Step | Hyp | Ref
| Expression |
1 | | 1nprm 16013 |
. . . 4
⊢ ¬ 1
∈ ℙ |
2 | | simpr 488 |
. . . . . . . . 9
⊢ (((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) ∧ 𝐵 ⊆
{(0g‘𝐺)})
→ 𝐵 ⊆
{(0g‘𝐺)}) |
3 | | cygctb.1 |
. . . . . . . . . . . 12
⊢ 𝐵 = (Base‘𝐺) |
4 | | eqid 2798 |
. . . . . . . . . . . 12
⊢
(0g‘𝐺) = (0g‘𝐺) |
5 | 3, 4 | grpidcl 18123 |
. . . . . . . . . . 11
⊢ (𝐺 ∈ Grp →
(0g‘𝐺)
∈ 𝐵) |
6 | 5 | snssd 4702 |
. . . . . . . . . 10
⊢ (𝐺 ∈ Grp →
{(0g‘𝐺)}
⊆ 𝐵) |
7 | 6 | ad2antrr 725 |
. . . . . . . . 9
⊢ (((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) ∧ 𝐵 ⊆
{(0g‘𝐺)})
→ {(0g‘𝐺)} ⊆ 𝐵) |
8 | 2, 7 | eqssd 3932 |
. . . . . . . 8
⊢ (((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) ∧ 𝐵 ⊆
{(0g‘𝐺)})
→ 𝐵 =
{(0g‘𝐺)}) |
9 | 8 | fveq2d 6649 |
. . . . . . 7
⊢ (((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) ∧ 𝐵 ⊆
{(0g‘𝐺)})
→ (♯‘𝐵) =
(♯‘{(0g‘𝐺)})) |
10 | | fvex 6658 |
. . . . . . . 8
⊢
(0g‘𝐺) ∈ V |
11 | | hashsng 13726 |
. . . . . . . 8
⊢
((0g‘𝐺) ∈ V →
(♯‘{(0g‘𝐺)}) = 1) |
12 | 10, 11 | ax-mp 5 |
. . . . . . 7
⊢
(♯‘{(0g‘𝐺)}) = 1 |
13 | 9, 12 | eqtrdi 2849 |
. . . . . 6
⊢ (((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) ∧ 𝐵 ⊆
{(0g‘𝐺)})
→ (♯‘𝐵) =
1) |
14 | | simplr 768 |
. . . . . 6
⊢ (((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) ∧ 𝐵 ⊆
{(0g‘𝐺)})
→ (♯‘𝐵)
∈ ℙ) |
15 | 13, 14 | eqeltrrd 2891 |
. . . . 5
⊢ (((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) ∧ 𝐵 ⊆
{(0g‘𝐺)})
→ 1 ∈ ℙ) |
16 | 15 | ex 416 |
. . . 4
⊢ ((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) → (𝐵 ⊆
{(0g‘𝐺)}
→ 1 ∈ ℙ)) |
17 | 1, 16 | mtoi 202 |
. . 3
⊢ ((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) → ¬ 𝐵
⊆ {(0g‘𝐺)}) |
18 | | nss 3977 |
. . 3
⊢ (¬
𝐵 ⊆
{(0g‘𝐺)}
↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ {(0g‘𝐺)})) |
19 | 17, 18 | sylib 221 |
. 2
⊢ ((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) → ∃𝑥(𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ {(0g‘𝐺)})) |
20 | | eqid 2798 |
. . 3
⊢
(od‘𝐺) =
(od‘𝐺) |
21 | | simpll 766 |
. . 3
⊢ (((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) ∧ (𝑥 ∈
𝐵 ∧ ¬ 𝑥 ∈
{(0g‘𝐺)}))
→ 𝐺 ∈
Grp) |
22 | | simprl 770 |
. . 3
⊢ (((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) ∧ (𝑥 ∈
𝐵 ∧ ¬ 𝑥 ∈
{(0g‘𝐺)}))
→ 𝑥 ∈ 𝐵) |
23 | | simprr 772 |
. . . . 5
⊢ (((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) ∧ (𝑥 ∈
𝐵 ∧ ¬ 𝑥 ∈
{(0g‘𝐺)}))
→ ¬ 𝑥 ∈
{(0g‘𝐺)}) |
24 | 20, 4, 3 | odeq1 18679 |
. . . . . . 7
⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (((od‘𝐺)‘𝑥) = 1 ↔ 𝑥 = (0g‘𝐺))) |
25 | 21, 22, 24 | syl2anc 587 |
. . . . . 6
⊢ (((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) ∧ (𝑥 ∈
𝐵 ∧ ¬ 𝑥 ∈
{(0g‘𝐺)}))
→ (((od‘𝐺)‘𝑥) = 1 ↔ 𝑥 = (0g‘𝐺))) |
26 | | velsn 4541 |
. . . . . 6
⊢ (𝑥 ∈
{(0g‘𝐺)}
↔ 𝑥 =
(0g‘𝐺)) |
