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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 | ⊢ ((𝐺 RegUSGraph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑃𝐿𝑁) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1133 | . . 3 ⊢ ((𝐺 RegUSGraph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑃 ∈ 𝑉) | |
2 | nnnn0 11905 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
3 | 2 | 3ad2ant3 1131 | . . 3 ⊢ ((𝐺 RegUSGraph 0 ∧ 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 27749 | . . 3 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃})) |
7 | 1, 3, 6 | syl2anc 586 | . 2 ⊢ ((𝐺 RegUSGraph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑃𝐿𝑁) = (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃})) |
8 | rusgrusgr 27346 | . . . . . . . . . 10 ⊢ (𝐺 RegUSGraph 0 → 𝐺 ∈ USGraph) | |
9 | usgr0edg0rusgr 27357 | . . . . . . . . . . 11 ⊢ (𝐺 ∈ USGraph → (𝐺 RegUSGraph 0 ↔ (Edg‘𝐺) = ∅)) | |
10 | 9 | biimpcd 251 | . . . . . . . . . 10 ⊢ (𝐺 RegUSGraph 0 → (𝐺 ∈ USGraph → (Edg‘𝐺) = ∅)) |
11 | 8, 10 | mpd 15 | . . . . . . . . 9 ⊢ (𝐺 RegUSGraph 0 → (Edg‘𝐺) = ∅) |
12 | 0enwwlksnge1 27642 | . . . . . . . . 9 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝑁 ∈ ℕ) → (𝑁 WWalksN 𝐺) = ∅) | |
13 | 11, 12 | sylan 582 | . . . . . . . 8 ⊢ ((𝐺 RegUSGraph 0 ∧ 𝑁 ∈ ℕ) → (𝑁 WWalksN 𝐺) = ∅) |
14 | eleq2 2901 | . . . . . . . . 9 ⊢ ((𝑁 WWalksN 𝐺) = ∅ → (𝑤 ∈ (𝑁 WWalksN 𝐺) ↔ 𝑤 ∈ ∅)) | |
15 | noel 4296 | . . . . . . . . . 10 ⊢ ¬ 𝑤 ∈ ∅ | |
16 | 15 | pm2.21i 119 | . . . . . . . . 9 ⊢ (𝑤 ∈ ∅ → ¬ (𝑤‘0) = 𝑃) |
17 | 14, 16 | syl6bi 255 | . . . . . . . 8 ⊢ ((𝑁 WWalksN 𝐺) = ∅ → (𝑤 ∈ (𝑁 WWalksN 𝐺) → ¬ (𝑤‘0) = 𝑃)) |
18 | 13, 17 | syl 17 | . . . . . . 7 ⊢ ((𝐺 RegUSGraph 0 ∧ 𝑁 ∈ ℕ) → (𝑤 ∈ (𝑁 WWalksN 𝐺) → ¬ (𝑤‘0) = 𝑃)) |
19 | 18 | 3adant2 1127 | . . . . . 6 ⊢ ((𝐺 RegUSGraph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑤 ∈ (𝑁 WWalksN 𝐺) → ¬ (𝑤‘0) = 𝑃)) |
20 | 19 | ralrimiv 3181 | . . . . 5 ⊢ ((𝐺 RegUSGraph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ∀𝑤 ∈ (𝑁 WWalksN 𝐺) ¬ (𝑤‘0) = 𝑃) |
21 | rabeq0 4338 | . . . . 5 ⊢ ({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} = ∅ ↔ ∀𝑤 ∈ (𝑁 WWalksN 𝐺) ¬ (𝑤‘0) = 𝑃) | |
22 | 20, 21 | sylibr 236 | . . . 4 ⊢ ((𝐺 RegUSGraph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} = ∅) |
23 | 22 | fveq2d 6674 | . . 3 ⊢ ((𝐺 RegUSGraph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (♯‘∅)) |
24 | hash0 13729 | . . 3 ⊢ (♯‘∅) = 0 | |
25 | 23, 24 | syl6eq 2872 | . 2 ⊢ ((𝐺 RegUSGraph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = 0) |
26 | 7, 25 | eqtrd 2856 | 1 ⊢ ((𝐺 RegUSGraph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑃𝐿𝑁) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 {crab 3142 ∅c0 4291 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 0cc0 10537 ℕcn 11638 ℕ0cn0 11898 ♯chash 13691 Vtxcvtx 26781 Edgcedg 26832 USGraphcusgr 26934 RegUSGraph crusgr 27338 WWalksN cwwlksn 27604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-xadd 12509 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-edg 26833 df-uhgr 26843 df-upgr 26867 df-uspgr 26935 df-usgr 26936 df-vtxdg 27248 df-rgr 27339 df-rusgr 27340 df-wwlks 27608 df-wwlksn 27609 |
This theorem is referenced by: (None) |
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