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Mirrors > Home > MPE Home > Th. List > wlklenvp1 | Structured version Visualization version GIF version |
Description: The number of vertices of a walk (in an undirected graph) is the number of its edges plus 1. (Contributed by Alexander van der Vekens, 29-Jun-2018.) (Revised by AV, 1-May-2021.) |
Ref | Expression |
---|---|
wlklenvp1 | ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝑃) = ((♯‘𝐹) + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wlkcl 26914 | . 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) | |
2 | eqid 2826 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
3 | 2 | wlkp 26915 | . 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) |
4 | ffz0hash 13521 | . 2 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (♯‘𝑃) = ((♯‘𝐹) + 1)) | |
5 | 1, 3, 4 | syl2anc 581 | 1 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝑃) = ((♯‘𝐹) + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 class class class wbr 4874 ⟶wf 6120 ‘cfv 6124 (class class class)co 6906 0cc0 10253 1c1 10254 + caddc 10256 ℕ0cn0 11619 ...cfz 12620 ♯chash 13411 Vtxcvtx 26295 Walkscwlks 26895 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-ifp 1092 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-map 8125 df-pm 8126 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-card 9079 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-n0 11620 df-z 11706 df-uz 11970 df-fz 12621 df-fzo 12762 df-hash 13412 df-word 13576 df-wlks 26898 |
This theorem is referenced by: wlklenvm1 26920 wlklenvclwlk 26953 wlkonwlk1l 26961 usgr2wlkspthlem2 27061 wlklnwwlkln1 27168 wlknwwlksnbij 27193 umgrwwlks2on 27287 elwwlks2 27296 elwspths2spth 27297 clwlkclwwlkf1lem2 27337 clwlkclwwlkf1lem2OLD 27338 clwlkclwwlkfOLD 27342 clwlkclwwlkf 27346 clwwlknonclwlknonf1o 27761 clwwlknonclwlknonf1oOLD 27762 dlwwlknondlwlknonf1olem1 27765 dlwwlknonclwlknonf1olem1OLD 27766 |
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