pfxcl - Metamath Proof Explorer

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Theorem pfxcl 14660

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 4347	. . . 4 ⊢ ((𝑆 prefix 𝐿) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑆 prefix 𝐿))
3		df-pfx 14654	. . . . . 6 ⊢ prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ₀ ↦ (𝑠 substr ⟨0, 𝑙⟩))
4	3	elmpocl2 7664	. . . . 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 14656	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ ℕ₀) → (𝑆 prefix 𝐿) = (𝑆 substr ⟨0, 𝐿⟩))
8		swrdcl 14628	. . . . 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 14522	. . 3 ⊢ ∅ ∈ Word 𝐴
13	12	a1i 11	. 2 ⊢ (𝑆 ∈ Word 𝐴 → ∅ ∈ Word 𝐴)
14	1, 11, 13	pm2.61ne 3024	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 2937 Vcvv 3471 ∅c0 4323 ⟨cop 4635 (class class class)co 7420 0cc0 11139 ℕ₀cn0 12503 Word cword 14497 substr csubstr 14623 prefix cpfx 14653

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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216

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 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-fzo 13661 df-hash 14323 df-word 14498 df-substr 14624 df-pfx 14654

This theorem is referenced by: pfxfvlsw 14678 pfxeq 14679 ccatpfx 14684 lenrevpfxcctswrd 14695 wrdind 14705 wrd2ind 14706 pfxccatin12 14716 splcl 14735 spllen 14737 splfv1 14738 splfv2a 14739 splval2 14740 repswpfx 14768 cshwcl 14781 cshwlen 14782 cshwidxmod 14786 pfx2 14931 gsumspl 18796 psgnunilem5 19449 efgsres 19693 efgredleme 19698 efgredlemc 19700 efgcpbllemb 19710 frgpuplem 19727 wwlksm1edg 29705 wwlksnred 29716 wwlksnextwrd 29721 clwlkclwwlk 29825 clwwlkinwwlk 29863 clwwlkf 29870 wwlksubclwwlk 29881 pfxlsw2ccat 32686 wrdt2ind 32687 splfv3 32692 cycpmco2f1 32858 cycpmco2rn 32859 cycpmco2lem2 32861 cycpmco2lem3 32862 cycpmco2lem4 32863 cycpmco2lem5 32864 cycpmco2lem6 32865 cycpmco2 32867 signsvtn0 34202 signstfveq0 34209 revpfxsfxrev 34725 swrdrevpfx 34726 pfxwlk 34733 swrdwlk 34736

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