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Theorem rusgrpropnb 29439

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

**Hypothesis**
Ref	Expression
rusgrpropnb.v	⊢ 𝑉 = (Vtx‘𝐺)

**Assertion**
Ref	Expression
rusgrpropnb	⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾))

Distinct variable groups: 𝑣,𝐺 𝑣,𝐾 Allowed substitution hint: 𝑉(𝑣)

**Proof of Theorem rusgrpropnb**
Step	Hyp	Ref	Expression
1		rusgrpropnb.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		eqid 2725	. . 3 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
3	1, 2	rusgrprop0 29423	. 2 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
4		simp1 1133	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → 𝐺 ∈ USGraph)
5		simp2 1134	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → 𝐾 ∈ ℕ₀^*)
6	1	hashnbusgrvd 29384	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑣 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣))
7	6	adantlr 713	. . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^*) ∧ 𝑣 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣))
8		eqeq2 2737	. . . . . . 7 ⊢ (𝐾 = ((VtxDeg‘𝐺)‘𝑣) → ((♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾 ↔ (♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣)))
9	8	eqcoms 2733	. . . . . 6 ⊢ (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → ((♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾 ↔ (♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣)))
10	7, 9	syl5ibrcom 246	. . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^*) ∧ 𝑣 ∈ 𝑉) → (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾))
11	10	ralimdva 3157	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^*) → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → ∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾))
12	11	3impia 1114	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → ∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾)
13	4, 5, 12	3jca 1125	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾))
14	3, 13	syl 17	1 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3051 class class class wbr 5143 ‘cfv 6542 (class class class)co 7415 ℕ₀^*cxnn0 12572 ♯chash 14319 Vtxcvtx 28851 USGraphcusgr 29004 NeighbVtx cnbgr 29187 VtxDegcvtxdg 29321 RegUSGraph crusgr 29412

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-oadd 8487 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-dju 9922 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-n0 12501 df-xnn0 12573 df-z 12587 df-uz 12851 df-xadd 13123 df-fz 13515 df-hash 14320 df-edg 28903 df-uhgr 28913 df-ushgr 28914 df-upgr 28937 df-umgr 28938 df-uspgr 29005 df-usgr 29006 df-nbgr 29188 df-vtxdg 29322 df-rgr 29413 df-rusgr 29414

This theorem is referenced by: rusgrpropedg 29440 rusgrpropadjvtx 29441 rusgr1vtx 29444 numclwwlk1 30213

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