Step | Hyp | Ref
| Expression |
1 | | oveq1 7150 |
. . . . 5
⊢ (𝑥 = 0 → (𝑥(.g‘(mulGrp‘ℂfld))𝐴) =
(0(.g‘(mulGrp‘ℂfld))𝐴)) |
2 | | oveq2 7151 |
. . . . 5
⊢ (𝑥 = 0 → (𝐴↑𝑥) = (𝐴↑0)) |
3 | 1, 2 | eqeq12d 2775 |
. . . 4
⊢ (𝑥 = 0 → ((𝑥(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑥)
↔ (0(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑0))) |
4 | 3 | imbi2d 345 |
. . 3
⊢ (𝑥 = 0 → ((𝐴 ∈ ℂ → (𝑥(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑥))
↔ (𝐴 ∈ ℂ →
(0(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑0)))) |
5 | | oveq1 7150 |
. . . . 5
⊢ (𝑥 = 𝑦 → (𝑥(.g‘(mulGrp‘ℂfld))𝐴) = (𝑦(.g‘(mulGrp‘ℂfld))𝐴)) |
6 | | oveq2 7151 |
. . . . 5
⊢ (𝑥 = 𝑦 → (𝐴↑𝑥) = (𝐴↑𝑦)) |
7 | 5, 6 | eqeq12d 2775 |
. . . 4
⊢ (𝑥 = 𝑦 → ((𝑥(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑥)
↔ (𝑦(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑦))) |
8 | 7 | imbi2d 345 |
. . 3
⊢ (𝑥 = 𝑦 → ((𝐴 ∈ ℂ → (𝑥(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑥))
↔ (𝐴 ∈ ℂ → (𝑦(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑦)))) |
9 | | oveq1 7150 |
. . . . 5
⊢ (𝑥 = (𝑦 + 1) → (𝑥(.g‘(mulGrp‘ℂfld))𝐴) = ((𝑦 +
1)(.g‘(mulGrp‘ℂfld))𝐴)) |
10 | | oveq2 7151 |
. . . . 5
⊢ (𝑥 = (𝑦 + 1) → (𝐴↑𝑥) = (𝐴↑(𝑦 + 1))) |
11 | 9, 10 | eqeq12d 2775 |
. . . 4
⊢ (𝑥 = (𝑦 + 1) → ((𝑥(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑥)
↔ ((𝑦 +
1)(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑(𝑦 +
1)))) |
12 | 11 | imbi2d 345 |
. . 3
⊢ (𝑥 = (𝑦 + 1) → ((𝐴 ∈ ℂ → (𝑥(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑥))
↔ (𝐴 ∈ ℂ → ((𝑦 +
1)(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑(𝑦 +
1))))) |
13 | | oveq1 7150 |
. . . . 5
⊢ (𝑥 = 𝐵 → (𝑥(.g‘(mulGrp‘ℂfld))𝐴) = (𝐵(.g‘(mulGrp‘ℂfld))𝐴)) |
14 | | oveq2 7151 |
. . . . 5
⊢ (𝑥 = 𝐵 → (𝐴↑𝑥) = (𝐴↑𝐵)) |
15 | 13, 14 | eqeq12d 2775 |
. . . 4
⊢ (𝑥 = 𝐵 → ((𝑥(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑥)
↔ (𝐵(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝐵))) |
16 | 15 | imbi2d 345 |
. . 3
⊢ (𝑥 = 𝐵 → ((𝐴 ∈ ℂ → (𝑥(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑥))
↔ (𝐴 ∈ ℂ → (𝐵(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝐵)))) |
17 | | eqid 2759 |
. . . . . 6
⊢
(mulGrp‘ℂfld) =
(mulGrp‘ℂfld) |
18 | | cnfldbas 20155 |
. . . . . 6
⊢ ℂ =
(Base‘ℂfld) |
19 | 17, 18 | mgpbas 19298 |
. . . . 5
⊢ ℂ =
(Base‘(mulGrp‘ℂfld)) |
20 | | cnfld1 20176 |
. . . . . 6
⊢ 1 =
(1r‘ℂfld) |
21 | 17, 20 | ringidval 19306 |
. . . . 5
⊢ 1 =
(0g‘(mulGrp‘ℂfld)) |
22 | | eqid 2759 |
. . . . 5
