Theorem clwwlknun 30474

Description: The set of closed walks of fixed length 𝑁 in a simple graph 𝐺 is the union of the closed walks of the fixed length 𝑁 on each of the vertices of graph 𝐺. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 3-Mar-2022.) (Proof shortened by AV, 28-Mar-2022.)

Hypothesis

Ref	Expression
clwwlknun.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
clwwlknun	⊢ (𝐺 ∈ USGraph → (𝑁 ClWWalksN 𝐺) = ∪ 𝑥 ∈ 𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁))

Distinct variable groups:   𝑥,𝐺   𝑥,𝑁   𝑥,𝑉

Proof of Theorem clwwlknun

Dummy variables 𝑦 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 4959	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁) ↔ ∃𝑥 ∈ 𝑉 𝑦 ∈ (𝑥(ClWWalksNOn‘𝐺)𝑁))
2		isclwwlknon 30453	. . . . 5 ⊢ (𝑦 ∈ (𝑥(ClWWalksNOn‘𝐺)𝑁) ↔ (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥))
3	2	rexbii 3111	. . . 4 ⊢ (∃𝑥 ∈ 𝑉 𝑦 ∈ (𝑥(ClWWalksNOn‘𝐺)𝑁) ↔ ∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥))
4		simpl 487	. . . . . 6 ⊢ ((𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥) → 𝑦 ∈ (𝑁 ClWWalksN 𝐺))
5	4	rexlimivw 3161	. . . . 5 ⊢ (∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥) → 𝑦 ∈ (𝑁 ClWWalksN 𝐺))
6		clwwlknun.v	. . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺)
7		eqid 2762	. . . . . . . . 9 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
8	6, 7	clwwlknp 30399	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑁 ClWWalksN 𝐺) → ((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺)))
9	8	anim2i 628	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)) → (𝐺 ∈ USGraph ∧ ((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺))))
10	7, 6	usgrpredgv 29558	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USGraph ∧ {(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺)) → ((lastS‘𝑦) ∈ 𝑉 ∧ (𝑦‘0) ∈ 𝑉))
11	10	ex 417	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USGraph → ({(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺) → ((lastS‘𝑦) ∈ 𝑉 ∧ (𝑦‘0) ∈ 𝑉)))
12		simpr 489	. . . . . . . . . . . 12 ⊢ (((lastS‘𝑦) ∈ 𝑉 ∧ (𝑦‘0) ∈ 𝑉) → (𝑦‘0) ∈ 𝑉)
13	11, 12	syl6com 38	. . . . . . . . . . 11 ⊢ ({(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺) → (𝐺 ∈ USGraph → (𝑦‘0) ∈ 𝑉))
14	13	3ad2ant3 1152	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺)) → (𝐺 ∈ USGraph → (𝑦‘0) ∈ 𝑉))
15	14	impcom 412	. . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ ((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺))) → (𝑦‘0) ∈ 𝑉)
16		simpr 489	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USGraph ∧ ((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺))) ∧ 𝑥 = (𝑦‘0)) → 𝑥 = (𝑦‘0))
17	16	eqcomd 2768	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USGraph ∧ ((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺))) ∧ 𝑥 = (𝑦‘0)) → (𝑦‘0) = 𝑥)
18	17	biantrud 540	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USGraph ∧ ((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺))) ∧ 𝑥 = (𝑦‘0)) → (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥)))
19	18	bicomd 226	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ ((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺))) ∧ 𝑥 = (𝑦‘0)) → ((𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥) ↔ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)))
20	15, 19	rspcedv 3573	. . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ ((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺))) → (𝑦 ∈ (𝑁 ClWWalksN 𝐺) → ∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥)))
21	20	adantld 495	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ ((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺))) → ((𝐺 ∈ USGraph ∧ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)) → ∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥)))
22	9, 21	mpcom 39	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)) → ∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥))
23	22	ex 417	. . . . 5 ⊢ (𝐺 ∈ USGraph → (𝑦 ∈ (𝑁 ClWWalksN 𝐺) → ∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥)))
24	5, 23	impbid2 229	. . . 4 ⊢ (𝐺 ∈ USGraph → (∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥) ↔ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)))
25	3, 24	bitrid 286	. . 3 ⊢ (𝐺 ∈ USGraph → (∃𝑥 ∈ 𝑉 𝑦 ∈ (𝑥(ClWWalksNOn‘𝐺)𝑁) ↔ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)))
26	1, 25	bitr2id 287	. 2 ⊢ (𝐺 ∈ USGraph → (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁)))
27	26	eqrdv 2760	1 ⊢ (𝐺 ∈ USGraph → (𝑁 ClWWalksN 𝐺) = ∪ 𝑥 ∈ 𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ∀wral 3078  ∃wrex 3088  {cpr 4590  ∪ ciun 4955  ‘cfv 6536  (class class class)co 7412  0cc0 11106  1c1 11107   + caddc 11109   − cmin 11447  ..^cfzo 13689  ♯chash 14373  Word cword 14557  lastSclsw 14606  Vtxcvtx 29357  Edgcedg 29408  USGraphcusgr 29510   ClWWalksN cclwwlkn 30386  ClWWalksNOncclwwlknon 30449

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-oadd 8455  df-er 8692  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9894  df-card 9932  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-nn 12240  df-2 12309  df-n0 12511  df-xnn0 12584  df-z 12598  df-uz 12869  df-fz 13542  df-fzo 13690  df-hash 14374  df-word 14558  df-edg 29409  df-umgr 29444  df-usgr 29512  df-clwwlk 30344  df-clwwlkn 30387  df-clwwlknon 30450

This theorem is used by:  numclwwlk4  30748