Theorem swrdrevpfx 35311

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 13551	. . . . . 6 ⊢ (𝐹 ∈ (0...𝐿) → (𝐿 − 𝐹) ∈ (0...𝐿))
2		pfxcl 14601	. . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 prefix 𝐿) ∈ Word 𝑉)
3		revcl 14684	. . . . . . . . 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 1133	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿 − 𝐹) ∈ (0...𝐿)) → (reverse‘(𝑊 prefix 𝐿)) ∈ Word 𝑉)
6		simp3 1138	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿 − 𝐹) ∈ (0...𝐿)) → (𝐿 − 𝐹) ∈ (0...𝐿))
7		revlen 14685	. . . . . . . . . . . . 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 14607	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 prefix 𝐿)) = 𝐿)
11	9, 10	eqtrd 2771	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(reverse‘(𝑊 prefix 𝐿))) = 𝐿)
12	11	3adant3 1132	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿 − 𝐹) ∈ (0...𝐿)) → (♯‘(reverse‘(𝑊 prefix 𝐿))) = 𝐿)
13	12	oveq2d 7374	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿 − 𝐹) ∈ (0...𝐿)) → (0...(♯‘(reverse‘(𝑊 prefix 𝐿)))) = (0...𝐿))
14	6, 13	eleqtrrd 2839	. . . . . . 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 1165	. . . . 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 1126	. . . 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 35310	. . . 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 14690	. . . . . 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 7373	. . . 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 1133	. . 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 7373	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿 − 𝐹)) = (𝐿 − (𝐿 − 𝐹)))
25	24	3adant2 1131	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿 − 𝐹)) = (𝐿 − (𝐿 − 𝐹)))
26		elfzel2 13438	. . . . . . . . 9 ⊢ (𝐹 ∈ (0...𝐿) → 𝐿 ∈ ℤ)
27	26	zcnd 12597	. . . . . . . 8 ⊢ (𝐹 ∈ (0...𝐿) → 𝐿 ∈ ℂ)
28		elfzelz 13440	. . . . . . . . 9 ⊢ (𝐹 ∈ (0...𝐿) → 𝐹 ∈ ℤ)
29	28	zcnd 12597	. . . . . . . 8 ⊢ (𝐹 ∈ (0...𝐿) → 𝐹 ∈ ℂ)
30	27, 29	nncand 11497	. . . . . . 7 ⊢ (𝐹 ∈ (0...𝐿) → (𝐿 − (𝐿 − 𝐹)) = 𝐹)
31	30	3ad2ant2 1134	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝐿 − (𝐿 − 𝐹)) = 𝐹)
32	25, 31	eqtrd 2771	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿 − 𝐹)) = 𝐹)
33	11	3adant2 1131	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(reverse‘(𝑊 prefix 𝐿))) = 𝐿)
34	32, 33	opeq12d 4837	. . . 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 7374	. . 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 2775	. 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 13437	. . . . . . . 8 ⊢ (𝐹 ∈ (0...𝐿) → 𝐿 ∈ (ℤ≥‘𝐹))
38		eluzfz2 13448	. . . . . . . 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 1134	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (𝐹...𝐿)))
42		swrdpfx 14630	. . . . 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 1131	. . 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 2772	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 1086   = wceq 1541   ∈ wcel 2113  ⟨cop 4586  ‘cfv 6492  (class class class)co 7358  0cc0 11026   − cmin 11364  ℤ_≥cuz 12751  ...cfz 13423  ♯chash 14253  Word cword 14436   substr csubstr 14564   prefix cpfx 14594  reversecreverse 14681

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  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 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-n0 12402  df-z 12489  df-uz 12752  df-fz 13424  df-fzo 13571  df-hash 14254  df-word 14437  df-substr 14565  df-pfx 14595  df-reverse 14682

This theorem is referenced by:  swrdwlk  35321