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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 11892 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
2 | ccatws1lenp1b 13963 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) → ((♯‘(𝑊 ++ 〈“(𝑊‘0)”〉)) = (𝑁 + 1) ↔ (♯‘𝑊) = 𝑁)) | |
3 | 1, 2 | sylan2 592 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → ((♯‘(𝑊 ++ 〈“(𝑊‘0)”〉)) = (𝑁 + 1) ↔ (♯‘𝑊) = 𝑁)) |
4 | 3 | anbi2d 628 | . . 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 483 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑊 ∈ Word 𝑉) | |
6 | eleq1 2897 | . . . . . . . . . 10 ⊢ ((♯‘𝑊) = 𝑁 → ((♯‘𝑊) ∈ ℕ ↔ 𝑁 ∈ ℕ)) | |
7 | len0nnbi 13891 | . . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 ≠ ∅ ↔ (♯‘𝑊) ∈ ℕ)) | |
8 | 7 | biimprcd 251 | . . . . . . . . . 10 ⊢ ((♯‘𝑊) ∈ ℕ → (𝑊 ∈ Word 𝑉 → 𝑊 ≠ ∅)) |
9 | 6, 8 | syl6bir 255 | . . . . . . . . 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 418 | . . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (♯‘𝑊) = 𝑁) → 𝑊 ≠ ∅) |
12 | clwwlkwwlksb.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
13 | 12 | clwwlkwwlksb 27760 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (𝑊 ∈ (ClWWalks‘𝐺) ↔ (𝑊 ++ 〈“(𝑊‘0)”〉) ∈ (WWalks‘𝐺))) |
14 | 5, 11, 13 | syl2an2r 681 | . . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (♯‘𝑊) = 𝑁) → (𝑊 ∈ (ClWWalks‘𝐺) ↔ (𝑊 ++ 〈“(𝑊‘0)”〉) ∈ (WWalks‘𝐺))) |
15 | 14 | bicomd 224 | . . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (♯‘𝑊) = 𝑁) → ((𝑊 ++ 〈“(𝑊‘0)”〉) ∈ (WWalks‘𝐺) ↔ 𝑊 ∈ (ClWWalks‘𝐺))) |
16 | 15 | ex 413 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → ((♯‘𝑊) = 𝑁 → ((𝑊 ++ 〈“(𝑊‘0)”〉) ∈ (WWalks‘𝐺) ↔ 𝑊 ∈ (ClWWalks‘𝐺)))) |
17 | 16 | pm5.32rd 578 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (((𝑊 ++ 〈“(𝑊‘0)”〉) ∈ (WWalks‘𝐺) ∧ (♯‘𝑊) = 𝑁) ↔ (𝑊 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑊) = 𝑁))) |
18 | 4, 17 | bitrd 280 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (((𝑊 ++ 〈“(𝑊‘0)”〉) ∈ (WWalks‘𝐺) ∧ (♯‘(𝑊 ++ 〈“(𝑊‘0)”〉)) = (𝑁 + 1)) ↔ (𝑊 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑊) = 𝑁))) |
19 | 1 | adantl 482 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0) |
20 | iswwlksn 27543 | . . 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 27732 | . . 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 313 | 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 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∅c0 4288 ‘cfv 6348 (class class class)co 7145 0cc0 10525 1c1 10526 + caddc 10528 ℕcn 11626 ℕ0cn0 11885 ♯chash 13678 Word cword 13849 ++ cconcat 13910 〈“cs1 13937 Vtxcvtx 26708 WWalkscwwlks 27530 WWalksN cwwlksn 27531 ClWWalkscclwwlk 27686 ClWWalksN cclwwlkn 27729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-hash 13679 df-word 13850 df-lsw 13903 df-concat 13911 df-s1 13938 df-wwlks 27535 df-wwlksn 27536 df-clwwlk 27687 df-clwwlkn 27730 |
This theorem is referenced by: clwwlknonwwlknonb 27812 |
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