⟨cop 4631 (class class class)co 7415 0cc0 11133 ℕ₀cn0 12497 Word cword 14491 substr csubstr 14617 prefix cpfx 14647

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-n0 12498 df-z 12584 df-uz 12848 df-fz 13512 df-fzo 13655 df-hash 14317 df-word 14492 df-substr 14618 df-pfx 14648

This theorem is referenced by: pfxfvlsw 14672 pfxeq 14673 ccatpfx 14678 lenrevpfxcctswrd 14689 wrdind 14699 wrd2ind 14700 pfxccatin12 14710 splcl 14729 spllen 14731 splfv1 14732 splfv2a 14733 splval2 14734 repswpfx 14762 cshwcl 14775 cshwlen 14776 cshwidxmod 14780 pfx2 14925 gsumspl 18790 psgnunilem5 19443 efgsres 19687 efgredleme 19692 efgredlemc 19694 efgcpbllemb 19704 frgpuplem 19721 wwlksm1edg 29686 wwlksnred 29697 wwlksnextwrd 29702 clwlkclwwlk 29806 clwwlkinwwlk 29844 clwwlkf 29851 wwlksubclwwlk 29862 pfxlsw2ccat 32668 wrdt2ind 32669 splfv3 32674 cycpmco2f1 32840 cycpmco2rn 32841 cycpmco2lem2 32843 cycpmco2lem3 32844 cycpmco2lem4 32845 cycpmco2lem5 32846 cycpmco2lem6 32847 cycpmco2 32849 signsvtn0 34197 signstfveq0 34204 revpfxsfxrev 34720 swrdrevpfx 34721 pfxwlk 34728 swrdwlk 34731

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