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Mirrors > Home > MPE Home > Th. List > swrds2m | Structured version Visualization version GIF version |
Description: Extract two adjacent symbols from a word in reverse direction. (Contributed by AV, 11-May-2022.) |
Ref | Expression |
---|---|
swrds2m | ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (2...(♯‘𝑊))) → (𝑊 substr ⟨(𝑁 − 2), 𝑁⟩) = ⟨“(𝑊‘(𝑁 − 2))(𝑊‘(𝑁 − 1))”⟩) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzelz 13498 | . . . . . . . 8 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → 𝑁 ∈ ℤ) | |
2 | 1 | zcnd 12664 | . . . . . . 7 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → 𝑁 ∈ ℂ) |
3 | 2cnd 12287 | . . . . . . 7 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → 2 ∈ ℂ) | |
4 | 2, 3 | npcand 11572 | . . . . . 6 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → ((𝑁 − 2) + 2) = 𝑁) |
5 | 4 | eqcomd 2739 | . . . . 5 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → 𝑁 = ((𝑁 − 2) + 2)) |
6 | 5 | opeq2d 4880 | . . . 4 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → ⟨(𝑁 − 2), 𝑁⟩ = ⟨(𝑁 − 2), ((𝑁 − 2) + 2)⟩) |
7 | 6 | oveq2d 7422 | . . 3 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → (𝑊 substr ⟨(𝑁 − 2), 𝑁⟩) = (𝑊 substr ⟨(𝑁 − 2), ((𝑁 − 2) + 2)⟩)) |
8 | 7 | adantl 483 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (2...(♯‘𝑊))) → (𝑊 substr ⟨(𝑁 − 2), 𝑁⟩) = (𝑊 substr ⟨(𝑁 − 2), ((𝑁 − 2) + 2)⟩)) |
9 | simpl 484 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (2...(♯‘𝑊))) → 𝑊 ∈ Word 𝑉) | |
10 | elfzuz 13494 | . . . . 5 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → 𝑁 ∈ (ℤ≥‘2)) | |
11 | uznn0sub 12858 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 − 2) ∈ ℕ0) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → (𝑁 − 2) ∈ ℕ0) |
13 | 12 | adantl 483 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (2...(♯‘𝑊))) → (𝑁 − 2) ∈ ℕ0) |
14 | 1cnd 11206 | . . . . . . 7 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → 1 ∈ ℂ) | |
15 | 2, 3, 14 | subsubd 11596 | . . . . . 6 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → (𝑁 − (2 − 1)) = ((𝑁 − 2) + 1)) |
16 | 2m1e1 12335 | . . . . . . 7 ⊢ (2 − 1) = 1 | |
17 | 16 | oveq2i 7417 | . . . . . 6 ⊢ (𝑁 − (2 − 1)) = (𝑁 − 1) |
18 | 15, 17 | eqtr3di 2788 | . . . . 5 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → ((𝑁 − 2) + 1) = (𝑁 − 1)) |
19 | 2eluzge1 12875 | . . . . . . . 8 ⊢ 2 ∈ (ℤ≥‘1) | |
20 | fzss1 13537 | . . . . . . . 8 ⊢ (2 ∈ (ℤ≥‘1) → (2...(♯‘𝑊)) ⊆ (1...(♯‘𝑊))) | |
21 | 19, 20 | ax-mp 5 | . . . . . . 7 ⊢ (2...(♯‘𝑊)) ⊆ (1...(♯‘𝑊)) |
22 | 21 | sseli 3978 | . . . . . 6 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → 𝑁 ∈ (1...(♯‘𝑊))) |
23 | fz1fzo0m1 13677 | . . . . . 6 ⊢ (𝑁 ∈ (1...(♯‘𝑊)) → (𝑁 − 1) ∈ (0..^(♯‘𝑊))) | |
24 | 22, 23 | syl 17 | . . . . 5 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → (𝑁 − 1) ∈ (0..^(♯‘𝑊))) |
25 | 18, 24 | eqeltrd 2834 | . . . 4 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → ((𝑁 − 2) + 1) ∈ (0..^(♯‘𝑊))) |
26 | 25 | adantl 483 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (2...(♯‘𝑊))) → ((𝑁 − 2) + 1) ∈ (0..^(♯‘𝑊))) |
27 | swrds2 14888 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑁 − 2) ∈ ℕ0 ∧ ((𝑁 − 2) + 1) ∈ (0..^(♯‘𝑊))) → (𝑊 substr ⟨(𝑁 − 2), ((𝑁 − 2) + 2)⟩) = ⟨“(𝑊‘(𝑁 − 2))(𝑊‘((𝑁 − 2) + 1))”⟩) | |
28 | 9, 13, 26, 27 | syl3anc 1372 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (2...(♯‘𝑊))) → (𝑊 substr ⟨(𝑁 − 2), ((𝑁 − 2) + 2)⟩) = ⟨“(𝑊‘(𝑁 − 2))(𝑊‘((𝑁 − 2) + 1))”⟩) |
29 | eqidd 2734 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (2...(♯‘𝑊))) → (𝑊‘(𝑁 − 2)) = (𝑊‘(𝑁 − 2))) | |
30 | 18 | fveq2d 6893 | . . . 4 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → (𝑊‘((𝑁 − 2) + 1)) = (𝑊‘(𝑁 − 1))) |
31 | 30 | adantl 483 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (2...(♯‘𝑊))) → (𝑊‘((𝑁 − 2) + 1)) = (𝑊‘(𝑁 − 1))) |
32 | 29, 31 | s2eqd 14811 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (2...(♯‘𝑊))) → ⟨“(𝑊‘(𝑁 − 2))(𝑊‘((𝑁 − 2) + 1))”⟩ = ⟨“(𝑊‘(𝑁 − 2))(𝑊‘(𝑁 − 1))”⟩) |
33 | 8, 28, 32 | 3eqtrd 2777 | 1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (2...(♯‘𝑊))) → (𝑊 substr ⟨(𝑁 − 2), 𝑁⟩) = ⟨“(𝑊‘(𝑁 − 2))(𝑊‘(𝑁 − 1))”⟩) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3948 ⟨cop 4634 ‘cfv 6541 (class class class)co 7406 0cc0 11107 1c1 11108 + caddc 11110 − cmin 11441 2c2 12264 ℕ0cn0 12469 ℤ≥cuz 12819 ...cfz 13481 ..^cfzo 13624 ♯chash 14287 Word cword 14461 substr csubstr 14587 ⟨“cs2 14789 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-fzo 13625 df-hash 14288 df-word 14462 df-concat 14518 df-s1 14543 df-substr 14588 df-s2 14796 |
This theorem is referenced by: 2clwwlk2clwwlklem 29589 |
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