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Mirrors > Home > MPE Home > Th. List > fusgrregdegfi | Structured version Visualization version GIF version |
Description: In a nonempty finite simple graph, the degree of each vertex is finite. (Contributed by Alexander van der Vekens, 6-Mar-2018.) (Revised by AV, 19-Dec-2020.) |
Ref | Expression |
---|---|
isrusgr0.v | ⊢ 𝑉 = (Vtx‘𝐺) |
isrusgr0.d | ⊢ 𝐷 = (VtxDeg‘𝐺) |
Ref | Expression |
---|---|
fusgrregdegfi | ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 → 𝐾 ∈ ℕ0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isrusgr0.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | vtxdgfusgr 27558 | . . 3 ⊢ (𝐺 ∈ FinUSGraph → ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0) |
3 | r19.26 3085 | . . . . . 6 ⊢ (∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0 ∧ (𝐷‘𝑣) = 𝐾) ↔ (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0 ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)) | |
4 | isrusgr0.d | . . . . . . . . . . . 12 ⊢ 𝐷 = (VtxDeg‘𝐺) | |
5 | 4 | fveq1i 6707 | . . . . . . . . . . 11 ⊢ (𝐷‘𝑣) = ((VtxDeg‘𝐺)‘𝑣) |
6 | 5 | eqeq1i 2739 | . . . . . . . . . 10 ⊢ ((𝐷‘𝑣) = 𝐾 ↔ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) |
7 | eleq1 2821 | . . . . . . . . . 10 ⊢ (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0 ↔ 𝐾 ∈ ℕ0)) | |
8 | 6, 7 | sylbi 220 | . . . . . . . . 9 ⊢ ((𝐷‘𝑣) = 𝐾 → (((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0 ↔ 𝐾 ∈ ℕ0)) |
9 | 8 | biimpac 482 | . . . . . . . 8 ⊢ ((((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0 ∧ (𝐷‘𝑣) = 𝐾) → 𝐾 ∈ ℕ0) |
10 | 9 | ralimi 3076 | . . . . . . 7 ⊢ (∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0 ∧ (𝐷‘𝑣) = 𝐾) → ∀𝑣 ∈ 𝑉 𝐾 ∈ ℕ0) |
11 | rspn0 4257 | . . . . . . 7 ⊢ (𝑉 ≠ ∅ → (∀𝑣 ∈ 𝑉 𝐾 ∈ ℕ0 → 𝐾 ∈ ℕ0)) | |
12 | 10, 11 | syl5com 31 | . . . . . 6 ⊢ (∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0 ∧ (𝐷‘𝑣) = 𝐾) → (𝑉 ≠ ∅ → 𝐾 ∈ ℕ0)) |
13 | 3, 12 | sylbir 238 | . . . . 5 ⊢ ((∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0 ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾) → (𝑉 ≠ ∅ → 𝐾 ∈ ℕ0)) |
14 | 13 | ex 416 | . . . 4 ⊢ (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0 → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 → (𝑉 ≠ ∅ → 𝐾 ∈ ℕ0))) |
15 | 14 | com23 86 | . . 3 ⊢ (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0 → (𝑉 ≠ ∅ → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 → 𝐾 ∈ ℕ0))) |
16 | 2, 15 | syl 17 | . 2 ⊢ (𝐺 ∈ FinUSGraph → (𝑉 ≠ ∅ → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 → 𝐾 ∈ ℕ0))) |
17 | 16 | imp 410 | 1 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 → 𝐾 ∈ ℕ0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 ∀wral 3054 ∅c0 4227 ‘cfv 6369 ℕ0cn0 12073 Vtxcvtx 27059 FinUSGraphcfusgr 27376 VtxDegcvtxdg 27525 |
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 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
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 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-2o 8192 df-oadd 8195 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-dju 9500 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-n0 12074 df-xnn0 12146 df-z 12160 df-uz 12422 df-xadd 12688 df-fz 13079 df-hash 13880 df-vtx 27061 df-iedg 27062 df-edg 27111 df-uhgr 27121 df-upgr 27145 df-umgr 27146 df-uspgr 27213 df-usgr 27214 df-fusgr 27377 df-vtxdg 27526 |
This theorem is referenced by: fusgrn0eqdrusgr 27630 frusgrnn0 27631 fusgreghash2wsp 28393 frrusgrord0lem 28394 |
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