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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 13561 | . . . . . . . 8 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → 𝑁 ∈ ℤ) | |
2 | 1 | zcnd 12721 | . . . . . . 7 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → 𝑁 ∈ ℂ) |
3 | 2cnd 12342 | . . . . . . 7 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → 2 ∈ ℂ) | |
4 | 2, 3 | npcand 11622 | . . . . . 6 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → ((𝑁 − 2) + 2) = 𝑁) |
5 | 4 | eqcomd 2741 | . . . . 5 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → 𝑁 = ((𝑁 − 2) + 2)) |
6 | 5 | opeq2d 4885 | . . . 4 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → 〈(𝑁 − 2), 𝑁〉 = 〈(𝑁 − 2), ((𝑁 − 2) + 2)〉) |
7 | 6 | oveq2d 7447 | . . 3 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → (𝑊 substr 〈(𝑁 − 2), 𝑁〉) = (𝑊 substr 〈(𝑁 − 2), ((𝑁 − 2) + 2)〉)) |
8 | 7 | adantl 481 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (2...(♯‘𝑊))) → (𝑊 substr 〈(𝑁 − 2), 𝑁〉) = (𝑊 substr 〈(𝑁 − 2), ((𝑁 − 2) + 2)〉)) |
9 | simpl 482 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (2...(♯‘𝑊))) → 𝑊 ∈ Word 𝑉) | |
10 | elfzuz 13557 | . . . . 5 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → 𝑁 ∈ (ℤ≥‘2)) | |
11 | uznn0sub 12915 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 − 2) ∈ ℕ0) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → (𝑁 − 2) ∈ ℕ0) |
13 | 12 | adantl 481 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (2...(♯‘𝑊))) → (𝑁 − 2) ∈ ℕ0) |
14 | 1cnd 11254 | . . . . . . 7 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → 1 ∈ ℂ) | |
15 | 2, 3, 14 | subsubd 11646 | . . . . . 6 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → (𝑁 − (2 − 1)) = ((𝑁 − 2) + 1)) |
16 | 2m1e1 12390 | . . . . . . 7 ⊢ (2 − 1) = 1 | |
17 | 16 | oveq2i 7442 | . . . . . 6 ⊢ (𝑁 − (2 − 1)) = (𝑁 − 1) |
18 | 15, 17 | eqtr3di 2790 | . . . . 5 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → ((𝑁 − 2) + 1) = (𝑁 − 1)) |
19 | 2eluzge1 12934 | . . . . . . . 8 ⊢ 2 ∈ (ℤ≥‘1) | |
20 | fzss1 13600 | . . . . . . . 8 ⊢ (2 ∈ (ℤ≥‘1) → (2...(♯‘𝑊)) ⊆ (1...(♯‘𝑊))) | |
21 | 19, 20 | ax-mp 5 | . . . . . . 7 ⊢ (2...(♯‘𝑊)) ⊆ (1...(♯‘𝑊)) |
22 | 21 | sseli 3991 | . . . . . 6 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → 𝑁 ∈ (1...(♯‘𝑊))) |
23 | fz1fzo0m1 13747 | . . . . . 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 2839 | . . . 4 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → ((𝑁 − 2) + 1) ∈ (0..^(♯‘𝑊))) |
26 | 25 | adantl 481 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (2...(♯‘𝑊))) → ((𝑁 − 2) + 1) ∈ (0..^(♯‘𝑊))) |
27 | swrds2 14976 | . . 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 1370 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (2...(♯‘𝑊))) → (𝑊 substr 〈(𝑁 − 2), ((𝑁 − 2) + 2)〉) = 〈“(𝑊‘(𝑁 − 2))(𝑊‘((𝑁 − 2) + 1))”〉) |
29 | eqidd 2736 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (2...(♯‘𝑊))) → (𝑊‘(𝑁 − 2)) = (𝑊‘(𝑁 − 2))) | |
30 | 18 | fveq2d 6911 | . . . 4 ⊢ (𝑁 ∈ (2...(♯‘𝑊)) → (𝑊‘((𝑁 − 2) + 1)) = (𝑊‘(𝑁 − 1))) |
31 | 30 | adantl 481 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (2...(♯‘𝑊))) → (𝑊‘((𝑁 − 2) + 1)) = (𝑊‘(𝑁 − 1))) |
32 | 29, 31 | s2eqd 14899 | . 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 2779 | 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 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 〈cop 4637 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 + caddc 11156 − cmin 11490 2c2 12319 ℕ0cn0 12524 ℤ≥cuz 12876 ...cfz 13544 ..^cfzo 13691 ♯chash 14366 Word cword 14549 substr csubstr 14675 〈“cs2 14877 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-concat 14606 df-s1 14631 df-substr 14676 df-s2 14884 |
This theorem is referenced by: 2clwwlk2clwwlklem 30375 |
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