swrdrevpfx - Mathbox for BTernaryTau

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Theorem swrdrevpfx 34635

Description: A subword expressed in terms of reverses and prefixes. (Contributed by BTernaryTau, 3-Dec-2023.)

**Assertion**
Ref	Expression
swrdrevpfx	⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝑊 substr ⟨𝐹, 𝐿⟩) = (reverse‘((reverse‘(𝑊 prefix 𝐿)) prefix (𝐿 − 𝐹))))

**Proof of Theorem swrdrevpfx**
Step	Hyp	Ref	Expression
1		fznn0sub2 13614	. . . . . 6 ⊢ (𝐹 ∈ (0...𝐿) → (𝐿 − 𝐹) ∈ (0...𝐿))
2		pfxcl 14633	. . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 prefix 𝐿) ∈ Word 𝑉)
3		revcl 14717	. . . . . . . . 9 ⊢ ((𝑊 prefix 𝐿) ∈ Word 𝑉 → (reverse‘(𝑊 prefix 𝐿)) ∈ Word 𝑉)
4	2, 3	syl 17	. . . . . . . 8 ⊢ (𝑊 ∈ Word 𝑉 → (reverse‘(𝑊 prefix 𝐿)) ∈ Word 𝑉)
5	4	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿 − 𝐹) ∈ (0...𝐿)) → (reverse‘(𝑊 prefix 𝐿)) ∈ Word 𝑉)
6		simp3 1135	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿 − 𝐹) ∈ (0...𝐿)) → (𝐿 − 𝐹) ∈ (0...𝐿))
7		revlen 14718	. . . . . . . . . . . . 13 ⊢ ((𝑊 prefix 𝐿) ∈ Word 𝑉 → (♯‘(reverse‘(𝑊 prefix 𝐿))) = (♯‘(𝑊 prefix 𝐿)))
8	2, 7	syl 17	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘(reverse‘(𝑊 prefix 𝐿))) = (♯‘(𝑊 prefix 𝐿)))
9	8	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(reverse‘(𝑊 prefix 𝐿))) = (♯‘(𝑊 prefix 𝐿)))
10		pfxlen 14639	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 prefix 𝐿)) = 𝐿)
11	9, 10	eqtrd 2766	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(reverse‘(𝑊 prefix 𝐿))) = 𝐿)
12	11	3adant3 1129	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿 − 𝐹) ∈ (0...𝐿)) → (♯‘(reverse‘(𝑊 prefix 𝐿))) = 𝐿)
13	12	oveq2d 7421	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿 − 𝐹) ∈ (0...𝐿)) → (0...(♯‘(reverse‘(𝑊 prefix 𝐿)))) = (0...𝐿))
14	6, 13	eleqtrrd 2830	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿 − 𝐹) ∈ (0...𝐿)) → (𝐿 − 𝐹) ∈ (0...(♯‘(reverse‘(𝑊 prefix 𝐿)))))
15	5, 14	jca 511	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿 − 𝐹) ∈ (0...𝐿)) → ((reverse‘(𝑊 prefix 𝐿)) ∈ Word 𝑉 ∧ (𝐿 − 𝐹) ∈ (0...(♯‘(reverse‘(𝑊 prefix 𝐿))))))
16	1, 15	syl3an3 1162	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐹 ∈ (0...𝐿)) → ((reverse‘(𝑊 prefix 𝐿)) ∈ Word 𝑉 ∧ (𝐿 − 𝐹) ∈ (0...(♯‘(reverse‘(𝑊 prefix 𝐿))))))
17	16	3com23 1123	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((reverse‘(𝑊 prefix 𝐿)) ∈ Word 𝑉 ∧ (𝐿 − 𝐹) ∈ (0...(♯‘(reverse‘(𝑊 prefix 𝐿))))))
18		revpfxsfxrev 34634	. . . 4 ⊢ (((reverse‘(𝑊 prefix 𝐿)) ∈ Word 𝑉 ∧ (𝐿 − 𝐹) ∈ (0...(♯‘(reverse‘(𝑊 prefix 𝐿))))) → (reverse‘((reverse‘(𝑊 prefix 𝐿)) prefix (𝐿 − 𝐹))) = ((reverse‘(reverse‘(𝑊 prefix 𝐿))) substr ⟨((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿 − 𝐹)), (♯‘(reverse‘(𝑊 prefix 𝐿)))⟩))
19	17, 18	syl 17	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (reverse‘((reverse‘(𝑊 prefix 𝐿)) prefix (𝐿 − 𝐹))) = ((reverse‘(reverse‘(𝑊 prefix 𝐿))) substr ⟨((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿 − 𝐹)), (♯‘(reverse‘(𝑊 prefix 𝐿)))⟩))
20		revrev 14723	. . . . . 6 ⊢ ((𝑊 prefix 𝐿) ∈ Word 𝑉 → (reverse‘(reverse‘(𝑊 prefix 𝐿))) = (𝑊 prefix 𝐿))
21	2, 20	syl 17	. . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (reverse‘(reverse‘(𝑊 prefix 𝐿))) = (𝑊 prefix 𝐿))
