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Theorem 2clwwlk 29333
Description: Value of operation 𝐢, mapping a vertex v and an integer n greater than 1 to the "closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) with v(n-2) = v" according to definition 6 in [Huneke] p. 2. Such closed walks are "double loops" consisting of a closed (n-2)-walk v = v(0) ... v(n-2) = v and a closed 2-walk v = v(n-2) v(n-1) v(n) = v, see 2clwwlk2clwwlk 29336. (𝑋𝐢𝑁) is called the "set of double loops of length 𝑁 on vertex 𝑋 " in the following. (Contributed by Alexander van der Vekens, 14-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 20-Apr-2022.)
Hypothesis
Ref Expression
2clwwlk.c 𝐢 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})
Assertion
Ref Expression
2clwwlk ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (𝑋𝐢𝑁) = {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋})
Distinct variable groups:   𝑛,𝐺,𝑣,𝑀   𝑛,𝑁,𝑣,𝑀   𝑛,𝑉,𝑣   𝑛,𝑋,𝑣,𝑀
Allowed substitution hints:   𝐢(𝑀,𝑣,𝑛)   𝑉(𝑀)

Proof of Theorem 2clwwlk
StepHypRef Expression
1 oveq12 7371 . . 3 ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) β†’ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) = (𝑋(ClWWalksNOnβ€˜πΊ)𝑁))
2 fvoveq1 7385 . . . . 5 (𝑛 = 𝑁 β†’ (π‘€β€˜(𝑛 βˆ’ 2)) = (π‘€β€˜(𝑁 βˆ’ 2)))
32adantl 483 . . . 4 ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) β†’ (π‘€β€˜(𝑛 βˆ’ 2)) = (π‘€β€˜(𝑁 βˆ’ 2)))
4 simpl 484 . . . 4 ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) β†’ 𝑣 = 𝑋)
53, 4eqeq12d 2753 . . 3 ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) β†’ ((π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣 ↔ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋))
61, 5rabeqbidv 3427 . 2 ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) β†’ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣} = {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋})
7 2clwwlk.c . 2 𝐢 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})
8 ovex 7395 . . 3 (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∈ V
98rabex 5294 . 2 {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋} ∈ V
106, 7, 9ovmpoa 7515 1 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (𝑋𝐢𝑁) = {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {crab 3410  β€˜cfv 6501  (class class class)co 7362   ∈ cmpo 7364   βˆ’ cmin 11392  2c2 12215  β„€β‰₯cuz 12770  ClWWalksNOncclwwlknon 29073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367
This theorem is referenced by:  2clwwlk2  29334  2clwwlkel  29335  extwwlkfab  29338  numclwwlk3lem2lem  29369  numclwwlk3lem2  29370
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