pfxcl - Metamath Proof Explorer

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Theorem pfxcl 14466

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 2824	. 2 ⊢ ((𝑆 prefix 𝐿) = ∅ → ((𝑆 prefix 𝐿) ∈ Word 𝐴 ↔ ∅ ∈ Word 𝐴))
2		n0 4290	. . . 4 ⊢ ((𝑆 prefix 𝐿) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑆 prefix 𝐿))
3		df-pfx 14460	. . . . . 6 ⊢ prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ₀ ↦ (𝑠 substr ⟨0, 𝑙⟩))
4	3	elmpocl2 7554	. . . . 5 ⊢ (𝑥 ∈ (𝑆 prefix 𝐿) → 𝐿 ∈ ℕ₀)
5	4	exlimiv 1932	. . . 4 ⊢ (∃𝑥 𝑥 ∈ (𝑆 prefix 𝐿) → 𝐿 ∈ ℕ₀)
6	2, 5	sylbi 216	. . 3 ⊢ ((𝑆 prefix 𝐿) ≠ ∅ → 𝐿 ∈ ℕ₀)
7		pfxval 14462	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ ℕ₀) → (𝑆 prefix 𝐿) = (𝑆 substr ⟨0, 𝐿⟩))
8		swrdcl 14434	. . . . 5 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨0, 𝐿⟩) ∈ Word 𝐴)
9	8	adantr 481	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ ℕ₀) → (𝑆 substr ⟨0, 𝐿⟩) ∈ Word 𝐴)
10	7, 9	eqeltrd 2837	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ ℕ₀) → (𝑆 prefix 𝐿) ∈ Word 𝐴)
11	6, 10	sylan2 593	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑆 prefix 𝐿) ≠ ∅) → (𝑆 prefix 𝐿) ∈ Word 𝐴)
12		wrd0 14320	. . 3 ⊢ ∅ ∈ Word 𝐴
13	12	a1i 11	. 2 ⊢ (𝑆 ∈ Word 𝐴 → ∅ ∈ Word 𝐴)
14	1, 11, 13	pm2.61ne 3027	1 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 prefix 𝐿) ∈ Word 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∃wex 1780 ∈ wcel 2105 ≠ wne 2940 Vcvv 3440 ∅c0 4266 ⟨cop 4576 (class class class)co 7316 0cc0 10950 ℕ₀cn0 12312 Word cword 14295 substr csubstr 14429 prefix cpfx 14459

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5223 ax-sep 5237 ax-nul 5244 ax-pow 5302 ax-pr 5366 ax-un 7629 ax-cnex 11006 ax-resscn 11007 ax-1cn 11008 ax-icn 11009 ax-addcl 11010 ax-addrcl 11011 ax-mulcl 11012 ax-mulrcl 11013 ax-mulcom 11014 ax-addass 11015 ax-mulass 11016 ax-distr 11017 ax-i2m1 11018 ax-1ne0 11019 ax-1rid 11020 ax-rnegex 11021 ax-rrecex 11022 ax-cnre 11023 ax-pre-lttri 11024 ax-pre-lttrn 11025 ax-pre-ltadd 11026 ax-pre-mulgt0 11027

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3442 df-sbc 3726 df-csb 3842 df-dif 3899 df-un 3901 df-in 3903 df-ss 3913 df-pss 3915 df-nul 4267 df-if 4471 df-pw 4546 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4850 df-int 4892 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5170 df-tr 5204 df-id 5506 df-eprel 5512 df-po 5520 df-so 5521 df-fr 5562 df-we 5564 df-xp 5613 df-rel 5614 df-cnv 5615 df-co 5616 df-dm 5617 df-rn 5618 df-res 5619 df-ima 5620 df-pred 6224 df-ord 6291 df-on 6292 df-lim 6293 df-suc 6294 df-iota 6417 df-fun 6467 df-fn 6468 df-f 6469 df-f1 6470 df-fo 6471 df-f1o 6472 df-fv 6473 df-riota 7273 df-ov 7319 df-oprab 7320 df-mpo 7321 df-om 7759 df-1st 7877 df-2nd 7878 df-frecs 8145 df-wrecs 8176 df-recs 8250 df-rdg 8289 df-1o 8345 df-er 8547 df-en 8783 df-dom 8784 df-sdom 8785 df-fin 8786 df-card 9774 df-pnf 11090 df-mnf 11091 df-xr 11092 df-ltxr 11093 df-le 11094 df-sub 11286 df-neg 11287 df-nn 12053 df-n0 12313 df-z 12399 df-uz 12662 df-fz 13319 df-fzo 13462 df-hash 14124 df-word 14296 df-substr 14430 df-pfx 14460

This theorem is referenced by: pfxfvlsw 14484 pfxeq 14485 ccatpfx 14490 lenrevpfxcctswrd 14501 wrdind 14511 wrd2ind 14512 pfxccatin12 14522 splcl 14541 spllen 14543 splfv1 14544 splfv2a 14545 splval2 14546 repswpfx 14574 cshwcl 14587 cshwlen 14588 cshwidxmod 14592 pfx2 14736 gsumspl 18556 psgnunilem5 19175 efgsres 19416 efgredleme 19421 efgredlemc 19423 efgcpbllemb 19433 frgpuplem 19450 wwlksm1edg 28378 wwlksnred 28389 wwlksnextwrd 28394 clwlkclwwlk 28498 clwwlkinwwlk 28536 clwwlkf 28543 wwlksubclwwlk 28554 pfxlsw2ccat 31355 wrdt2ind 31356 splfv3 31361 cycpmco2f1 31522 cycpmco2rn 31523 cycpmco2lem2 31525 cycpmco2lem3 31526 cycpmco2lem4 31527 cycpmco2lem5 31528 cycpmco2lem6 31529 cycpmco2 31531 signsvtn0 32685 signstfveq0 32692 revpfxsfxrev 33212 swrdrevpfx 33213 pfxwlk 33220 swrdwlk 33223

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