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Mirrors > Home > MPE Home > Th. List > clwwlknwwlksnb | Structured version Visualization version GIF version |
Description: A word over vertices represents a closed walk of a fixed length 𝑁 greater than zero iff the word concatenated with its first symbol represents a walk of length 𝑁. This theorem would not hold for 𝑁 = 0 and 𝑊 = ∅, because (𝑊 ++ 〈“(𝑊‘0)”〉) = 〈“∅”〉 ∈ (0 WWalksN 𝐺) could be true, but not 𝑊 ∈ (0 ClWWalksN 𝐺) ↔ ∅ ∈ ∅. (Contributed by AV, 4-Mar-2022.) (Proof shortened by AV, 22-Mar-2022.) |
Ref | Expression |
---|---|
clwwlkwwlksb.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
clwwlknwwlksnb | ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 ++ 〈“(𝑊‘0)”〉) ∈ (𝑁 WWalksN 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnnn0 12379 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
2 | ccatws1lenp1b 14463 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) → ((♯‘(𝑊 ++ 〈“(𝑊‘0)”〉)) = (𝑁 + 1) ↔ (♯‘𝑊) = 𝑁)) | |
3 | 1, 2 | sylan2 594 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → ((♯‘(𝑊 ++ 〈“(𝑊‘0)”〉)) = (𝑁 + 1) ↔ (♯‘𝑊) = 𝑁)) |
4 | 3 | anbi2d 630 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (((𝑊 ++ 〈“(𝑊‘0)”〉) ∈ (WWalks‘𝐺) ∧ (♯‘(𝑊 ++ 〈“(𝑊‘0)”〉)) = (𝑁 + 1)) ↔ ((𝑊 ++ 〈“(𝑊‘0)”〉) ∈ (WWalks‘𝐺) ∧ (♯‘𝑊) = 𝑁))) |
5 | simpl 484 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑊 ∈ Word 𝑉) | |
6 | eleq1 2826 | . . . . . . . . . 10 ⊢ ((♯‘𝑊) = 𝑁 → ((♯‘𝑊) ∈ ℕ ↔ 𝑁 ∈ ℕ)) | |
7 | len0nnbi 14393 | . . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 ≠ ∅ ↔ (♯‘𝑊) ∈ ℕ)) | |
8 | 7 | biimprcd 250 | . . . . . . . . . 10 ⊢ ((♯‘𝑊) ∈ ℕ → (𝑊 ∈ Word 𝑉 → 𝑊 ≠ ∅)) |
9 | 6, 8 | syl6bir 254 | . . . . . . . . 9 ⊢ ((♯‘𝑊) = 𝑁 → (𝑁 ∈ ℕ → (𝑊 ∈ Word 𝑉 → 𝑊 ≠ ∅))) |
10 | 9 | com13 88 | . . . . . . . 8 ⊢ (𝑊 ∈ Word 𝑉 → (𝑁 ∈ ℕ → ((♯‘𝑊) = 𝑁 → 𝑊 ≠ ∅))) |
11 | 10 | imp31 419 | . . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (♯‘𝑊) = 𝑁) → 𝑊 ≠ ∅) |
12 | clwwlkwwlksb.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
13 | 12 | clwwlkwwlksb 28827 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (𝑊 ∈ (ClWWalks‘𝐺) ↔ (𝑊 ++ 〈“(𝑊‘0)”〉) ∈ (WWalks‘𝐺))) |
14 | 5, 11, 13 | syl2an2r 684 | . . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (♯‘𝑊) = 𝑁) → (𝑊 ∈ (ClWWalks‘𝐺) ↔ (𝑊 ++ 〈“(𝑊‘0)”〉) ∈ (WWalks‘𝐺))) |
15 | 14 | bicomd 222 | . . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (♯‘𝑊) = 𝑁) → ((𝑊 ++ 〈“(𝑊‘0)”〉) ∈ (WWalks‘𝐺) ↔ 𝑊 ∈ (ClWWalks‘𝐺))) |
16 | 15 | ex 414 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → ((♯‘𝑊) = 𝑁 → ((𝑊 ++ 〈“(𝑊‘0)”〉) ∈ (WWalks‘𝐺) ↔ 𝑊 ∈ (ClWWalks‘𝐺)))) |
17 | 16 | pm5.32rd 579 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (((𝑊 ++ 〈“(𝑊‘0)”〉) ∈ (WWalks‘𝐺) ∧ (♯‘𝑊) = 𝑁) ↔ (𝑊 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑊) = 𝑁))) |
18 | 4, 17 | bitrd 279 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (((𝑊 ++ 〈“(𝑊‘0)”〉) ∈ (WWalks‘𝐺) ∧ (♯‘(𝑊 ++ 〈“(𝑊‘0)”〉)) = (𝑁 + 1)) ↔ (𝑊 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑊) = 𝑁))) |
19 | 1 | adantl 483 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0) |
20 | iswwlksn 28612 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑊 ++ 〈“(𝑊‘0)”〉) ∈ (𝑁 WWalksN 𝐺) ↔ ((𝑊 ++ 〈“(𝑊‘0)”〉) ∈ (WWalks‘𝐺) ∧ (♯‘(𝑊 ++ 〈“(𝑊‘0)”〉)) = (𝑁 + 1)))) | |
21 | 19, 20 | syl 17 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑊 ++ 〈“(𝑊‘0)”〉) ∈ (𝑁 WWalksN 𝐺) ↔ ((𝑊 ++ 〈“(𝑊‘0)”〉) ∈ (WWalks‘𝐺) ∧ (♯‘(𝑊 ++ 〈“(𝑊‘0)”〉)) = (𝑁 + 1)))) |
22 | isclwwlkn 28800 | . . 3 ⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑊) = 𝑁)) | |
23 | 22 | a1i 11 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑊) = 𝑁))) |
24 | 18, 21, 23 | 3bitr4rd 312 | 1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 ++ 〈“(𝑊‘0)”〉) ∈ (𝑁 WWalksN 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2942 ∅c0 4281 ‘cfv 6494 (class class class)co 7352 0cc0 11010 1c1 11011 + caddc 11013 ℕcn 12112 ℕ0cn0 12372 ♯chash 14184 Word cword 14356 ++ cconcat 14412 〈“cs1 14437 Vtxcvtx 27776 WWalkscwwlks 28599 WWalksN cwwlksn 28600 ClWWalkscclwwlk 28754 ClWWalksN cclwwlkn 28797 |
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 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-1st 7914 df-2nd 7915 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-1o 8405 df-oadd 8409 df-er 8607 df-map 8726 df-en 8843 df-dom 8844 df-sdom 8845 df-fin 8846 df-dju 9796 df-card 9834 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-nn 12113 df-2 12175 df-n0 12373 df-xnn0 12445 df-z 12459 df-uz 12723 df-fz 13380 df-fzo 13523 df-hash 14185 df-word 14357 df-lsw 14405 df-concat 14413 df-s1 14438 df-wwlks 28604 df-wwlksn 28605 df-clwwlk 28755 df-clwwlkn 28798 |
This theorem is referenced by: clwwlknonwwlknonb 28879 |
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