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Mirrors > Home > MPE Home > Th. List > Mathboxes > gpg5order | Structured version Visualization version GIF version |
Description: The order of a generalized Petersen graph G(5,K), which is either the Petersen graph G(5,2) or the 5-prism G(5,1), is 10. (Contributed by AV, 26-Aug-2025.) |
Ref | Expression |
---|---|
gpg5order | ⊢ (𝐾 ∈ (1...2) → (♯‘(Vtx‘(5 gPetersenGr 𝐾))) = ;10) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 5nn 12349 | . . 3 ⊢ 5 ∈ ℕ | |
2 | 2z 12646 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
3 | fzval3 13769 | . . . . . . 7 ⊢ (2 ∈ ℤ → (1...2) = (1..^(2 + 1))) | |
4 | 2, 3 | ax-mp 5 | . . . . . 6 ⊢ (1...2) = (1..^(2 + 1)) |
5 | 2p1e3 12405 | . . . . . . . 8 ⊢ (2 + 1) = 3 | |
6 | ceil5half3 47279 | . . . . . . . 8 ⊢ (⌈‘(5 / 2)) = 3 | |
7 | 5, 6 | eqtr4i 2765 | . . . . . . 7 ⊢ (2 + 1) = (⌈‘(5 / 2)) |
8 | 7 | oveq2i 7441 | . . . . . 6 ⊢ (1..^(2 + 1)) = (1..^(⌈‘(5 / 2))) |
9 | 4, 8 | eqtri 2762 | . . . . 5 ⊢ (1...2) = (1..^(⌈‘(5 / 2))) |
10 | 9 | eleq2i 2830 | . . . 4 ⊢ (𝐾 ∈ (1...2) ↔ 𝐾 ∈ (1..^(⌈‘(5 / 2)))) |
11 | 10 | biimpi 216 | . . 3 ⊢ (𝐾 ∈ (1...2) → 𝐾 ∈ (1..^(⌈‘(5 / 2)))) |
12 | eqid 2734 | . . . 4 ⊢ (1..^(⌈‘(5 / 2))) = (1..^(⌈‘(5 / 2))) | |
13 | 12 | gpgorder 47947 | . . 3 ⊢ ((5 ∈ ℕ ∧ 𝐾 ∈ (1..^(⌈‘(5 / 2)))) → (♯‘(Vtx‘(5 gPetersenGr 𝐾))) = (2 · 5)) |
14 | 1, 11, 13 | sylancr 587 | . 2 ⊢ (𝐾 ∈ (1...2) → (♯‘(Vtx‘(5 gPetersenGr 𝐾))) = (2 · 5)) |
15 | 5cn 12351 | . . 3 ⊢ 5 ∈ ℂ | |
16 | 2cn 12338 | . . 3 ⊢ 2 ∈ ℂ | |
17 | 5t2e10 12830 | . . 3 ⊢ (5 · 2) = ;10 | |
18 | 15, 16, 17 | mulcomli 11267 | . 2 ⊢ (2 · 5) = ;10 |
19 | 14, 18 | eqtrdi 2790 | 1 ⊢ (𝐾 ∈ (1...2) → (♯‘(Vtx‘(5 gPetersenGr 𝐾))) = ;10) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 ‘cfv 6562 (class class class)co 7430 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 / cdiv 11917 ℕcn 12263 2c2 12318 3c3 12319 5c5 12321 ℤcz 12610 ;cdc 12730 ...cfz 13543 ..^cfzo 13690 ⌈cceil 13827 ♯chash 14365 Vtxcvtx 29027 gPetersenGr cgpg 47934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-inf 9480 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-xnn0 12597 df-z 12611 df-dec 12731 df-uz 12876 df-rp 13032 df-fz 13544 df-fzo 13691 df-fl 13828 df-ceil 13829 df-mod 13906 df-hash 14366 df-struct 17180 df-slot 17215 df-ndx 17227 df-base 17245 df-edgf 29018 df-vtx 29029 df-gpg 47935 |
This theorem is referenced by: gpg5grlic 47974 |
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