swrdcl - Metamath Proof Explorer

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Theorem swrdcl 14007

Description: Closure of the subword extractor. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

**Assertion**
Ref	Expression
swrdcl	⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝐹, 𝐿⟩) ∈ Word 𝐴)

Proof of Theorem swrdcl Dummy variables 𝑠 𝑏 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2900	. 2 ⊢ ((𝑆 substr ⟨𝐹, 𝐿⟩) = ∅ → ((𝑆 substr ⟨𝐹, 𝐿⟩) ∈ Word 𝐴 ↔ ∅ ∈ Word 𝐴))
2		n0 4310	. . . 4 ⊢ ((𝑆 substr ⟨𝐹, 𝐿⟩) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑆 substr ⟨𝐹, 𝐿⟩))
3		df-substr 14003	. . . . . . 7 ⊢ substr = (𝑠 ∈ V, 𝑏 ∈ (ℤ × ℤ) ↦ if(((1^st ‘𝑏)..^(2^nd ‘𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2^nd ‘𝑏) − (1^st ‘𝑏))) ↦ (𝑠‘(𝑥 + (1^st ‘𝑏)))), ∅))
4	3	elmpocl2 7389	. . . . . 6 ⊢ (𝑥 ∈ (𝑆 substr ⟨𝐹, 𝐿⟩) → ⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ))
5		opelxp 5591	. . . . . 6 ⊢ (⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ) ↔ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
6	4, 5	sylib 220	. . . . 5 ⊢ (𝑥 ∈ (𝑆 substr ⟨𝐹, 𝐿⟩) → (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
7	6	exlimiv 1931	. . . 4 ⊢ (∃𝑥 𝑥 ∈ (𝑆 substr ⟨𝐹, 𝐿⟩) → (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
8	2, 7	sylbi 219	. . 3 ⊢ ((𝑆 substr ⟨𝐹, 𝐿⟩) ≠ ∅ → (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
9		swrdval 14005	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 substr ⟨𝐹, 𝐿⟩) = if((𝐹..^𝐿) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))), ∅))
10		wrdf 13867	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Word 𝐴 → 𝑆:(0..^(♯‘𝑆))⟶𝐴)
11	10	3ad2ant1 1129	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → 𝑆:(0..^(♯‘𝑆))⟶𝐴)
12	11	ad2antrr 724	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹..^𝐿) ⊆ dom 𝑆) ∧ 𝑥 ∈ (0..^(𝐿 − 𝐹))) → 𝑆:(0..^(♯‘𝑆))⟶𝐴)
13		simplr 767	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹..^𝐿) ⊆ dom 𝑆) ∧ 𝑥 ∈ (0..^(𝐿 − 𝐹))) → (𝐹..^𝐿) ⊆ dom 𝑆)
14		simpr 487	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹..^𝐿) ⊆ dom 𝑆) ∧ 𝑥 ∈ (0..^(𝐿 − 𝐹))) → 𝑥 ∈ (0..^(𝐿 − 𝐹)))
15		simpll3 1210	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹..^𝐿) ⊆ dom 𝑆) ∧ 𝑥 ∈ (0..^(𝐿 − 𝐹))) → 𝐿 ∈ ℤ)
16		simpll2 1209	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹..^𝐿) ⊆ dom 𝑆) ∧ 𝑥 ∈ (0..^(𝐿 − 𝐹))) → 𝐹 ∈ ℤ)
17		fzoaddel2 13094	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (0..^(𝐿 − 𝐹)) ∧ 𝐿 ∈ ℤ ∧ 𝐹 ∈ ℤ) → (𝑥 + 𝐹) ∈ (𝐹..^𝐿))
18	14, 15, 16, 17	syl3anc 1367	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹..^𝐿) ⊆ dom 𝑆) ∧ 𝑥 ∈ (0..^(𝐿 − 𝐹))) → (𝑥 + 𝐹) ∈ (𝐹..^𝐿))
19	13, 18	sseldd 3968	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹..^𝐿) ⊆ dom 𝑆) ∧ 𝑥 ∈ (0..^(𝐿 − 𝐹))) → (𝑥 + 𝐹) ∈ dom 𝑆)
20	12	fdmd 6523	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹..^𝐿) ⊆ dom 𝑆) ∧ 𝑥 ∈ (0..^(𝐿 − 𝐹))) → dom 𝑆 = (0..^(♯‘𝑆)))