22	21	oveq1d 7420	. . . 4 ⊢ (𝑊 ∈ Word 𝑉 → ((reverse‘(reverse‘(𝑊 prefix 𝐿))) substr ⟨((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿 − 𝐹)), (♯‘(reverse‘(𝑊 prefix 𝐿)))⟩) = ((𝑊 prefix 𝐿) substr ⟨((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿 − 𝐹)), (♯‘(reverse‘(𝑊 prefix 𝐿)))⟩))
23	22	3ad2ant1 1130	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((reverse‘(reverse‘(𝑊 prefix 𝐿))) substr ⟨((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿 − 𝐹)), (♯‘(reverse‘(𝑊 prefix 𝐿)))⟩) = ((𝑊 prefix 𝐿) substr ⟨((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿 − 𝐹)), (♯‘(reverse‘(𝑊 prefix 𝐿)))⟩))
24	11	oveq1d 7420	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿 − 𝐹)) = (𝐿 − (𝐿 − 𝐹)))
25	24	3adant2 1128	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿 − 𝐹)) = (𝐿 − (𝐿 − 𝐹)))
26		elfzel2 13505	. . . . . . . . 9 ⊢ (𝐹 ∈ (0...𝐿) → 𝐿 ∈ ℤ)
27	26	zcnd 12671	. . . . . . . 8 ⊢ (𝐹 ∈ (0...𝐿) → 𝐿 ∈ ℂ)
28		elfzelz 13507	. . . . . . . . 9 ⊢ (𝐹 ∈ (0...𝐿) → 𝐹 ∈ ℤ)
29	28	zcnd 12671	. . . . . . . 8 ⊢ (𝐹 ∈ (0...𝐿) → 𝐹 ∈ ℂ)
30	27, 29	nncand 11580	. . . . . . 7 ⊢ (𝐹 ∈ (0...𝐿) → (𝐿 − (𝐿 − 𝐹)) = 𝐹)
31	30	3ad2ant2 1131	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝐿 − (𝐿 − 𝐹)) = 𝐹)
32	25, 31	eqtrd 2766	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿 − 𝐹)) = 𝐹)
33	11	3adant2 1128	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(reverse‘(𝑊 prefix 𝐿))) = 𝐿)
34	32, 33	opeq12d 4876	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ⟨((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿 − 𝐹)), (♯‘(reverse‘(𝑊 prefix 𝐿)))⟩ = ⟨𝐹, 𝐿⟩)
35	34	oveq2d 7421	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((𝑊 prefix 𝐿) substr ⟨((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿 − 𝐹)), (♯‘(reverse‘(𝑊 prefix 𝐿)))⟩) = ((𝑊 prefix 𝐿) substr ⟨𝐹, 𝐿⟩))
36	19, 23, 35	3eqtrd 2770	. 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (reverse‘((reverse‘(𝑊 prefix 𝐿)) prefix (𝐿 − 𝐹))) = ((𝑊 prefix 𝐿) substr ⟨𝐹, 𝐿⟩))
37		elfzuz3 13504	. . . . . . . 8 ⊢ (𝐹 ∈ (0...𝐿) → 𝐿 ∈ (ℤ_≥‘𝐹))
38		eluzfz2 13515	. . . . . . . 8 ⊢ (𝐿 ∈ (ℤ_≥‘𝐹) → 𝐿 ∈ (𝐹...𝐿))
39	37, 38	syl 17	. . . . . . 7 ⊢ (𝐹 ∈ (0...𝐿) → 𝐿 ∈ (𝐹...𝐿))
40	39	ancli 548	. . . . . 6 ⊢ (𝐹 ∈ (0...𝐿) → (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (𝐹...𝐿)))
41	40	3ad2ant2 1131	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (𝐹...𝐿)))
42		swrdpfx 14663	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (𝐹...𝐿)) → ((𝑊 prefix 𝐿) substr ⟨𝐹, 𝐿⟩) = (𝑊 substr ⟨𝐹, 𝐿⟩)))
43	41, 42	syl5 34	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((𝑊 prefix 𝐿) substr ⟨𝐹, 𝐿⟩) = (𝑊 substr ⟨𝐹, 𝐿⟩)))
44	43	3adant2 1128	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((𝑊 prefix 𝐿) substr ⟨𝐹, 𝐿⟩) = (𝑊 substr ⟨𝐹, 𝐿⟩)))
45	44	pm2.43i 52	. 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((𝑊 prefix 𝐿) substr ⟨𝐹, 𝐿⟩) = (𝑊 substr ⟨𝐹, 𝐿⟩))
46	36, 45	eqtr2d 2767	1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝑊 substr ⟨𝐹, 𝐿⟩) = (reverse‘((reverse‘(𝑊 prefix 𝐿)) prefix (𝐿 − 𝐹))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⟨cop 4629 ‘cfv 6537 (class class class)co 7405 0cc0 11112 − cmin 11448 ℤ_≥cuz 12826 ...cfz 13490 ♯chash 14295 Word cword 14470 substr csubstr 14596 prefix cpfx 14626 reversecreverse 14714

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-hash 14296 df-word 14471 df-substr 14597 df-pfx 14627 df-reverse 14715

This theorem is referenced by: swrdwlk 34645

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