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| Mirrors > Home > MPE Home > Th. List > numclwwlkovh0 | Structured version Visualization version GIF version | ||
| Description: Value of operation 𝐻, 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 7 in [Huneke] p. 2. (Contributed by AV, 1-May-2022.) | 
| Ref | Expression | 
|---|---|
| numclwwlkovh.h | ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣}) | 
| Ref | Expression | 
|---|---|
| numclwwlkovh0 | ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑋𝐻𝑁) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) ≠ 𝑋}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | oveq12 7440 | . . 3 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → (𝑣(ClWWalksNOn‘𝐺)𝑛) = (𝑋(ClWWalksNOn‘𝐺)𝑁)) | |
| 2 | oveq1 7438 | . . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑛 − 2) = (𝑁 − 2)) | |
| 3 | 2 | adantl 481 | . . . . 5 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → (𝑛 − 2) = (𝑁 − 2)) | 
| 4 | 3 | fveq2d 6910 | . . . 4 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → (𝑤‘(𝑛 − 2)) = (𝑤‘(𝑁 − 2))) | 
| 5 | simpl 482 | . . . 4 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → 𝑣 = 𝑋) | |
| 6 | 4, 5 | neeq12d 3002 | . . 3 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → ((𝑤‘(𝑛 − 2)) ≠ 𝑣 ↔ (𝑤‘(𝑁 − 2)) ≠ 𝑋)) | 
| 7 | 1, 6 | rabeqbidv 3455 | . 2 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣} = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) ≠ 𝑋}) | 
| 8 | numclwwlkovh.h | . 2 ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣}) | |
| 9 | ovex 7464 | . . 3 ⊢ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∈ V | |
| 10 | 9 | rabex 5339 | . 2 ⊢ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) ≠ 𝑋} ∈ V | 
| 11 | 7, 8, 10 | ovmpoa 7588 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑋𝐻𝑁) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) ≠ 𝑋}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 {crab 3436 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 − cmin 11492 2c2 12321 ℤ≥cuz 12878 ClWWalksNOncclwwlknon 30106 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 | 
| This theorem is referenced by: numclwwlkovh 30392 numclwwlk3lem2lem 30402 numclwwlk3lem2 30403 | 
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