| Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
| 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 12352 | . . 3 ⊢ 5 ∈ ℕ | |
| 2 | 2z 12649 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
| 3 | fzval3 13773 | . . . . . . 7 ⊢ (2 ∈ ℤ → (1...2) = (1..^(2 + 1))) | |
| 4 | 2, 3 | ax-mp 5 | . . . . . 6 ⊢ (1...2) = (1..^(2 + 1)) |
| 5 | 2p1e3 12408 | . . . . . . . 8 ⊢ (2 + 1) = 3 | |
| 6 | ceil5half3 47342 | . . . . . . . 8 ⊢ (⌈‘(5 / 2)) = 3 | |
| 7 | 5, 6 | eqtr4i 2768 | . . . . . . 7 ⊢ (2 + 1) = (⌈‘(5 / 2)) |
| 8 | 7 | oveq2i 7442 | . . . . . 6 ⊢ (1..^(2 + 1)) = (1..^(⌈‘(5 / 2))) |
| 9 | 4, 8 | eqtri 2765 | . . . . 5 ⊢ (1...2) = (1..^(⌈‘(5 / 2))) |
| 10 | 9 | eleq2i 2833 | . . . 4 ⊢ (𝐾 ∈ (1...2) ↔ 𝐾 ∈ (1..^(⌈‘(5 / 2)))) |
| 11 | 10 | biimpi 216 | . . 3 ⊢ (𝐾 ∈ (1...2) → 𝐾 ∈ (1..^(⌈‘(5 / 2)))) |
| 12 | eqid 2737 | . . . 4 ⊢ (1..^(⌈‘(5 / 2))) = (1..^(⌈‘(5 / 2))) | |
| 13 | 12 | gpgorder 48013 | . . 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 12354 | . . 3 ⊢ 5 ∈ ℂ | |
| 16 | 2cn 12341 | . . 3 ⊢ 2 ∈ ℂ | |
| 17 | 5t2e10 12833 | . . 3 ⊢ (5 · 2) = ;10 | |
| 18 | 15, 16, 17 | mulcomli 11270 | . 2 ⊢ (2 · 5) = ;10 |
| 19 | 14, 18 | eqtrdi 2793 | 1 ⊢ (𝐾 ∈ (1...2) → (♯‘(Vtx‘(5 gPetersenGr 𝐾))) = ;10) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 / cdiv 11920 ℕcn 12266 2c2 12321 3c3 12322 5c5 12324 ℤcz 12613 ;cdc 12733 ...cfz 13547 ..^cfzo 13694 ⌈cceil 13831 ♯chash 14369 Vtxcvtx 29013 gPetersenGr cgpg 47999 |
| 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 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-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-fl 13832 df-ceil 13833 df-mod 13910 df-hash 14370 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-edgf 29004 df-vtx 29015 df-gpg 48000 |
| This theorem is referenced by: gpg5grlic 48047 |
| Copyright terms: Public domain | W3C validator |