![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rusgr0edg | Structured version Visualization version GIF version |
Description: Special case for graphs without edges: There are no walks of length greater than 0. (Contributed by Alexander van der Vekens, 26-Jul-2018.) (Revised by AV, 7-May-2021.) |
Ref | Expression |
---|---|
rusgrnumwwlk.v | ⊢ 𝑉 = (Vtx‘𝐺) |
rusgrnumwwlk.l | ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})) |
Ref | Expression |
---|---|
rusgr0edg | ⊢ ((𝐺RegUSGraph0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑃𝐿𝑁) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1130 | . . 3 ⊢ ((𝐺RegUSGraph0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑃 ∈ 𝑉) | |
2 | nnnn0 11752 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
3 | 2 | 3ad2ant3 1128 | . . 3 ⊢ ((𝐺RegUSGraph0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0) |
4 | rusgrnumwwlk.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
5 | rusgrnumwwlk.l | . . . 4 ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})) | |
6 | 4, 5 | rusgrnumwwlklem 27436 | . . 3 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃})) |
7 | 1, 3, 6 | syl2anc 584 | . 2 ⊢ ((𝐺RegUSGraph0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑃𝐿𝑁) = (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃})) |
8 | rusgrusgr 27029 | . . . . . . . . . 10 ⊢ (𝐺RegUSGraph0 → 𝐺 ∈ USGraph) | |
9 | usgr0edg0rusgr 27040 | . . . . . . . . . . 11 ⊢ (𝐺 ∈ USGraph → (𝐺RegUSGraph0 ↔ (Edg‘𝐺) = ∅)) | |
10 | 9 | biimpcd 250 | . . . . . . . . . 10 ⊢ (𝐺RegUSGraph0 → (𝐺 ∈ USGraph → (Edg‘𝐺) = ∅)) |
11 | 8, 10 | mpd 15 | . . . . . . . . 9 ⊢ (𝐺RegUSGraph0 → (Edg‘𝐺) = ∅) |
12 | 0enwwlksnge1 27329 | . . . . . . . . 9 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝑁 ∈ ℕ) → (𝑁 WWalksN 𝐺) = ∅) | |
13 | 11, 12 | sylan 580 | . . . . . . . 8 ⊢ ((𝐺RegUSGraph0 ∧ 𝑁 ∈ ℕ) → (𝑁 WWalksN 𝐺) = ∅) |
14 | eleq2 2871 | . . . . . . . . 9 ⊢ ((𝑁 WWalksN 𝐺) = ∅ → (𝑤 ∈ (𝑁 WWalksN 𝐺) ↔ 𝑤 ∈ ∅)) | |
15 | noel 4216 | . . . . . . . . . 10 ⊢ ¬ 𝑤 ∈ ∅ | |
16 | 15 | pm2.21i 119 | . . . . . . . . 9 ⊢ (𝑤 ∈ ∅ → ¬ (𝑤‘0) = 𝑃) |
17 | 14, 16 | syl6bi 254 | . . . . . . . 8 ⊢ ((𝑁 WWalksN 𝐺) = ∅ → (𝑤 ∈ (𝑁 WWalksN 𝐺) → ¬ (𝑤‘0) = 𝑃)) |
18 | 13, 17 | syl 17 | . . . . . . 7 ⊢ ((𝐺RegUSGraph0 ∧ 𝑁 ∈ ℕ) → (𝑤 ∈ (𝑁 WWalksN 𝐺) → ¬ (𝑤‘0) = 𝑃)) |
19 | 18 | 3adant2 1124 | . . . . . 6 ⊢ ((𝐺RegUSGraph0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑤 ∈ (𝑁 WWalksN 𝐺) → ¬ (𝑤‘0) = 𝑃)) |
20 | 19 | ralrimiv 3148 | . . . . 5 ⊢ ((𝐺RegUSGraph0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ∀𝑤 ∈ (𝑁 WWalksN 𝐺) ¬ (𝑤‘0) = 𝑃) |
21 | rabeq0 4258 | . . . . 5 ⊢ ({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} = ∅ ↔ ∀𝑤 ∈ (𝑁 WWalksN 𝐺) ¬ (𝑤‘0) = 𝑃) | |
22 | 20, 21 | sylibr 235 | . . . 4 ⊢ ((𝐺RegUSGraph0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} = ∅) |
23 | 22 | fveq2d 6542 | . . 3 ⊢ ((𝐺RegUSGraph0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (♯‘∅)) |
24 | hash0 13578 | . . 3 ⊢ (♯‘∅) = 0 | |
25 | 23, 24 | syl6eq 2847 | . 2 ⊢ ((𝐺RegUSGraph0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = 0) |
26 | 7, 25 | eqtrd 2831 | 1 ⊢ ((𝐺RegUSGraph0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑃𝐿𝑁) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ∀wral 3105 {crab 3109 ∅c0 4211 class class class wbr 4962 ‘cfv 6225 (class class class)co 7016 ∈ cmpo 7018 0cc0 10383 ℕcn 11486 ℕ0cn0 11745 ♯chash 13540 Vtxcvtx 26464 Edgcedg 26515 USGraphcusgr 26617 RegUSGraphcrusgr 27021 WWalksN cwwlksn 27291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-er 8139 df-map 8258 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-card 9214 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-2 11548 df-n0 11746 df-xnn0 11816 df-z 11830 df-uz 12094 df-xadd 12358 df-fz 12743 df-fzo 12884 df-hash 13541 df-word 13708 df-edg 26516 df-uhgr 26526 df-upgr 26550 df-uspgr 26618 df-usgr 26619 df-vtxdg 26931 df-rgr 27022 df-rusgr 27023 df-wwlks 27295 df-wwlksn 27296 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |