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Theorem rusgrpropadjvtx 29570

Description: The properties of a k-regular simple graph expressed with adjacent vertices. (Contributed by Alexander van der Vekens, 26-Jul-2018.) (Revised by AV, 27-Dec-2020.)

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

**Assertion**
Ref	Expression
rusgrpropadjvtx	⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (♯‘{𝑘 ∈ 𝑉 ∣ {𝑣, 𝑘} ∈ (Edg‘𝐺)}) = 𝐾))

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

**Proof of Theorem rusgrpropadjvtx**
Step	Hyp	Ref	Expression
1		rusgrpropnb.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	rusgrpropnb 29568	. 2 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾))
3		simp1 1136	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾) → 𝐺 ∈ USGraph)
4		simp2 1137	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾) → 𝐾 ∈ ℕ₀^*)
5		eqid 2736	. . . . . . . . . . . 12 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
6	1, 5	nbusgrvtx 29332	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ USGraph ∧ 𝑣 ∈ 𝑉) → (𝐺 NeighbVtx 𝑣) = {𝑘 ∈ 𝑉 ∣ {𝑣, 𝑘} ∈ (Edg‘𝐺)})
7	6	fveq2d 6885	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ 𝑣 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑣)) = (♯‘{𝑘 ∈ 𝑉 ∣ {𝑣, 𝑘} ∈ (Edg‘𝐺)}))
8	7	eqcomd 2742	. . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ 𝑣 ∈ 𝑉) → (♯‘{𝑘 ∈ 𝑉 ∣ {𝑣, 𝑘} ∈ (Edg‘𝐺)}) = (♯‘(𝐺 NeighbVtx 𝑣)))
9	8	adantr 480	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑣 ∈ 𝑉) ∧ (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾) → (♯‘{𝑘 ∈ 𝑉 ∣ {𝑣, 𝑘} ∈ (Edg‘𝐺)}) = (♯‘(𝐺 NeighbVtx 𝑣)))
10		simpr 484	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑣 ∈ 𝑉) ∧ (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾) → (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾)
11	9, 10	eqtrd 2771	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑣 ∈ 𝑉) ∧ (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾) → (♯‘{𝑘 ∈ 𝑉 ∣ {𝑣, 𝑘} ∈ (Edg‘𝐺)}) = 𝐾)
12	11	ex 412	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑣 ∈ 𝑉) → ((♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾 → (♯‘{𝑘 ∈ 𝑉 ∣ {𝑣, 𝑘} ∈ (Edg‘𝐺)}) = 𝐾))
13	12	ralimdva 3153	. . . . 5 ⊢ (𝐺 ∈ USGraph → (∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾 → ∀𝑣 ∈ 𝑉 (♯‘{𝑘 ∈ 𝑉 ∣ {𝑣, 𝑘} ∈ (Edg‘𝐺)}) = 𝐾))
14	13	imp 406	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ ∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾) → ∀𝑣 ∈ 𝑉 (♯‘{𝑘 ∈ 𝑉 ∣ {𝑣, 𝑘} ∈ (Edg‘𝐺)}) = 𝐾)
15	14	3adant2 1131	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾) → ∀𝑣 ∈ 𝑉 (♯‘{𝑘 ∈ 𝑉 ∣ {𝑣, 𝑘} ∈ (Edg‘𝐺)}) = 𝐾)
16	3, 4, 15	3jca 1128	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾) → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (♯‘{𝑘 ∈ 𝑉 ∣ {𝑣, 𝑘} ∈ (Edg‘𝐺)}) = 𝐾))
17	2, 16	syl 17	1 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 (♯‘{𝑘 ∈ 𝑉 ∣ {𝑣, 𝑘} ∈ (Edg‘𝐺)}) = 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3052 {crab 3420 {cpr 4608 class class class wbr 5124 ‘cfv 6536 (class class class)co 7410 ℕ₀^*cxnn0 12579 ♯chash 14353 Vtxcvtx 28980 Edgcedg 29031 USGraphcusgr 29133 NeighbVtx cnbgr 29316 RegUSGraph crusgr 29541

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-dju 9920 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-n0 12507 df-xnn0 12580 df-z 12594 df-uz 12858 df-xadd 13134 df-fz 13530 df-hash 14354 df-edg 29032 df-uhgr 29042 df-ushgr 29043 df-upgr 29066 df-umgr 29067 df-uspgr 29134 df-usgr 29135 df-nbgr 29317 df-vtxdg 29451 df-rgr 29542 df-rusgr 29543

This theorem is referenced by: rusgrnumwrdl2 29571 rusgrnumwwlks 29961

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