rusgrprop0 - Metamath Proof Explorer

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Theorem rusgrprop0 29654

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 29649	. 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 29647	. . . 4 ⊢ (𝐺 RegGraph 𝐾 → (𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))
5	4	anim2i 623	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾) → (𝐺 ∈ USGraph ∧ (𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))
6		3anass 1100	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾) ↔ (𝐺 ∈ USGraph ∧ (𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))
7	5, 6	sylibr 235	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾) → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))
8	1, 7	syl 17	1 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ∀wral 3053 class class class wbr 5072 ‘cfv 6485 ℕ₀^*cxnn0 12501 Vtxcvtx 29083 USGraphcusgr 29236 VtxDegcvtxdg 29552 RegGraph crgr 29642 RegUSGraph crusgr 29643

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-ext 2711 ax-sep 5218 ax-pr 5362

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-sb 2074 df-clab 2718 df-cleq 2731 df-clel 2814 df-ral 3054 df-rex 3064 df-rab 3392 df-v 3433 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-br 5073 df-opab 5135 df-xp 5624 df-iota 6441 df-fv 6493 df-rgr 29644 df-rusgr 29645

This theorem is referenced by: frusgrnn0 29658 cusgrm1rusgr 29669 rusgrpropnb 29670 frgrreg 30482 frgrregord013 30483

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