| Step | Hyp | Ref
| Expression |
| 1 | | oveq1 7438 |
. . . . 5
⊢ (𝑥 = 0 → (𝑥(.g‘(mulGrp‘ℂfld))𝐴) =
(0(.g‘(mulGrp‘ℂfld))𝐴)) |
| 2 | | oveq2 7439 |
. . . . 5
⊢ (𝑥 = 0 → (𝐴↑𝑥) = (𝐴↑0)) |
| 3 | 1, 2 | eqeq12d 2753 |
. . . 4
⊢ (𝑥 = 0 → ((𝑥(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑥)
↔ (0(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑0))) |
| 4 | 3 | imbi2d 340 |
. . 3
⊢ (𝑥 = 0 → ((𝐴 ∈ ℂ → (𝑥(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑥))
↔ (𝐴 ∈ ℂ →
(0(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑0)))) |
| 5 | | oveq1 7438 |
. . . . 5
⊢ (𝑥 = 𝑦 → (𝑥(.g‘(mulGrp‘ℂfld))𝐴) = (𝑦(.g‘(mulGrp‘ℂfld))𝐴)) |
| 6 | | oveq2 7439 |
. . . . 5
⊢ (𝑥 = 𝑦 → (𝐴↑𝑥) = (𝐴↑𝑦)) |
| 7 | 5, 6 | eqeq12d 2753 |
. . . 4
⊢ (𝑥 = 𝑦 → ((𝑥(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑥)
↔ (𝑦(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑦))) |
| 8 | 7 | imbi2d 340 |
. . 3
⊢ (𝑥 = 𝑦 → ((𝐴 ∈ ℂ → (𝑥(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑥))
↔ (𝐴 ∈ ℂ → (𝑦(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑦)))) |
| 9 | | oveq1 7438 |
. . . . 5
⊢ (𝑥 = (𝑦 + 1) → (𝑥(.g‘(mulGrp‘ℂfld))𝐴) = ((𝑦 +
1)(.g‘(mulGrp‘ℂfld))𝐴)) |
| 10 | | oveq2 7439 |
. . . . 5
⊢ (𝑥 = (𝑦 + 1) → (𝐴↑𝑥) = (𝐴↑(𝑦 + 1))) |
| 11 | 9, 10 | eqeq12d 2753 |
. . . 4
⊢ (𝑥 = (𝑦 + 1) → ((𝑥(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑥)
↔ ((𝑦 +
1)(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑(𝑦 +
1)))) |
| 12 | 11 | imbi2d 340 |
. . 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 7438 |
. . . . 5
⊢ (𝑥 = 𝐵 → (𝑥(.g‘(mulGrp‘ℂfld))𝐴) = (𝐵(.g‘(mulGrp‘ℂfld))𝐴)) |
| 14 | | oveq2 7439 |
. . . . 5
⊢ (𝑥 = 𝐵 → (𝐴↑𝑥) = (𝐴↑𝐵)) |
| 15 | 13, 14 | eqeq12d 2753 |
. . . 4
⊢ (𝑥 = 𝐵 → ((𝑥(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑥)
↔ (𝐵(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝐵))) |
| 16 | 15 | imbi2d 340 |
. . 3
⊢ (𝑥 = 𝐵 → ((𝐴 ∈ ℂ → (𝑥(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑥))
↔ (𝐴 ∈ ℂ → (𝐵(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝐵)))) |
| 17 | | eqid 2737 |
. . . . . 6
⊢
(mulGrp‘ℂfld) =
(mulGrp‘ℂfld) |
| 18 | | cnfldbas 21368 |
. . . . . 6
⊢ ℂ =
(Base‘ℂfld) |
| 19 | 17, 18 | mgpbas 20142 |
. . . . 5
⊢ ℂ =
(Base‘(mulGrp‘ℂfld)) |
| 20 | | cnfld1 21406 |
. . . . . 6
⊢ 1 =
(1r‘ℂfld) |
| 21 | 17, 20 | ringidval 20180 |
. . . . 5
⊢ 1 =
(0g‘(mulGrp‘ℂfld)) |
| 22 | | eqid 2737 |
. . . . 5
⊢
(.g‘(mulGrp‘ℂfld)) =
(.g‘(mulGrp‘ℂfld)) |
| 23 | 19, 21, 22 | mulg0 19092 |
. . . 4
⊢ (𝐴 ∈ ℂ →
(0(.g‘(mulGrp‘ℂfld))𝐴) = 1) |
| 24 | | exp0 14106 |
. . . 4
⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1) |
| 25 | 23, 24 | eqtr4d 2780 |
. . 3
⊢ (𝐴 ∈ ℂ →
(0(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑0)) |
| 26 | | oveq1 7438 |
. . . . . 6
⊢ ((𝑦(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑦)
→ ((𝑦(.g‘(mulGrp‘ℂfld))𝐴) · 𝐴) = ((𝐴↑𝑦)
· 𝐴)) |
| 27 | | cnring 21403 |
. . . . . . . . . 10
⊢
ℂfld ∈ Ring |
| 28 | 17 | ringmgp 20236 |
. . . . . . . . . 10
⊢
(ℂfld ∈ Ring →
(mulGrp‘ℂfld) ∈ Mnd) |
| 29 | 27, 28 | ax-mp 5 |
. . . . . . . . 9
⊢
(mulGrp‘ℂfld) ∈ Mnd |
| 30 | | cnfldmul 21372 |
. . . . . . . . . . 11
⊢ ·
= (.r‘ℂfld) |
| 31 | 17, 30 | mgpplusg 20141 |
. . . . . . . . . 10
⊢ ·
= (+g‘(mulGrp‘ℂfld)) |
| 32 | 19, 22, 31 | mulgnn0p1 19103 |
. . . . . . . . 9
⊢
(((mulGrp‘ℂfld) ∈ Mnd ∧ 𝑦 ∈ ℕ0
∧ 𝐴 ∈ ℂ)
→ ((𝑦 +
1)(.g‘(mulGrp‘ℂfld))𝐴) = ((𝑦(.g‘(mulGrp‘ℂfld))𝐴) · 𝐴)) |
| 33 | 29, 32 | mp3an1 1450 |
. . . . . . . 8
⊢ ((𝑦 ∈ ℕ0
∧ 𝐴 ∈ ℂ)
→ ((𝑦 +
1)(.g‘(mulGrp‘ℂfld))𝐴) = ((𝑦(.g‘(mulGrp‘ℂfld))𝐴) · 𝐴)) |
| 34 | 33 | ancoms 458 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℕ0)
→ ((𝑦 +
1)(.g‘(mulGrp‘ℂfld))𝐴) = ((𝑦(.g‘(mulGrp‘ℂfld))𝐴) · 𝐴)) |
| 35 | | expp1 14109 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℕ0)
→ (𝐴↑(𝑦 + 1)) = ((𝐴↑𝑦) · 𝐴)) |
| 36 | 34, 35 | eqeq12d 2753 |
. . . . . 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 | imbitrrid 246 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℕ0)
→ ((𝑦(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝑦)
→ ((𝑦 +
1)(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑(𝑦 +
1)))) |
| 38 | 37 | expcom 413 |
. . . 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 12713 |
. 2
⊢ (𝐵 ∈ ℕ0
→ (𝐴 ∈ ℂ
→ (𝐵(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝐵))) |
| 41 | 40 | impcom 407 |
1
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ0)
→ (𝐵(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝐵)) |