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Mirrors > Home > MPE Home > Th. List > isrusgr0 | Structured version Visualization version GIF version |
Description: The property of being a k-regular simple graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.) (Revised by AV, 26-Dec-2020.) |
Ref | Expression |
---|---|
isrusgr0.v | ⊢ 𝑉 = (Vtx‘𝐺) |
isrusgr0.d | ⊢ 𝐷 = (VtxDeg‘𝐺) |
Ref | Expression |
---|---|
isrusgr0 | ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isrusgr 28507 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (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 | isrgr 28505 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))) |
5 | 4 | anbi2d 629 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → ((𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾) ↔ (𝐺 ∈ USGraph ∧ (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))) |
6 | 3anass 1095 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾) ↔ (𝐺 ∈ USGraph ∧ (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))) | |
7 | 5, 6 | bitr4di 288 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → ((𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾) ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))) |
8 | 1, 7 | bitrd 278 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3064 class class class wbr 5105 ‘cfv 6496 ℕ0*cxnn0 12484 Vtxcvtx 27945 USGraphcusgr 28098 VtxDegcvtxdg 28411 RegGraph crgr 28501 RegUSGraph crusgr 28502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3065 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-iota 6448 df-fv 6504 df-rgr 28503 df-rusgr 28504 |
This theorem is referenced by: usgreqdrusgr 28514 cusgrrusgr 28527 rgrusgrprc 28535 rusgrprc 28536 |
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