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Mirrors > Home > MPE Home > Th. List > 2clwwlk | Structured version Visualization version GIF version |
Description: Value of operation 𝐶, 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 28128. (𝑋𝐶𝑁) 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.) |
Ref | Expression |
---|---|
2clwwlk.c | ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) |
Ref | Expression |
---|---|
2clwwlk | ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑋𝐶𝑁) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq12 7164 | . . 3 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → (𝑣(ClWWalksNOn‘𝐺)𝑛) = (𝑋(ClWWalksNOn‘𝐺)𝑁)) | |
2 | fvoveq1 7178 | . . . . 5 ⊢ (𝑛 = 𝑁 → (𝑤‘(𝑛 − 2)) = (𝑤‘(𝑁 − 2))) | |
3 | 2 | adantl 484 | . . . 4 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → (𝑤‘(𝑛 − 2)) = (𝑤‘(𝑁 − 2))) |
4 | simpl 485 | . . . 4 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → 𝑣 = 𝑋) | |
5 | 3, 4 | eqeq12d 2837 | . . 3 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → ((𝑤‘(𝑛 − 2)) = 𝑣 ↔ (𝑤‘(𝑁 − 2)) = 𝑋)) |
6 | 1, 5 | rabeqbidv 3485 | . 2 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣} = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}) |
7 | 2clwwlk.c | . 2 ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) | |
8 | ovex 7188 | . . 3 ⊢ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∈ V | |
9 | 8 | rabex 5234 | . 2 ⊢ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋} ∈ V |
10 | 6, 7, 9 | ovmpoa 7304 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑋𝐶𝑁) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 ‘cfv 6354 (class class class)co 7155 ∈ cmpo 7157 − cmin 10869 2c2 11691 ℤ≥cuz 12242 ClWWalksNOncclwwlknon 27865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-iota 6313 df-fun 6356 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 |
This theorem is referenced by: 2clwwlk2 28126 2clwwlkel 28127 extwwlkfab 28130 numclwwlk3lem2lem 28161 numclwwlk3lem2 28162 |
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