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Mirrors > Home > MPE Home > Th. List > wwlknllvtx | Structured version Visualization version GIF version |
Description: If a word 𝑊 represents a walk of a fixed length 𝑁, then the first and the last symbol of the word is a vertex. (Contributed by AV, 14-Mar-2022.) |
Ref | Expression |
---|---|
wwlknllvtx.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
wwlknllvtx | ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑊‘0) ∈ 𝑉 ∧ (𝑊‘𝑁) ∈ 𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wwlknbp1 27952 | . . 3 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1))) | |
2 | wwlknvtx 27953 | . . 3 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → ∀𝑥 ∈ (0...𝑁)(𝑊‘𝑥) ∈ (Vtx‘𝐺)) | |
3 | 0elfz 13233 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁)) | |
4 | fveq2 6735 | . . . . . . . 8 ⊢ (𝑥 = 0 → (𝑊‘𝑥) = (𝑊‘0)) | |
5 | 4 | eleq1d 2823 | . . . . . . 7 ⊢ (𝑥 = 0 → ((𝑊‘𝑥) ∈ (Vtx‘𝐺) ↔ (𝑊‘0) ∈ (Vtx‘𝐺))) |
6 | 5 | adantl 485 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 = 0) → ((𝑊‘𝑥) ∈ (Vtx‘𝐺) ↔ (𝑊‘0) ∈ (Vtx‘𝐺))) |
7 | 3, 6 | rspcdv 3541 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (∀𝑥 ∈ (0...𝑁)(𝑊‘𝑥) ∈ (Vtx‘𝐺) → (𝑊‘0) ∈ (Vtx‘𝐺))) |
8 | nn0fz0 13234 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (0...𝑁)) | |
9 | 8 | biimpi 219 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (0...𝑁)) |
10 | fveq2 6735 | . . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑊‘𝑥) = (𝑊‘𝑁)) | |
11 | 10 | eleq1d 2823 | . . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝑊‘𝑥) ∈ (Vtx‘𝐺) ↔ (𝑊‘𝑁) ∈ (Vtx‘𝐺))) |
12 | 11 | adantl 485 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 = 𝑁) → ((𝑊‘𝑥) ∈ (Vtx‘𝐺) ↔ (𝑊‘𝑁) ∈ (Vtx‘𝐺))) |
13 | 9, 12 | rspcdv 3541 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (∀𝑥 ∈ (0...𝑁)(𝑊‘𝑥) ∈ (Vtx‘𝐺) → (𝑊‘𝑁) ∈ (Vtx‘𝐺))) |
14 | 7, 13 | jcad 516 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (∀𝑥 ∈ (0...𝑁)(𝑊‘𝑥) ∈ (Vtx‘𝐺) → ((𝑊‘0) ∈ (Vtx‘𝐺) ∧ (𝑊‘𝑁) ∈ (Vtx‘𝐺)))) |
15 | 14 | 3ad2ant1 1135 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1)) → (∀𝑥 ∈ (0...𝑁)(𝑊‘𝑥) ∈ (Vtx‘𝐺) → ((𝑊‘0) ∈ (Vtx‘𝐺) ∧ (𝑊‘𝑁) ∈ (Vtx‘𝐺)))) |
16 | 1, 2, 15 | sylc 65 | . 2 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑊‘0) ∈ (Vtx‘𝐺) ∧ (𝑊‘𝑁) ∈ (Vtx‘𝐺))) |
17 | wwlknllvtx.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
18 | 17 | eleq2i 2830 | . . 3 ⊢ ((𝑊‘0) ∈ 𝑉 ↔ (𝑊‘0) ∈ (Vtx‘𝐺)) |
19 | 17 | eleq2i 2830 | . . 3 ⊢ ((𝑊‘𝑁) ∈ 𝑉 ↔ (𝑊‘𝑁) ∈ (Vtx‘𝐺)) |
20 | 18, 19 | anbi12i 630 | . 2 ⊢ (((𝑊‘0) ∈ 𝑉 ∧ (𝑊‘𝑁) ∈ 𝑉) ↔ ((𝑊‘0) ∈ (Vtx‘𝐺) ∧ (𝑊‘𝑁) ∈ (Vtx‘𝐺))) |
21 | 16, 20 | sylibr 237 | 1 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑊‘0) ∈ 𝑉 ∧ (𝑊‘𝑁) ∈ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2111 ∀wral 3062 ‘cfv 6397 (class class class)co 7231 0cc0 10753 1c1 10754 + caddc 10756 ℕ0cn0 12114 ...cfz 13119 ♯chash 13920 Word cword 14093 Vtxcvtx 27111 WWalksN cwwlksn 27934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-cnex 10809 ax-resscn 10810 ax-1cn 10811 ax-icn 10812 ax-addcl 10813 ax-addrcl 10814 ax-mulcl 10815 ax-mulrcl 10816 ax-mulcom 10817 ax-addass 10818 ax-mulass 10819 ax-distr 10820 ax-i2m1 10821 ax-1ne0 10822 ax-1rid 10823 ax-rnegex 10824 ax-rrecex 10825 ax-cnre 10826 ax-pre-lttri 10827 ax-pre-lttrn 10828 ax-pre-ltadd 10829 ax-pre-mulgt0 10830 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-int 4874 df-iun 4920 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-pred 6175 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-riota 7188 df-ov 7234 df-oprab 7235 df-mpo 7236 df-om 7663 df-1st 7779 df-2nd 7780 df-wrecs 8067 df-recs 8128 df-rdg 8166 df-1o 8222 df-er 8411 df-map 8530 df-en 8647 df-dom 8648 df-sdom 8649 df-fin 8650 df-card 9579 df-pnf 10893 df-mnf 10894 df-xr 10895 df-ltxr 10896 df-le 10897 df-sub 11088 df-neg 11089 df-nn 11855 df-n0 12115 df-z 12201 df-uz 12463 df-fz 13120 df-fzo 13263 df-hash 13921 df-word 14094 df-wwlks 27938 df-wwlksn 27939 |
This theorem is referenced by: iswwlksnon 27961 |
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