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Theorem rusgrpropnb 29621

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 2740	. . 3 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
3	1, 2	rusgrprop0 29605	. 2 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
4		simp1 1136	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → 𝐺 ∈ USGraph)
5		simp2 1137	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → 𝐾 ∈ ℕ₀^*)
6	1	hashnbusgrvd 29566	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑣 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣))
7	6	adantlr 714	. . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^*) ∧ 𝑣 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣))
8		eqeq2 2752	. . . . . . 7 ⊢ (𝐾 = ((VtxDeg‘𝐺)‘𝑣) → ((♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾 ↔ (♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣)))
9	8	eqcoms 2748	. . . . . 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 3173	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^*) → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → ∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾))
12	11	3impia 1117	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → ∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾)
13	4, 5, 12	3jca 1128	. 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 1537 ∈ wcel 2108 ∀wral 3067 class class class wbr 5166 ‘cfv 6575 (class class class)co 7450 ℕ₀^*cxnn0 12627 ♯chash 14381 Vtxcvtx 29033 USGraphcusgr 29186 NeighbVtx cnbgr 29369 VtxDegcvtxdg 29503 RegUSGraph crusgr 29594

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-1st 8032 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-2o 8525 df-oadd 8528 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-dju 9972 df-card 10010 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-nn 12296 df-2 12358 df-n0 12556 df-xnn0 12628 df-z 12642 df-uz 12906 df-xadd 13178 df-fz 13570 df-hash 14382 df-edg 29085 df-uhgr 29095 df-ushgr 29096 df-upgr 29119 df-umgr 29120 df-uspgr 29187 df-usgr 29188 df-nbgr 29370 df-vtxdg 29504 df-rgr 29595 df-rusgr 29596

This theorem is referenced by: rusgrpropedg 29622 rusgrpropadjvtx 29623 rusgr1vtx 29626 numclwwlk1 30395

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