Theorem clwlknon2num 30354

Description: There are k walks of length 2 on each vertex 𝑋 in a k-regular simple graph. Variant of clwwlknon2num 30091, 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 ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘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 29549	. . . . . 6 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)
2		usgruspgr 29164	. . . . . 6 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
3	1, 2	syl 17	. . . . 5 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USPGraph)
4	3	3ad2ant2 1134	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → 𝐺 ∈ USPGraph)
5		clwlknon2num.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
6	5	eleq2i 2827	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 ↔ 𝑋 ∈ (Vtx‘𝐺))
7	6	biimpi 216	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ (Vtx‘𝐺))
8	7	3ad2ant3 1135	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (Vtx‘𝐺))
9		2nn 12318	. . . . 5 ⊢ 2 ∈ ℕ
10	9	a1i 11	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → 2 ∈ ℕ)
11		clwwlknonclwlknonen 30349	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ (Vtx‘𝐺) ∧ 2 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)2))
12	4, 8, 10, 11	syl3anc 1373	. . 3 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)2))
13	1	anim2i 617	. . . . . . . . 9 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (𝑉 ∈ Fin ∧ 𝐺 ∈ USGraph))
14	13	ancomd 461	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
15	5	isfusgr 29302	. . . . . . . 8 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
16	14, 15	sylibr 234	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph)
17	16	3adant3 1132	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → 𝐺 ∈ FinUSGraph)
18		2nn0 12523	. . . . . . 7 ⊢ 2 ∈ ℕ0
19	18	a1i 11	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → 2 ∈ ℕ0)
20		wlksnfi 29894	. . . . . 6 ⊢ ((𝐺 ∈ FinUSGraph ∧ 2 ∈ ℕ0) → {𝑤 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 2} ∈ Fin)
21	17, 19, 20	syl2anc 584	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → {𝑤 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 2} ∈ Fin)
22		clwlkswks 29763	. . . . . . 7 ⊢ (ClWalks‘𝐺) ⊆ (Walks‘𝐺)
23	22	a1i 11	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → (ClWalks‘𝐺) ⊆ (Walks‘𝐺))
24		simp2l 1200	. . . . . 6 ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋) ∧ 𝑤 ∈ (ClWalks‘𝐺)) → (♯‘(1st ‘𝑤)) = 2)
25	23, 24	rabssrabd 4063	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} ⊆ {𝑤 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 2})
26	21, 25	ssfid 9278	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} ∈ Fin)
27	5	clwwlknonfin 30080	. . . . 5 ⊢ (𝑉 ∈ Fin → (𝑋(ClWWalksNOn‘𝐺)2) ∈ Fin)
28	27	3ad2ant1 1133	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑋(ClWWalksNOn‘𝐺)2) ∈ Fin)
29		hashen 14370	. . . 4 ⊢ (({𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} ∈ Fin ∧ (𝑋(ClWWalksNOn‘𝐺)2) ∈ Fin) → ((♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}) = (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) ↔ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)2)))
30	26, 28, 29	syl2anc 584	. . 3 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → ((♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}) = (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) ↔ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)2)))
31	12, 30	mpbird 257	. 2 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}) = (♯‘(𝑋(ClWWalksNOn‘𝐺)2)))
32	7	anim2i 617	. . . 4 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ (Vtx‘𝐺)))
33	32	3adant1 1130	. . 3 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ (Vtx‘𝐺)))
34		clwwlknon2num 30091	. . 3 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ (Vtx‘𝐺)) → (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) = 𝐾)
35	33, 34	syl 17	. 2 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) = 𝐾)
36	31, 35	eqtrd 2771	1 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}) = 𝐾)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {crab 3420   ⊆ wss 3931   class class class wbr 5124  ‘cfv 6536  (class class class)co 7410  1^st c1st 7991  2^nd c2nd 7992   ≈ cen 8961  Fincfn 8964  0cc0 11134  ℕcn 12245  2c2 12300  ℕ₀cn0 12506  ♯chash 14353  Vtxcvtx 28980  USPGraphcuspgr 29132  USGraphcusgr 29133  FinUSGraphcfusgr 29300   RegUSGraph crusgr 29541  Walkscwlks 29581  ClWalkscclwlks 29757  ClWWalksNOncclwwlknon 30073

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-ifp 1063  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-map 8847  df-pm 8848  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-rp 13014  df-xadd 13134  df-fz 13530  df-fzo 13677  df-seq 14025  df-exp 14085  df-hash 14354  df-word 14537  df-lsw 14586  df-concat 14594  df-s1 14619  df-substr 14664  df-pfx 14694  df-edg 29032  df-uhgr 29042  df-ushgr 29043  df-upgr 29066  df-umgr 29067  df-uspgr 29134  df-usgr 29135  df-fusgr 29301  df-nbgr 29317  df-vtxdg 29451  df-rgr 29542  df-rusgr 29543  df-wlks 29584  df-clwlks 29758  df-wwlks 29817  df-wwlksn 29818  df-clwwlk 29968  df-clwwlkn 30011  df-clwwlknon 30074

This theorem is referenced by:  numclwlk1lem1  30355