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Theorem rusgrpropnb 29627

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 2737	. . 3 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
3	1, 2	rusgrprop0 29611	. 2 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
4		simp1 1137	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → 𝐺 ∈ USGraph)
5		simp2 1138	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → 𝐾 ∈ ℕ₀^*)
6	1	hashnbusgrvd 29572	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑣 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣))
7	6	adantlr 715	. . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^*) ∧ 𝑣 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣))
8		eqeq2 2749	. . . . . . 7 ⊢ (𝐾 = ((VtxDeg‘𝐺)‘𝑣) → ((♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾 ↔ (♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣)))
9	8	eqcoms 2745	. . . . . 6 ⊢ (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → ((♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾 ↔ (♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣)))
10	7, 9	syl5ibrcom 247	. . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^*) ∧ 𝑣 ∈ 𝑉) → (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾))
11	10	ralimdva 3167	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^*) → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → ∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾))
12	11	3impia 1118	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → ∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾)
13	4, 5, 12	3jca 1129	. 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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1539 ∈ wcel 2108 ∀wral 3061 class class class wbr 5151 ‘cfv 6569 (class class class)co 7438 ℕ₀^*cxnn0 12606 ♯chash 14375 Vtxcvtx 29039 USGraphcusgr 29192 NeighbVtx cnbgr 29375 VtxDegcvtxdg 29509 RegUSGraph crusgr 29600

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5288 ax-sep 5305 ax-nul 5315 ax-pow 5374 ax-pr 5441 ax-un 7761 ax-cnex 11218 ax-resscn 11219 ax-1cn 11220 ax-icn 11221 ax-addcl 11222 ax-addrcl 11223 ax-mulcl 11224 ax-mulrcl 11225 ax-mulcom 11226 ax-addass 11227 ax-mulass 11228 ax-distr 11229 ax-i2m1 11230 ax-1ne0 11231 ax-1rid 11232 ax-rnegex 11233 ax-rrecex 11234 ax-cnre 11235 ax-pre-lttri 11236 ax-pre-lttrn 11237 ax-pre-ltadd 11238 ax-pre-mulgt0 11239

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3483 df-sbc 3795 df-csb 3912 df-dif 3969 df-un 3971 df-in 3973 df-ss 3983 df-pss 3986 df-nul 4343 df-if 4535 df-pw 4610 df-sn 4635 df-pr 4637 df-op 4641 df-uni 4916 df-int 4955 df-iun 5001 df-br 5152 df-opab 5214 df-mpt 5235 df-tr 5269 df-id 5587 df-eprel 5593 df-po 5601 df-so 5602 df-fr 5645 df-we 5647 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-rn 5704 df-res 5705 df-ima 5706 df-pred 6329 df-ord 6395 df-on 6396 df-lim 6397 df-suc 6398 df-iota 6522 df-fun 6571 df-fn 6572 df-f 6573 df-f1 6574 df-fo 6575 df-f1o 6576 df-fv 6577 df-riota 7395 df-ov 7441 df-oprab 7442 df-mpo 7443 df-om 7895 df-1st 8022 df-2nd 8023 df-frecs 8314 df-wrecs 8345 df-recs 8419 df-rdg 8458 df-1o 8514 df-2o 8515 df-oadd 8518 df-er 8753 df-en 8994 df-dom 8995 df-sdom 8996 df-fin 8997 df-dju 9948 df-card 9986 df-pnf 11304 df-mnf 11305 df-xr 11306 df-ltxr 11307 df-le 11308 df-sub 11501 df-neg 11502 df-nn 12274 df-2 12336 df-n0 12534 df-xnn0 12607 df-z 12621 df-uz 12886 df-xadd 13162 df-fz 13554 df-hash 14376 df-edg 29091 df-uhgr 29101 df-ushgr 29102 df-upgr 29125 df-umgr 29126 df-uspgr 29193 df-usgr 29194 df-nbgr 29376 df-vtxdg 29510 df-rgr 29601 df-rusgr 29602

This theorem is referenced by: rusgrpropedg 29628 rusgrpropadjvtx 29629 rusgr1vtx 29632 numclwwlk1 30406

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