Theorem swrdrevpfx 35431

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 13637	. . . . . 6 ⊢ (𝐹 ∈ (0...𝐿) → (𝐿 − 𝐹) ∈ (0...𝐿))
2		pfxcl 14688	. . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 prefix 𝐿) ∈ Word 𝑉)
3		revcl 14771	. . . . . . . . 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 1145	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿 − 𝐹) ∈ (0...𝐿)) → (reverse‘(𝑊 prefix 𝐿)) ∈ Word 𝑉)
6		simp3 1150	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿 − 𝐹) ∈ (0...𝐿)) → (𝐿 − 𝐹) ∈ (0...𝐿))
7		revlen 14772	. . . . . . . . . . . . 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 484	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(reverse‘(𝑊 prefix 𝐿))) = (♯‘(𝑊 prefix 𝐿)))
10		pfxlen 14694	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 prefix 𝐿)) = 𝐿)
11	9, 10	eqtrd 2796	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(reverse‘(𝑊 prefix 𝐿))) = 𝐿)
12	11	3adant3 1144	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿 − 𝐹) ∈ (0...𝐿)) → (♯‘(reverse‘(𝑊 prefix 𝐿))) = 𝐿)
13	12	oveq2d 7408	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿 − 𝐹) ∈ (0...𝐿)) → (0...(♯‘(reverse‘(𝑊 prefix 𝐿)))) = (0...𝐿))
14	6, 13	eleqtrrd 2864	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿 − 𝐹) ∈ (0...𝐿)) → (𝐿 − 𝐹) ∈ (0...(♯‘(reverse‘(𝑊 prefix 𝐿)))))
15	5, 14	jca 519	. . . . . 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 1177	. . . . 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 1138	. . . 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 35430	. . . 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 14777	. . . . . 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 7407	. . . 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 1145	. . 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 7407	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿 − 𝐹)) = (𝐿 − (𝐿 − 𝐹)))
25	24	3adant2 1143	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿 − 𝐹)) = (𝐿 − (𝐿 − 𝐹)))
26		elfzel2 13524	. . . . . . . . 9 ⊢ (𝐹 ∈ (0...𝐿) → 𝐿 ∈ ℤ)
27	26	zcnd 12675	. . . . . . . 8 ⊢ (𝐹 ∈ (0...𝐿) → 𝐿 ∈ ℂ)
28		elfzelz 13526	. . . . . . . . 9 ⊢ (𝐹 ∈ (0...𝐿) → 𝐹 ∈ ℤ)
29	28	zcnd 12675	. . . . . . . 8 ⊢ (𝐹 ∈ (0...𝐿) → 𝐹 ∈ ℂ)
30	27, 29	nncand 11544	. . . . . . 7 ⊢ (𝐹 ∈ (0...𝐿) → (𝐿 − (𝐿 − 𝐹)) = 𝐹)
31	30	3ad2ant2 1146	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝐿 − (𝐿 − 𝐹)) = 𝐹)
32	25, 31	eqtrd 2796	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿 − 𝐹)) = 𝐹)
33	11	3adant2 1143	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(reverse‘(𝑊 prefix 𝐿))) = 𝐿)
34	32, 33	opeq12d 4838	. . . 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 7408	. . 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 2800	. 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 13523	. . . . . . . 8 ⊢ (𝐹 ∈ (0...𝐿) → 𝐿 ∈ (ℤ≥‘𝐹))
38		eluzfz2 13534	. . . . . . . 8 ⊢ (𝐿 ∈ (ℤ≥‘𝐹) → 𝐿 ∈ (𝐹...𝐿))
39	37, 38	syl 17	. . . . . . 7 ⊢ (𝐹 ∈ (0...𝐿) → 𝐿 ∈ (𝐹...𝐿))
40	39	ancli 556	. . . . . 6 ⊢ (𝐹 ∈ (0...𝐿) → (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (𝐹...𝐿)))
41	40	3ad2ant2 1146	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (𝐹...𝐿)))
42		swrdpfx 14717	. . . . 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 1143	. . 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 2797	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 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ⟨cop 4587  ‘cfv 6517  (class class class)co 7392  0cc0 11070   − cmin 11411  ℤ_≥cuz 12836  ...cfz 13509  ♯chash 14340  Word cword 14523   substr csubstr 14651   prefix cpfx 14681  reversecreverse 14768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8673  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-card 9894  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-n0 12479  df-z 12566  df-uz 12837  df-fz 13510  df-fzo 13657  df-hash 14341  df-word 14524  df-substr 14652  df-pfx 14682  df-reverse 14769

This theorem is referenced by:  swrdwlk  35441