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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 27341 | . . 3 ⊢ RegGraph = {〈𝑔, 𝑘〉 ∣ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)} | |
2 | 1 | bropaex12 5644 | . 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 27343 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))) |
6 | 5 | biimpd 231 | . 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 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 class class class wbr 5068 ‘cfv 6357 ℕ0*cxnn0 11970 Vtxcvtx 26783 VtxDegcvtxdg 27249 RegGraph crgr 27339 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-iota 6316 df-fv 6365 df-rgr 27341 |
This theorem is referenced by: rusgrprop0 27351 uhgr0edg0rgrb 27358 frrusgrord 28122 |
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