Theorem swrdrevpfx 35315

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 13580	. . . . . 6 ⊢ (𝐹 ∈ (0...𝐿) → (𝐿 − 𝐹) ∈ (0...𝐿))
2		pfxcl 14631	. . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 prefix 𝐿) ∈ Word 𝑉)
3		revcl 14714	. . . . . . . . 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 1134	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿 − 𝐹) ∈ (0...𝐿)) → (reverse‘(𝑊 prefix 𝐿)) ∈ Word 𝑉)
6		simp3 1139	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿 − 𝐹) ∈ (0...𝐿)) → (𝐿 − 𝐹) ∈ (0...𝐿))
7		revlen 14715	. . . . . . . . . . . . 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 14637	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 prefix 𝐿)) = 𝐿)
11	9, 10	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(reverse‘(𝑊 prefix 𝐿))) = 𝐿)
12	11	3adant3 1133	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿 − 𝐹) ∈ (0...𝐿)) → (♯‘(reverse‘(𝑊 prefix 𝐿))) = 𝐿)
13	12	oveq2d 7376	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿 − 𝐹) ∈ (0...𝐿)) → (0...(♯‘(reverse‘(𝑊 prefix 𝐿)))) = (0...𝐿))
14	6, 13	eleqtrrd 2840	. . . . . . 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 1166	. . . . 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 1127	. . . 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 35314	. . . 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 14720	. . . . . 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 7375	. . . 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 1134	. . 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 7375	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿 − 𝐹)) = (𝐿 − (𝐿 − 𝐹)))
25	24	3adant2 1132	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿 − 𝐹)) = (𝐿 − (𝐿 − 𝐹)))
26		elfzel2 13467	. . . . . . . . 9 ⊢ (𝐹 ∈ (0...𝐿) → 𝐿 ∈ ℤ)
27	26	zcnd 12625	. . . . . . . 8 ⊢ (𝐹 ∈ (0...𝐿) → 𝐿 ∈ ℂ)
28		elfzelz 13469	. . . . . . . . 9 ⊢ (𝐹 ∈ (0...𝐿) → 𝐹 ∈ ℤ)
29	28	zcnd 12625	. . . . . . . 8 ⊢ (𝐹 ∈ (0...𝐿) → 𝐹 ∈ ℂ)
30	27, 29	nncand 11501	. . . . . . 7 ⊢ (𝐹 ∈ (0...𝐿) → (𝐿 − (𝐿 − 𝐹)) = 𝐹)
31	30	3ad2ant2 1135	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝐿 − (𝐿 − 𝐹)) = 𝐹)
32	25, 31	eqtrd 2772	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿 − 𝐹)) = 𝐹)
33	11	3adant2 1132	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(reverse‘(𝑊 prefix 𝐿))) = 𝐿)
34	32, 33	opeq12d 4825	. . . 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 7376	. . 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 2776	. 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 13466	. . . . . . . 8 ⊢ (𝐹 ∈ (0...𝐿) → 𝐿 ∈ (ℤ≥‘𝐹))
38		eluzfz2 13477	. . . . . . . 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 1135	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (𝐹...𝐿)))
42		swrdpfx 14660	. . . . 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 1132	. . 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 2773	1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝑊 substr ⟨𝐹, 𝐿⟩) = (reverse‘((reverse‘(𝑊 prefix 𝐿)) prefix (𝐿 − 𝐹))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ⟨cop 4574  ‘cfv 6492  (class class class)co 7360  0cc0 11029   − cmin 11368  ℤ_≥cuz 12779  ...cfz 13452  ♯chash 14283  Word cword 14466   substr csubstr 14594   prefix cpfx 14624  reversecreverse 14711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-n0 12429  df-z 12516  df-uz 12780  df-fz 13453  df-fzo 13600  df-hash 14284  df-word 14467  df-substr 14595  df-pfx 14625  df-reverse 14712

This theorem is referenced by:  swrdwlk  35325