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Mirrors > Home > MPE Home > Th. List > rusgrprop0 | Structured version Visualization version GIF version |
Description: The properties of a k-regular simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.) |
Ref | Expression |
---|---|
isrusgr0.v | ⊢ 𝑉 = (Vtx‘𝐺) |
isrusgr0.d | ⊢ 𝐷 = (VtxDeg‘𝐺) |
Ref | Expression |
---|---|
rusgrprop0 | ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rusgrprop 29595 | . 2 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)) | |
2 | isrusgr0.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | isrusgr0.d | . . . . 5 ⊢ 𝐷 = (VtxDeg‘𝐺) | |
4 | 2, 3 | rgrprop 29593 | . . . 4 ⊢ (𝐺 RegGraph 𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)) |
5 | 4 | anim2i 617 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾) → (𝐺 ∈ USGraph ∧ (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))) |
6 | 3anass 1094 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾) ↔ (𝐺 ∈ USGraph ∧ (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))) | |
7 | 5, 6 | sylibr 234 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾) → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)) |
8 | 1, 7 | syl 17 | 1 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 class class class wbr 5148 ‘cfv 6563 ℕ0*cxnn0 12597 Vtxcvtx 29028 USGraphcusgr 29181 VtxDegcvtxdg 29498 RegGraph crgr 29588 RegUSGraph crusgr 29589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-iota 6516 df-fv 6571 df-rgr 29590 df-rusgr 29591 |
This theorem is referenced by: frusgrnn0 29604 cusgrm1rusgr 29615 rusgrpropnb 29616 frgrreg 30423 frgrregord013 30424 |
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