27 | 25, 26 | syl6bbr 292 |
. . . . 5
⊢ (((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) ∧ (𝑥 ∈
𝐵 ∧ ¬ 𝑥 ∈
{(0g‘𝐺)}))
→ (((od‘𝐺)‘𝑥) = 1 ↔ 𝑥 ∈ {(0g‘𝐺)})) |
28 | 23, 27 | mtbird 328 |
. . . 4
⊢ (((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) ∧ (𝑥 ∈
𝐵 ∧ ¬ 𝑥 ∈
{(0g‘𝐺)}))
→ ¬ ((od‘𝐺)‘𝑥) = 1) |
29 | | prmnn 16008 |
. . . . . . . . . 10
⊢
((♯‘𝐵)
∈ ℙ → (♯‘𝐵) ∈ ℕ) |
30 | 29 | ad2antlr 726 |
. . . . . . . . 9
⊢ (((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) ∧ (𝑥 ∈
𝐵 ∧ ¬ 𝑥 ∈
{(0g‘𝐺)}))
→ (♯‘𝐵)
∈ ℕ) |
31 | 30 | nnnn0d 11943 |
. . . . . . . 8
⊢ (((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) ∧ (𝑥 ∈
𝐵 ∧ ¬ 𝑥 ∈
{(0g‘𝐺)}))
→ (♯‘𝐵)
∈ ℕ0) |
32 | 3 | fvexi 6659 |
. . . . . . . . 9
⊢ 𝐵 ∈ V |
33 | | hashclb 13715 |
. . . . . . . . 9
⊢ (𝐵 ∈ V → (𝐵 ∈ Fin ↔
(♯‘𝐵) ∈
ℕ0)) |
34 | 32, 33 | ax-mp 5 |
. . . . . . . 8
⊢ (𝐵 ∈ Fin ↔
(♯‘𝐵) ∈
ℕ0) |
35 | 31, 34 | sylibr 237 |
. . . . . . 7
⊢ (((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) ∧ (𝑥 ∈
𝐵 ∧ ¬ 𝑥 ∈
{(0g‘𝐺)}))
→ 𝐵 ∈
Fin) |
36 | 3, 20 | oddvds2 18685 |
. . . . . . 7
⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ∧ 𝑥 ∈ 𝐵) → ((od‘𝐺)‘𝑥) ∥ (♯‘𝐵)) |
37 | 21, 35, 22, 36 | syl3anc 1368 |
. . . . . 6
⊢ (((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) ∧ (𝑥 ∈
𝐵 ∧ ¬ 𝑥 ∈
{(0g‘𝐺)}))
→ ((od‘𝐺)‘𝑥) ∥ (♯‘𝐵)) |
38 | | simplr 768 |
. . . . . . 7
⊢ (((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) ∧ (𝑥 ∈
𝐵 ∧ ¬ 𝑥 ∈
{(0g‘𝐺)}))
→ (♯‘𝐵)
∈ ℙ) |
39 | 3, 20 | odcl2 18684 |
. . . . . . . 8
⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ∧ 𝑥 ∈ 𝐵) → ((od‘𝐺)‘𝑥) ∈ ℕ) |
40 | 21, 35, 22, 39 | syl3anc 1368 |
. . . . . . 7
⊢ (((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) ∧ (𝑥 ∈
𝐵 ∧ ¬ 𝑥 ∈
{(0g‘𝐺)}))
→ ((od‘𝐺)‘𝑥) ∈ ℕ) |
41 | | dvdsprime 16021 |
. . . . . . 7
⊢
(((♯‘𝐵)
∈ ℙ ∧ ((od‘𝐺)‘𝑥) ∈ ℕ) → (((od‘𝐺)‘𝑥) ∥ (♯‘𝐵) ↔ (((od‘𝐺)‘𝑥) = (♯‘𝐵) ∨ ((od‘𝐺)‘𝑥) = 1))) |
42 | 38, 40, 41 | syl2anc 587 |
. . . . . 6
⊢ (((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) ∧ (𝑥 ∈
𝐵 ∧ ¬ 𝑥 ∈
{(0g‘𝐺)}))
→ (((od‘𝐺)‘𝑥) ∥ (♯‘𝐵) ↔ (((od‘𝐺)‘𝑥) = (♯‘𝐵) ∨ ((od‘𝐺)‘𝑥) = 1))) |
43 | 37, 42 | mpbid 235 |
. . . . 5
⊢ (((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) ∧ (𝑥 ∈
𝐵 ∧ ¬ 𝑥 ∈
{(0g‘𝐺)}))
→ (((od‘𝐺)‘𝑥) = (♯‘𝐵) ∨ ((od‘𝐺)‘𝑥) = 1)) |
44 | 43 | ord 861 |
. . . 4
⊢ (((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) ∧ (𝑥 ∈
𝐵 ∧ ¬ 𝑥 ∈
{(0g‘𝐺)}))
→ (¬ ((od‘𝐺)‘𝑥) = (♯‘𝐵) → ((od‘𝐺)‘𝑥) = 1)) |
45 | 28, 44 | mt3d 150 |
. . 3
⊢ (((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) ∧ (𝑥 ∈
𝐵 ∧ ¬ 𝑥 ∈
{(0g‘𝐺)}))
→ ((od‘𝐺)‘𝑥) = (♯‘𝐵)) |
46 | 3, 20, 21, 22, 45 | iscygodd 19000 |
. 2
⊢ (((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) ∧ (𝑥 ∈
𝐵 ∧ ¬ 𝑥 ∈
{(0g‘𝐺)}))
→ 𝐺 ∈
CycGrp) |
47 | 19, 46 | exlimddv 1936 |
1
⊢ ((𝐺 ∈ Grp ∧
(♯‘𝐵) ∈
ℙ) → 𝐺 ∈
CycGrp) |