Theorem clwwlknun 30140

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 4999	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁) ↔ ∃𝑥 ∈ 𝑉 𝑦 ∈ (𝑥(ClWWalksNOn‘𝐺)𝑁))
2		isclwwlknon 30119	. . . . 5 ⊢ (𝑦 ∈ (𝑥(ClWWalksNOn‘𝐺)𝑁) ↔ (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥))
3	2	rexbii 3091	. . . 4 ⊢ (∃𝑥 ∈ 𝑉 𝑦 ∈ (𝑥(ClWWalksNOn‘𝐺)𝑁) ↔ ∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥))
4		simpl 482	. . . . . 6 ⊢ ((𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥) → 𝑦 ∈ (𝑁 ClWWalksN 𝐺))
5	4	rexlimivw 3148	. . . . 5 ⊢ (∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥) → 𝑦 ∈ (𝑁 ClWWalksN 𝐺))
6		clwwlknun.v	. . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺)
7		eqid 2734	. . . . . . . . 9 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
8	6, 7	clwwlknp 30065	. . . . . . . 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 617	. . . . . . 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 29228	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USGraph ∧ {(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺)) → ((lastS‘𝑦) ∈ 𝑉 ∧ (𝑦‘0) ∈ 𝑉))
11	10	ex 412	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USGraph → ({(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺) → ((lastS‘𝑦) ∈ 𝑉 ∧ (𝑦‘0) ∈ 𝑉)))
12		simpr 484	. . . . . . . . . . . 12 ⊢ (((lastS‘𝑦) ∈ 𝑉 ∧ (𝑦‘0) ∈ 𝑉) → (𝑦‘0) ∈ 𝑉)
13	11, 12	syl6com 37	. . . . . . . . . . 11 ⊢ ({(lastS‘𝑦), (𝑦‘0)} ∈ (Edg‘𝐺) → (𝐺 ∈ USGraph → (𝑦‘0) ∈ 𝑉))
14	13	3ad2ant3 1134	. . . . . . . . . 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 407	. . . . . . . . 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 484	. . . . . . . . . . . 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 2740	. . . . . . . . . . 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 531	. . . . . . . . . 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 223	. . . . . . . . 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 3614	. . . . . . . 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 490	. . . . . . 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 38	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)) → ∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥))
23	22	ex 412	. . . . 5 ⊢ (𝐺 ∈ USGraph → (𝑦 ∈ (𝑁 ClWWalksN 𝐺) → ∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥)))
24	5, 23	impbid2 226	. . . 4 ⊢ (𝐺 ∈ USGraph → (∃𝑥 ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑦‘0) = 𝑥) ↔ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)))
25	3, 24	bitrid 283	. . 3 ⊢ (𝐺 ∈ USGraph → (∃𝑥 ∈ 𝑉 𝑦 ∈ (𝑥(ClWWalksNOn‘𝐺)𝑁) ↔ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)))
26	1, 25	bitr2id 284	. 2 ⊢ (𝐺 ∈ USGraph → (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁)))
27	26	eqrdv 2732	1 ⊢ (𝐺 ∈ USGraph → (𝑁 ClWWalksN 𝐺) = ∪ 𝑥 ∈ 𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1536   ∈ wcel 2105  ∀wral 3058  ∃wrex 3067  {cpr 4632  ∪ ciun 4995  ‘cfv 6562  (class class class)co 7430  0cc0 11152  1c1 11153   + caddc 11155   − cmin 11489  ..^cfzo 13690  ♯chash 14365  Word cword 14548  lastSclsw 14596  Vtxcvtx 29027  Edgcedg 29078  USGraphcusgr 29180   ClWWalksN cclwwlkn 30052  ClWWalksNOncclwwlknon 30115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-xnn0 12597  df-z 12611  df-uz 12876  df-fz 13544  df-fzo 13691  df-hash 14366  df-word 14549  df-edg 29079  df-umgr 29114  df-usgr 29182  df-clwwlk 30010  df-clwwlkn 30053  df-clwwlknon 30116

This theorem is referenced by:  numclwwlk4  30414