Theorem clwlknon2num 29312

Description: There are k walks of length 2 on each vertex 𝑋 in a k-regular simple graph. Variant of clwwlknon2num 29049, 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 28512	. . . . . 6 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)
2		usgruspgr 28129	. . . . . 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 2829	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 ↔ 𝑋 ∈ (Vtx‘𝐺))
7	6	biimpi 215	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ (Vtx‘𝐺))
8	7	3ad2ant3 1135	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (Vtx‘𝐺))
9		2nn 12226	. . . . 5 ⊢ 2 ∈ ℕ
10	9	a1i 11	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → 2 ∈ ℕ)
11		clwwlknonclwlknonen 29307	. . . 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 1371	. . 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 462	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
15	5	isfusgr 28266	. . . . . . . 8 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
16	14, 15	sylibr 233	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph)
17	16	3adant3 1132	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → 𝐺 ∈ FinUSGraph)
18		2nn0 12430	. . . . . . 7 ⊢ 2 ∈ ℕ0
19	18	a1i 11	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → 2 ∈ ℕ0)
20		wlksnfi 28852	. . . . . 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 28724	. . . . . . 7 ⊢ (ClWalks‘𝐺) ⊆ (Walks‘𝐺)
23	22	a1i 11	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → (ClWalks‘𝐺) ⊆ (Walks‘𝐺))
24		simp2l 1199	. . . . . 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 4041	. . . . 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 9211	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} ∈ Fin)
27	5	clwwlknonfin 29038	. . . . 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 14247	. . . 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 256	. 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 29049	. . 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 2776	1 ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 2 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}) = 𝐾)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {crab 3407   ⊆ wss 3910   class class class wbr 5105  ‘cfv 6496  (class class class)co 7357  1^st c1st 7919  2^nd c2nd 7920   ≈ cen 8880  Fincfn 8883  0cc0 11051  ℕcn 12153  2c2 12208  ℕ₀cn0 12413  ♯chash 14230  Vtxcvtx 27947  USPGraphcuspgr 28099  USGraphcusgr 28100  FinUSGraphcfusgr 28264   RegUSGraph crusgr 28504  Walkscwlks 28544  ClWalkscclwlks 28718  ClWWalksNOncclwwlknon 29031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-rp 12916  df-xadd 13034  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-word 14403  df-lsw 14451  df-concat 14459  df-s1 14484  df-substr 14529  df-pfx 14559  df-edg 27999  df-uhgr 28009  df-ushgr 28010  df-upgr 28033  df-umgr 28034  df-uspgr 28101  df-usgr 28102  df-fusgr 28265  df-nbgr 28281  df-vtxdg 28414  df-rgr 28505  df-rusgr 28506  df-wlks 28547  df-clwlks 28719  df-wwlks 28775  df-wwlksn 28776  df-clwwlk 28926  df-clwwlkn 28969  df-clwwlknon 29032

This theorem is referenced by:  numclwlk1lem1  29313