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Mirrors > Home > MPE Home > Th. List > numclwwlk3lem2lem | Structured version Visualization version GIF version |
Description: Lemma for numclwwlk3lem2 28163: 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 28151 | . . 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 28126 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑋𝐶𝑁) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}) |
5 | 2, 4 | uneq12d 4140 | . 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 4274 | . . 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 3026 | . . . . . 6 ⊢ ((𝑤‘(𝑁 − 2)) = 𝑋 ∨ (𝑤‘(𝑁 − 2)) ≠ 𝑋) | |
8 | orcom 866 | . . . . . 6 ⊢ (((𝑤‘(𝑁 − 2)) ≠ 𝑋 ∨ (𝑤‘(𝑁 − 2)) = 𝑋) ↔ ((𝑤‘(𝑁 − 2)) = 𝑋 ∨ (𝑤‘(𝑁 − 2)) ≠ 𝑋)) | |
9 | 7, 8 | mpbir 233 | . . . . 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 3678 | . . 3 ⊢ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ ((𝑤‘(𝑁 − 2)) ≠ 𝑋 ∨ (𝑤‘(𝑁 − 2)) = 𝑋)} = (𝑋(ClWWalksNOn‘𝐺)𝑁) |
12 | 6, 11 | eqtri 2844 | . 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 | syl6req 2873 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑋(ClWWalksNOn‘𝐺)𝑁) = ((𝑋𝐻𝑁) ∪ (𝑋𝐶𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 {crab 3142 ∪ cun 3934 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 − cmin 10870 2c2 11693 ℤ≥cuz 12244 ClWWalksNOncclwwlknon 27866 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 |
This theorem is referenced by: numclwwlk3lem2 28163 |
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