21	19, 20	eleqtrd 2915	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹..^𝐿) ⊆ dom 𝑆) ∧ 𝑥 ∈ (0..^(𝐿 − 𝐹))) → (𝑥 + 𝐹) ∈ (0..^(♯‘𝑆)))
22	12, 21	ffvelrnd 6852	. . . . . . . 8 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹..^𝐿) ⊆ dom 𝑆) ∧ 𝑥 ∈ (0..^(𝐿 − 𝐹))) → (𝑆‘(𝑥 + 𝐹)) ∈ 𝐴)
23	22	fmpttd 6879	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹..^𝐿) ⊆ dom 𝑆) → (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))):(0..^(𝐿 − 𝐹))⟶𝐴)
24		iswrdi 13866	. . . . . . 7 ⊢ ((𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))):(0..^(𝐿 − 𝐹))⟶𝐴 → (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))) ∈ Word 𝐴)
25	23, 24	syl 17	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝐹..^𝐿) ⊆ dom 𝑆) → (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))) ∈ Word 𝐴)
26		wrd0 13889	. . . . . . 7 ⊢ ∅ ∈ Word 𝐴
27	26	a1i 11	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ ¬ (𝐹..^𝐿) ⊆ dom 𝑆) → ∅ ∈ Word 𝐴)
28	25, 27	ifclda 4501	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → if((𝐹..^𝐿) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝐿 − 𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))), ∅) ∈ Word 𝐴)
29	9, 28	eqeltrd 2913	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 substr ⟨𝐹, 𝐿⟩) ∈ Word 𝐴)
30	29	3expb 1116	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ)) → (𝑆 substr ⟨𝐹, 𝐿⟩) ∈ Word 𝐴)
31	8, 30	sylan2 594	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝐹, 𝐿⟩) ≠ ∅) → (𝑆 substr ⟨𝐹, 𝐿⟩) ∈ Word 𝐴)
32	26	a1i 11	. 2 ⊢ (𝑆 ∈ Word 𝐴 → ∅ ∈ Word 𝐴)
33	1, 31, 32	pm2.61ne 3102	1 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝐹, 𝐿⟩) ∈ Word 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ⊆ wss 3936 ∅c0 4291 ifcif 4467 ⟨cop 4573 ↦ cmpt 5146 × cxp 5553 dom cdm 5555 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 1^st c1st 7687 2^nd c2nd 7688 0cc0 10537 + caddc 10540 − cmin 10870 ℤcz 11982 ..^cfzo 13034 ♯chash 13691 Word cword 13862 substr csubstr 14002

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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-substr 14003

This theorem is referenced by: swrdf 14012 swrdspsleq 14027 swrds1 14028 ccatswrd 14030 swrdccat2 14031 pfxcl 14039 ccatpfx 14063 swrdswrd 14067 lenrevpfxcctswrd 14074 pfxccatin12 14095 swrdccat 14097 swrdccat3blem 14101 splcl 14114 spllen 14116 splfv1 14117 splfv2a 14118 splval2 14119 cshwcl 14160 cshwlen 14161 cshwidxmod 14165 gsumspl 18009 psgnunilem2 18623 efgredleme 18869 efgredlemc 18871 efgcpbllemb 18881 frgpuplem 18898 wrdsplex 30614 splfv3 30632 cycpmco2f1 30766 cycpmco2rn 30767 cycpmco2lem2 30769 cycpmco2lem3 30770 cycpmco2lem4 30771 cycpmco2lem5 30772 cycpmco2lem6 30773 cycpmco2 30775 revpfxsfxrev 32362

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