rusgrprop0 - Metamath Proof Explorer

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Theorem rusgrprop0 30148

Description: The properties of a k-regular simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.)

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

**Assertion**
Ref	Expression
rusgrprop0	⊢ (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)

**Proof of Theorem rusgrprop0**
Step	Hyp	Ref	Expression
1		rusgrprop 30143	. 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 30141	. . . 4 ⊢ (𝐺 RegGraph 𝐾 → (𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))
5	4	anim2i 629	. . 3 ⊢ ((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	sylibr 237	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾) → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))
8	1, 7	syl 18	1 ⊢ (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 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3077 class class class wbr 5103 ‘cfv 6538 ℕ₀^*cxnn0 12679 Vtxcvtx 29574 USGraphcusgr 29730 VtxDegcvtxdg 30046 RegGraph crgr 30136 RegUSGraph crusgr 30137

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-rex 3088 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-xp 5657 df-iota 6494 df-fv 6546 df-rgr 30138 df-rusgr 30139

This theorem is used by: frusgrnn0 30152 cusgrm1rusgr 30163 rusgrpropnb 30164 frgrreg 30995 frgrregord013 30996

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