pfxcl - Metamath Proof Explorer

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Theorem pfxcl 14725

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 2832	. 2 ⊢ ((𝑆 prefix 𝐿) = ∅ → ((𝑆 prefix 𝐿) ∈ Word 𝐴 ↔ ∅ ∈ Word 𝐴))
2		n0 4376	. . . 4 ⊢ ((𝑆 prefix 𝐿) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑆 prefix 𝐿))
3		df-pfx 14719	. . . . . 6 ⊢ prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ₀ ↦ (𝑠 substr ⟨0, 𝑙⟩))
4	3	elmpocl2 7693	. . . . 5 ⊢ (𝑥 ∈ (𝑆 prefix 𝐿) → 𝐿 ∈ ℕ₀)
5	4	exlimiv 1929	. . . 4 ⊢ (∃𝑥 𝑥 ∈ (𝑆 prefix 𝐿) → 𝐿 ∈ ℕ₀)
6	2, 5	sylbi 217	. . 3 ⊢ ((𝑆 prefix 𝐿) ≠ ∅ → 𝐿 ∈ ℕ₀)
7		pfxval 14721	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ ℕ₀) → (𝑆 prefix 𝐿) = (𝑆 substr ⟨0, 𝐿⟩))
8		swrdcl 14693	. . . . 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 2844	. . 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 14587	. . 3 ⊢ ∅ ∈ Word 𝐴
13	12	a1i 11	. 2 ⊢ (𝑆 ∈ Word 𝐴 → ∅ ∈ Word 𝐴)
14	1, 11, 13	pm2.61ne 3033	1 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 prefix 𝐿) ∈ Word 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∃wex 1777 ∈ wcel 2108 ≠ wne 2946 Vcvv 3488 ∅c0 4352 ⟨cop 4654 (class class class)co 7448 0cc0 11184 ℕ₀cn0 12553 Word cword 14562 substr csubstr 14688 prefix cpfx 14718

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-n0 12554 df-z 12640 df-uz 12904 df-fz 13568 df-fzo 13712 df-hash 14380 df-word 14563 df-substr 14689 df-pfx 14719

This theorem is referenced by: pfxfvlsw 14743 pfxeq 14744 ccatpfx 14749 lenrevpfxcctswrd 14760 wrdind 14770 wrd2ind 14771 pfxccatin12 14781 splcl 14800 spllen 14802 splfv1 14803 splfv2a 14804 splval2 14805 repswpfx 14833 cshwcl 14846 cshwlen 14847 cshwidxmod 14851 pfx2 14996 gsumspl 18879 psgnunilem5 19536 efgsres 19780 efgredleme 19785 efgredlemc 19787 efgcpbllemb 19797 frgpuplem 19814 wwlksm1edg 29914 wwlksnred 29925 wwlksnextwrd 29930 clwlkclwwlk 30034 clwwlkinwwlk 30072 clwwlkf 30079 wwlksubclwwlk 30090 pfxlsw2ccat 32917 wrdt2ind 32920 splfv3 32925 pfxchn 32982 cycpmco2f1 33117 cycpmco2rn 33118 cycpmco2lem2 33120 cycpmco2lem3 33121 cycpmco2lem4 33122 cycpmco2lem5 33123 cycpmco2lem6 33124 cycpmco2 33126 1arithidomlem1 33528 signsvtn0 34547 signstfveq0 34554 revpfxsfxrev 35083 swrdrevpfx 35084 pfxwlk 35091 swrdwlk 35094

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