⊢
(.g‘(mulGrp‘ℂfld)) =
(.g‘(mulGrp‘ℂfld)) |
23 | 19, 21, 22 | mulg0 18283 |
. . . 4
⊢ (𝐴 ∈ ℂ →
(0(.g‘(mulGrp‘ℂfld))𝐴) = 1) |
24 | | exp0 13468 |
. . . 4
⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1) |
25 | 23, 24 | eqtr4d 2797 |
. . 3
⊢ (𝐴 ∈ ℂ →
(0(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑0)) |
26 | | oveq1 7150 |
. . . . . 6
⊢ ((𝑦(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑦)
→ ((𝑦(.g‘(mulGrp‘ℂfld))𝐴) · 𝐴) = ((𝐴↑𝑦)
· 𝐴)) |
27 | | cnring 20173 |
. . . . . . . . . 10
⊢
ℂfld ∈ Ring |
28 | 17 | ringmgp 19356 |
. . . . . . . . . 10
⊢
(ℂfld ∈ Ring →
(mulGrp‘ℂfld) ∈ Mnd) |
29 | 27, 28 | ax-mp 5 |
. . . . . . . . 9
⊢
(mulGrp‘ℂfld) ∈ Mnd |
30 | | cnfldmul 20157 |
. . . . . . . . . . 11
⊢ ·
= (.r‘ℂfld) |
31 | 17, 30 | mgpplusg 19296 |
. . . . . . . . . 10
⊢ ·
= (+g‘(mulGrp‘ℂfld)) |
32 | 19, 22, 31 | mulgnn0p1 18291 |
. . . . . . . . 9
⊢
(((mulGrp‘ℂfld) ∈ Mnd ∧ 𝑦 ∈ ℕ0
∧ 𝐴 ∈ ℂ)
→ ((𝑦 +
1)(.g‘(mulGrp‘ℂfld))𝐴) = ((𝑦(.g‘(mulGrp‘ℂfld))𝐴) · 𝐴)) |
33 | 29, 32 | mp3an1 1446 |
. . . . . . . 8
⊢ ((𝑦 ∈ ℕ0
∧ 𝐴 ∈ ℂ)
→ ((𝑦 +
1)(.g‘(mulGrp‘ℂfld))𝐴) = ((𝑦(.g‘(mulGrp‘ℂfld))𝐴) · 𝐴)) |
34 | 33 | ancoms 463 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℕ0)
→ ((𝑦 +
1)(.g‘(mulGrp‘ℂfld))𝐴) = ((𝑦(.g‘(mulGrp‘ℂfld))𝐴) · 𝐴)) |
35 | | expp1 13471 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℕ0)
→ (𝐴↑(𝑦 + 1)) = ((𝐴↑𝑦) · 𝐴)) |
36 | 34, 35 | eqeq12d 2775 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℕ0)
→ (((𝑦 +
1)(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑(𝑦 + 1)) ↔ ((𝑦(.g‘(mulGrp‘ℂfld))𝐴) · 𝐴) = ((𝐴↑𝑦)
· 𝐴))) |
37 | 26, 36 | syl5ibr 249 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℕ0)
→ ((𝑦(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑦)
→ ((𝑦 +
1)(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑(𝑦 +
1)))) |
38 | 37 | expcom 418 |
. . . 4
⊢ (𝑦 ∈ ℕ0
→ (𝐴 ∈ ℂ
→ ((𝑦(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑦)
→ ((𝑦 +
1)(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑(𝑦 +
1))))) |
39 | 38 | a2d 29 |
. . 3
⊢ (𝑦 ∈ ℕ0
→ ((𝐴 ∈ ℂ
→ (𝑦(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑦))
→ (𝐴 ∈ ℂ → ((𝑦 +
1)(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑(𝑦 +
1))))) |
40 | 4, 8, 12, 16, 25, 39 | nn0ind 12101 |
. 2
⊢ (𝐵 ∈ ℕ0
→ (𝐴 ∈ ℂ
→ (𝐵(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝐵))) |
41 | 40 | impcom 412 |
1
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ0)
→ (𝐵(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝐵)) |