isrusgr0 - Metamath Proof Explorer

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Theorem isrusgr0 30129

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.)

**Hypotheses**
Ref	Expression
isrusgr0.v	⊢ 𝑉 = (Vtx‘𝐺)
isrusgr0.d	⊢ 𝐷 = (VtxDeg‘𝐺)

**Assertion**
Ref	Expression
isrusgr0	⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))

Distinct variable groups: 𝑣,𝐺 𝑣,𝐾 Allowed substitution hints: 𝐷(𝑣) 𝑉(𝑣) 𝑊(𝑣) 𝑍(𝑣)

**Proof of Theorem isrusgr0**
Step	Hyp	Ref	Expression
1		isrusgr 30124	. 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 30122	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))
5	4	anbi2d 642	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → ((𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾) ↔ (𝐺 ∈ USGraph ∧ (𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))))
6		3anass 1111	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾) ↔ (𝐺 ∈ USGraph ∧ (𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))
7	5, 6	bitr4di 292	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → ((𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾) ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))
8	1, 7	bitrd 282	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3077 class class class wbr 5103 ‘cfv 6531 ℕ₀^*cxnn0 12660 Vtxcvtx 29556 USGraphcusgr 29712 VtxDegcvtxdg 30028 RegGraph crgr 30118 RegUSGraph crusgr 30119

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-ext 2733 ax-sep 5249 ax-pr 5391

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-sb 2100 df-clab 2740 df-cleq 2753 df-clel 2836 df-ral 3078 df-rab 3414 df-v 3453 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-iota 6487 df-fv 6539 df-rgr 30120 df-rusgr 30121

This theorem is used by: usgreqdrusgr 30131 cusgrrusgr 30144 rgrusgrprc 30152 rusgrprc 30153

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