clwlknon2num - Metamath Proof Explorer

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Theorem clwlknon2num 27911

Description: There are k walks of length 2 on each vertex 𝑋 in a k-regular simple graph. Variant of clwwlknon2num 27623, using the general definition of walks instead of walks as words. (Contributed by AV, 4-Jun-2022.)

**Hypothesis**
Ref	Expression
clwlknon2num.v	⊢ 𝑉 = (Vtx‘𝐺)

**Assertion**
Ref	Expression
clwlknon2num	⊢ ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾 ∧ 𝑋 ∈ 𝑉) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}) = 𝐾)

Distinct variable groups: 𝑤,𝐺 𝑤,𝑉 𝑤,𝑋 𝑤,𝐾

**Proof of Theorem clwlknon2num**
Step	Hyp	Ref	Expression
1		rusgrusgr 27039	. . . . . 6 ⊢ (𝐺RegUSGraph𝐾 → 𝐺 ∈ USGraph)
2		usgruspgr 26656	. . . . . 6 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
3	1, 2	syl 17	. . . . 5 ⊢ (𝐺RegUSGraph𝐾 → 𝐺 ∈ USPGraph)
4	3	3ad2ant2 1114	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾 ∧ 𝑋 ∈ 𝑉) → 𝐺 ∈ USPGraph)
5		clwlknon2num.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
6	5	eleq2i 2851	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 ↔ 𝑋 ∈ (Vtx‘𝐺))
7	6	biimpi 208	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ (Vtx‘𝐺))
8	7	3ad2ant3 1115	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (Vtx‘𝐺))
9		2nn 11506	. . . . 5 ⊢ 2 ∈ ℕ
10	9	a1i 11	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾 ∧ 𝑋 ∈ 𝑉) → 2 ∈ ℕ)
11		clwwlknonclwlknonen 27902	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ (Vtx‘𝐺) ∧ 2 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)2))
12	4, 8, 10, 11	syl3anc 1351	. . 3 ⊢ ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾 ∧ 𝑋 ∈ 𝑉) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)2))
13	1	anim2i 607	. . . . . . . . 9 ⊢ ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → (𝑉 ∈ Fin ∧ 𝐺 ∈ USGraph))
14	13	ancomd 454	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
15	5	isfusgr 26793	. . . . . . . 8 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
16	14, 15	sylibr 226	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → 𝐺 ∈ FinUSGraph)
17	16	3adant3 1112	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾 ∧ 𝑋 ∈ 𝑉) → 𝐺 ∈ FinUSGraph)
18		2nn0 11719	. . . . . . 7 ⊢ 2 ∈ ℕ₀
19	18	a1i 11	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾 ∧ 𝑋 ∈ 𝑉) → 2 ∈ ℕ₀)
20		wlksnfi 27397	. . . . . 6 ⊢ ((𝐺 ∈ FinUSGraph ∧ 2 ∈ ℕ₀) → {𝑤 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 2} ∈ Fin)
21	17, 19, 20	syl2anc 576	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾 ∧ 𝑋 ∈ 𝑉) → {𝑤 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 2} ∈ Fin)
22		clwlkswks 27255	. . . . . . 7 ⊢ (ClWalks‘𝐺) ⊆ (Walks‘𝐺)
23	22	a1i 11	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾 ∧ 𝑋 ∈ 𝑉) → (ClWalks‘𝐺) ⊆ (Walks‘𝐺))
24		simp2l 1179	. . . . . 6 ⊢ (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾 ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) ∧ 𝑤 ∈ (ClWalks‘𝐺)) → (♯‘(1^st ‘𝑤)) = 2)
25	23, 24	rabssrabd 3944	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾 ∧ 𝑋 ∈ 𝑉) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ⊆ {𝑤 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 2})
26	21, 25	ssfid 8528	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾 ∧ 𝑋 ∈ 𝑉) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ∈ Fin)
27	5	clwwlknonfin 27612	. . . . 5 ⊢ (𝑉 ∈ Fin → (𝑋(ClWWalksNOn‘𝐺)2) ∈ Fin)
28	27	3ad2ant1 1113	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑋(ClWWalksNOn‘𝐺)2) ∈ Fin)
29		hashen 13515	. . . 4 ⊢ (({𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ∈ Fin ∧ (𝑋(ClWWalksNOn‘𝐺)2) ∈ Fin) → ((♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}) = (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) ↔ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)2)))
30	26, 28, 29	syl2anc 576	. . 3 ⊢ ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾 ∧ 𝑋 ∈ 𝑉) → ((♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}) = (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) ↔ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)2)))
31	12, 30	mpbird 249	. 2 ⊢ ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾 ∧ 𝑋 ∈ 𝑉) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}) = (♯‘(𝑋(ClWWalksNOn‘𝐺)2)))
32	7	anim2i 607	. . . 4 ⊢ ((𝐺RegUSGraph𝐾 ∧ 𝑋 ∈ 𝑉) → (𝐺RegUSGraph𝐾 ∧ 𝑋 ∈ (Vtx‘𝐺)))
33	32	3adant1 1110	. . 3 ⊢ ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾 ∧ 𝑋 ∈ 𝑉) → (𝐺RegUSGraph𝐾 ∧ 𝑋 ∈ (Vtx‘𝐺)))
34		clwwlknon2num 27623	. . 3 ⊢ ((𝐺RegUSGraph𝐾 ∧ 𝑋 ∈ (Vtx‘𝐺)) → (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) = 𝐾)
35	33, 34	syl 17	. 2 ⊢ ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾 ∧ 𝑋 ∈ 𝑉) → (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) = 𝐾)
36	31, 35	eqtrd 2808	1 ⊢ ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾 ∧ 𝑋 ∈ 𝑉) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}) = 𝐾)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2048 {crab 3086 ⊆ wss 3825 class class class wbr 4923 ‘cfv 6182 (class class class)co 6970 1^st c1st 7492 2^nd c2nd 7493 ≈ cen 8295 Fincfn 8298 0cc0 10327 ℕcn 11431 2c2 11488 ℕ₀cn0 11700 ♯chash 13498 Vtxcvtx 26474 USPGraphcuspgr 26626 USGraphcusgr 26627 FinUSGraphcfusgr 26791 RegUSGraphcrusgr 27031 Walkscwlks 27071 ClWalkscclwlks 27249 ClWWalksNOncclwwlknon 27605

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-ifp 1044 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-2o 7898 df-oadd 7901 df-er 8081 df-map 8200 df-pm 8201 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-dju 9116 df-card 9154 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-n0 11701 df-xnn0 11773 df-z 11787 df-uz 12052 df-rp 12198 df-xadd 12318 df-fz 12702 df-fzo 12843 df-seq 13178 df-exp 13238 df-hash 13499 df-word 13663 df-lsw 13716 df-concat 13724 df-s1 13749 df-substr 13794 df-pfx 13843 df-edg 26526 df-uhgr 26536 df-ushgr 26537 df-upgr 26560 df-umgr 26561 df-uspgr 26628 df-usgr 26629 df-fusgr 26792 df-nbgr 26808 df-vtxdg 26941 df-rgr 27032 df-rusgr 27033 df-wlks 27074 df-clwlks 27250 df-wwlks 27306 df-wwlksn 27307 df-clwwlk 27478 df-clwwlkn 27530 df-clwwlknon 27606

This theorem is referenced by: numclwlk1lem1 27912

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