Step | Hyp | Ref
| Expression |
1 | | eqid 2738 |
. . 3
⊢
(Base‘𝐺) =
(Base‘𝐺) |
2 | | eqid 2738 |
. . 3
⊢
(.g‘𝐺) = (.g‘𝐺) |
3 | 1, 2 | iscyg3 19486 |
. 2
⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥))) |
4 | | eqidd 2739 |
. . . 4
⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) → (Base‘𝐺) = (Base‘𝐺)) |
5 | | eqidd 2739 |
. . . 4
⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) → (+g‘𝐺) = (+g‘𝐺)) |
6 | | simpll 764 |
. . . 4
⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) → 𝐺 ∈ Grp) |
7 | | oveq1 7282 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑖 → (𝑛(.g‘𝐺)𝑥) = (𝑖(.g‘𝐺)𝑥)) |
8 | 7 | eqeq2d 2749 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑖 → (𝑦 = (𝑛(.g‘𝐺)𝑥) ↔ 𝑦 = (𝑖(.g‘𝐺)𝑥))) |
9 | 8 | cbvrexvw 3384 |
. . . . . . . . 9
⊢
(∃𝑛 ∈
ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥) ↔ ∃𝑖 ∈ ℤ 𝑦 = (𝑖(.g‘𝐺)𝑥)) |
10 | 9 | biimpi 215 |
. . . . . . . 8
⊢
(∃𝑛 ∈
ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥) → ∃𝑖 ∈ ℤ 𝑦 = (𝑖(.g‘𝐺)𝑥)) |
11 | 10 | ralimi 3087 |
. . . . . . 7
⊢
(∀𝑦 ∈
(Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥) → ∀𝑦 ∈ (Base‘𝐺)∃𝑖 ∈ ℤ 𝑦 = (𝑖(.g‘𝐺)𝑥)) |
12 | 11 | adantl 482 |
. . . . . 6
⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) → ∀𝑦 ∈ (Base‘𝐺)∃𝑖 ∈ ℤ 𝑦 = (𝑖(.g‘𝐺)𝑥)) |
13 | 12 | 3ad2ant1 1132 |
. . . . 5
⊢ ((((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ∀𝑦 ∈ (Base‘𝐺)∃𝑖 ∈ ℤ 𝑦 = (𝑖(.g‘𝐺)𝑥)) |
14 | | simpll 764 |
. . . . . . . . 9
⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝐺 ∈ Grp) |
15 | | simpr 485 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ (Base‘𝐺)) |
16 | 15 | anim1ci 616 |
. . . . . . . . . 10
⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑥 ∈ (Base‘𝐺))) |
17 | | df-3an 1088 |
. . . . . . . . . 10
⊢ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑥 ∈ (Base‘𝐺)) ↔ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑥 ∈ (Base‘𝐺))) |
18 | 16, 17 | sylibr 233 |
. . . . . . . . 9
⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑥 ∈ (Base‘𝐺))) |
19 | | eqid 2738 |
. . . . . . . . . 10
⊢
(+g‘𝐺) = (+g‘𝐺) |
20 | 1, 2, 19 | mulgdir 18735 |
. . . . . . . . 9
⊢ ((𝐺 ∈ Grp ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑥 ∈ (Base‘𝐺))) → ((𝑚 + 𝑛)(.g‘𝐺)𝑥) = ((𝑚(.g‘𝐺)𝑥)(+g‘𝐺)(𝑛(.g‘𝐺)𝑥))) |
21 | 14, 18, 20 | syl2anc 584 |
. . . . . . . 8
⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚 + 𝑛)(.g‘𝐺)𝑥) = ((𝑚(.g‘𝐺)𝑥)(+g‘𝐺)(𝑛(.g‘𝐺)𝑥))) |
22 | 21 | ralrimivva 3123 |
. . . . . . 7
⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ∀𝑚 ∈ ℤ ∀𝑛 ∈ ℤ ((𝑚 + 𝑛)(.g‘𝐺)𝑥) = ((𝑚(.g‘𝐺)𝑥)(+g‘𝐺)(𝑛(.g‘𝐺)𝑥))) |
23 | 22 | adantr 481 |
. . . . . 6
⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) → ∀𝑚 ∈ ℤ ∀𝑛 ∈ ℤ ((𝑚 + 𝑛)(.g‘𝐺)𝑥) = ((𝑚(.g‘𝐺)𝑥)(+g‘𝐺)(𝑛(.g‘𝐺)𝑥))) |
24 | 23 | 3ad2ant1 1132 |
. . . . 5
⊢ ((((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ∀𝑚 ∈ ℤ ∀𝑛 ∈ ℤ ((𝑚 + 𝑛)(.g‘𝐺)𝑥) = ((𝑚(.g‘𝐺)𝑥)(+g‘𝐺)(𝑛(.g‘𝐺)𝑥))) |
25 | | simp2 1136 |
. . . . 5
⊢ ((((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑎 ∈ (Base‘𝐺)) |
26 | | simp3 1137 |
. . . . 5
⊢ ((((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑏 ∈ (Base‘𝐺)) |
27 | | zsscn 12327 |
. . . . . 6
⊢ ℤ
⊆ ℂ |
28 | 27 | a1i 11 |
. . . . 5
⊢ ((((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ℤ ⊆
ℂ) |
29 | 13, 24, 25, 26, 28 | cyccom 18822 |
. . . 4
⊢ ((((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g‘𝐺)𝑏) = (𝑏(+g‘𝐺)𝑎)) |
30 | 4, 5, 6, 29 | isabld 19400 |
. . 3
⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) → 𝐺 ∈ Abel) |
31 | 30 | r19.29an 3217 |
. 2
⊢ ((𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g‘𝐺)𝑥)) → 𝐺 ∈ Abel) |
32 | 3, 31 | sylbi 216 |
1
⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Abel) |