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Mirrors > Home > MPE Home > Th. List > numclwwlk3lem2lem | Structured version Visualization version GIF version |
Description: Lemma for numclwwlk3lem2 28169: The set of closed vertices of a fixed length 𝑁 on a fixed vertex 𝑉 is the union of the set of closed walks of length 𝑁 at 𝑉 with the last but one vertex being 𝑉 and the set of closed walks of length 𝑁 at 𝑉 with the last but one vertex not being 𝑉. (Contributed by AV, 1-May-2022.) |
Ref | Expression |
---|---|
numclwwlk3lem2.c | ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) |
numclwwlk3lem2.h | ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣}) |
Ref | Expression |
---|---|
numclwwlk3lem2lem | ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑋(ClWWalksNOn‘𝐺)𝑁) = ((𝑋𝐻𝑁) ∪ (𝑋𝐶𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numclwwlk3lem2.h | . . . 4 ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣}) | |
2 | 1 | numclwwlkovh0 28157 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑋𝐻𝑁) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) ≠ 𝑋}) |
3 | numclwwlk3lem2.c | . . . 4 ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) | |
4 | 3 | 2clwwlk 28132 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑋𝐶𝑁) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}) |
5 | 2, 4 | uneq12d 4091 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → ((𝑋𝐻𝑁) ∪ (𝑋𝐶𝑁)) = ({𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) ≠ 𝑋} ∪ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋})) |
6 | unrab 4226 | . . 3 ⊢ ({𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) ≠ 𝑋} ∪ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ ((𝑤‘(𝑁 − 2)) ≠ 𝑋 ∨ (𝑤‘(𝑁 − 2)) = 𝑋)} | |
7 | exmidne 2997 | . . . . . 6 ⊢ ((𝑤‘(𝑁 − 2)) = 𝑋 ∨ (𝑤‘(𝑁 − 2)) ≠ 𝑋) | |
8 | orcom 867 | . . . . . 6 ⊢ (((𝑤‘(𝑁 − 2)) ≠ 𝑋 ∨ (𝑤‘(𝑁 − 2)) = 𝑋) ↔ ((𝑤‘(𝑁 − 2)) = 𝑋 ∨ (𝑤‘(𝑁 − 2)) ≠ 𝑋)) | |
9 | 7, 8 | mpbir 234 | . . . . 5 ⊢ ((𝑤‘(𝑁 − 2)) ≠ 𝑋 ∨ (𝑤‘(𝑁 − 2)) = 𝑋) |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) → ((𝑤‘(𝑁 − 2)) ≠ 𝑋 ∨ (𝑤‘(𝑁 − 2)) = 𝑋)) |
11 | 10 | rabeqc 3626 | . . 3 ⊢ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ ((𝑤‘(𝑁 − 2)) ≠ 𝑋 ∨ (𝑤‘(𝑁 − 2)) = 𝑋)} = (𝑋(ClWWalksNOn‘𝐺)𝑁) |
12 | 6, 11 | eqtri 2821 | . 2 ⊢ ({𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) ≠ 𝑋} ∪ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}) = (𝑋(ClWWalksNOn‘𝐺)𝑁) |
13 | 5, 12 | eqtr2di 2850 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑋(ClWWalksNOn‘𝐺)𝑁) = ((𝑋𝐻𝑁) ∪ (𝑋𝐶𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 {crab 3110 ∪ cun 3879 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 − cmin 10859 2c2 11680 ℤ≥cuz 12231 ClWWalksNOncclwwlknon 27872 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 |
This theorem is referenced by: numclwwlk3lem2 28169 |
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