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Mirrors > Home > MPE Home > Th. List > rgrprop | Structured version Visualization version GIF version |
Description: The properties of a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.) |
Ref | Expression |
---|---|
isrgr.v | ⊢ 𝑉 = (Vtx‘𝐺) |
isrgr.d | ⊢ 𝐷 = (VtxDeg‘𝐺) |
Ref | Expression |
---|---|
rgrprop | ⊢ (𝐺 RegGraph 𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rgr 29590 | . . 3 ⊢ RegGraph = {〈𝑔, 𝑘〉 ∣ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)} | |
2 | 1 | bropaex12 5780 | . 2 ⊢ (𝐺 RegGraph 𝐾 → (𝐺 ∈ V ∧ 𝐾 ∈ V)) |
3 | isrgr.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
4 | isrgr.d | . . . 4 ⊢ 𝐷 = (VtxDeg‘𝐺) | |
5 | 3, 4 | isrgr 29592 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))) |
6 | 5 | biimpd 229 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺 RegGraph 𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))) |
7 | 2, 6 | mpcom 38 | 1 ⊢ (𝐺 RegGraph 𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 class class class wbr 5148 ‘cfv 6563 ℕ0*cxnn0 12597 Vtxcvtx 29028 VtxDegcvtxdg 29498 RegGraph crgr 29588 |
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 |
This theorem is referenced by: rusgrprop0 29600 uhgr0edg0rgrb 29607 frrusgrord 30370 |
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