rusgrprop0 - Metamath Proof Explorer

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Theorem rusgrprop0 27036

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	⊢ (𝐺RegUSGraph𝐾 → (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 27031	. 2 ⊢ (𝐺RegUSGraph𝐾 → (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾))
2		isrusgr0.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
3		isrusgr0.d	. . . . 5 ⊢ 𝐷 = (VtxDeg‘𝐺)
4	2, 3	rgrprop 27029	. . . 4 ⊢ (𝐺RegGraph𝐾 → (𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))
5	4	anim2i 616	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾) → (𝐺 ∈ USGraph ∧ (𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))
6		3anass 1088	. . 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 ∧ 𝐺RegGraph𝐾) → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))
8	1, 7	syl 17	1 ⊢ (𝐺RegUSGraph𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1080 = wceq 1525 ∈ wcel 2083 ∀wral 3107 class class class wbr 4968 ‘cfv 6232 ℕ₀^*cxnn0 11821 Vtxcvtx 26468 USGraphcusgr 26621 VtxDegcvtxdg 26934 RegGraphcrgr 27024 RegUSGraphcrusgr 27025

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pr 5228

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-xp 5456 df-iota 6196 df-fv 6240 df-rgr 27026 df-rusgr 27027

This theorem is referenced by: frusgrnn0 27040 cusgrm1rusgr 27051 rusgrpropnb 27052 frgrreg 27861 frgrregord013 27862

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