| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | ringexp0nn.2 | . . 3
⊢ (𝜑 → 𝑁 ∈ ℕ) | 
| 2 | 1 | ancli 548 | . 2
⊢ (𝜑 → (𝜑 ∧ 𝑁 ∈ ℕ)) | 
| 3 |  | oveq1 7439 | . . . 4
⊢ (𝑥 = 1 → (𝑥 ↑
(0g‘𝑅)) =
(1 ↑
(0g‘𝑅))) | 
| 4 | 3 | eqeq1d 2738 | . . 3
⊢ (𝑥 = 1 → ((𝑥 ↑
(0g‘𝑅)) =
(0g‘𝑅)
↔ (1 ↑
(0g‘𝑅)) =
(0g‘𝑅))) | 
| 5 |  | oveq1 7439 | . . . 4
⊢ (𝑥 = 𝑦 → (𝑥 ↑
(0g‘𝑅)) =
(𝑦 ↑
(0g‘𝑅))) | 
| 6 | 5 | eqeq1d 2738 | . . 3
⊢ (𝑥 = 𝑦 → ((𝑥 ↑
(0g‘𝑅)) =
(0g‘𝑅)
↔ (𝑦 ↑
(0g‘𝑅)) =
(0g‘𝑅))) | 
| 7 |  | oveq1 7439 | . . . 4
⊢ (𝑥 = (𝑦 + 1) → (𝑥 ↑
(0g‘𝑅)) =
((𝑦 + 1) ↑
(0g‘𝑅))) | 
| 8 | 7 | eqeq1d 2738 | . . 3
⊢ (𝑥 = (𝑦 + 1) → ((𝑥 ↑
(0g‘𝑅)) =
(0g‘𝑅)
↔ ((𝑦 + 1) ↑
(0g‘𝑅)) =
(0g‘𝑅))) | 
| 9 |  | oveq1 7439 | . . . 4
⊢ (𝑥 = 𝑁 → (𝑥 ↑
(0g‘𝑅)) =
(𝑁 ↑
(0g‘𝑅))) | 
| 10 | 9 | eqeq1d 2738 | . . 3
⊢ (𝑥 = 𝑁 → ((𝑥 ↑
(0g‘𝑅)) =
(0g‘𝑅)
↔ (𝑁 ↑
(0g‘𝑅)) =
(0g‘𝑅))) | 
| 11 |  | ringexp0nn.1 | . . . . . . 7
⊢ (𝜑 → 𝑅 ∈ Ring) | 
| 12 |  | ringmnd 20241 | . . . . . . 7
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd) | 
| 13 | 11, 12 | syl 17 | . . . . . 6
⊢ (𝜑 → 𝑅 ∈ Mnd) | 
| 14 |  | eqid 2736 | . . . . . . 7
⊢
(Base‘𝑅) =
(Base‘𝑅) | 
| 15 |  | eqid 2736 | . . . . . . 7
⊢
(0g‘𝑅) = (0g‘𝑅) | 
| 16 | 14, 15 | mndidcl 18763 | . . . . . 6
⊢ (𝑅 ∈ Mnd →
(0g‘𝑅)
∈ (Base‘𝑅)) | 
| 17 | 13, 16 | syl 17 | . . . . 5
⊢ (𝜑 → (0g‘𝑅) ∈ (Base‘𝑅)) | 
| 18 |  | eqid 2736 | . . . . . . 7
⊢
(mulGrp‘𝑅) =
(mulGrp‘𝑅) | 
| 19 | 18, 14 | mgpbas 20143 | . . . . . 6
⊢
(Base‘𝑅) =
(Base‘(mulGrp‘𝑅)) | 
| 20 | 19 | a1i 11 | . . . . 5
⊢ (𝜑 → (Base‘𝑅) =
(Base‘(mulGrp‘𝑅))) | 
| 21 | 17, 20 | eleqtrd 2842 | . . . 4
⊢ (𝜑 → (0g‘𝑅) ∈
(Base‘(mulGrp‘𝑅))) | 
| 22 |  | eqid 2736 | . . . . 5
⊢
(Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅)) | 
| 23 |  | ringexp0nn.3 | . . . . 5
⊢  ↑ =
(.g‘(mulGrp‘𝑅)) | 
| 24 | 22, 23 | mulg1 19100 | . . . 4
⊢
((0g‘𝑅) ∈ (Base‘(mulGrp‘𝑅)) → (1 ↑
(0g‘𝑅)) =
(0g‘𝑅)) | 
| 25 | 21, 24 | syl 17 | . . 3
⊢ (𝜑 → (1 ↑
(0g‘𝑅)) =
(0g‘𝑅)) | 
| 26 |  | simplr 768 | . . . . 5
⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑦 ↑
(0g‘𝑅)) =
(0g‘𝑅))
→ 𝑦 ∈
ℕ) | 
| 27 | 21 | ad2antrr 726 | . . . . 5
⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑦 ↑
(0g‘𝑅)) =
(0g‘𝑅))
→ (0g‘𝑅) ∈ (Base‘(mulGrp‘𝑅))) | 
| 28 |  | eqid 2736 | . . . . . 6
⊢
(+g‘(mulGrp‘𝑅)) =
(+g‘(mulGrp‘𝑅)) | 
| 29 | 22, 23, 28 | mulgnnp1 19101 | . . . . 5
⊢ ((𝑦 ∈ ℕ ∧
(0g‘𝑅)
∈ (Base‘(mulGrp‘𝑅))) → ((𝑦 + 1) ↑
(0g‘𝑅)) =
((𝑦 ↑
(0g‘𝑅))(+g‘(mulGrp‘𝑅))(0g‘𝑅))) | 
| 30 | 26, 27, 29 | syl2anc 584 | . . . 4
⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑦 ↑
(0g‘𝑅)) =
(0g‘𝑅))
→ ((𝑦 + 1) ↑
(0g‘𝑅)) =
((𝑦 ↑
(0g‘𝑅))(+g‘(mulGrp‘𝑅))(0g‘𝑅))) | 
| 31 |  | simpr 484 | . . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑦 ↑
(0g‘𝑅)) =
(0g‘𝑅))
→ (𝑦 ↑
(0g‘𝑅)) =
(0g‘𝑅)) | 
| 32 | 31 | oveq1d 7447 | . . . . 5
⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑦 ↑
(0g‘𝑅)) =
(0g‘𝑅))
→ ((𝑦 ↑
(0g‘𝑅))(+g‘(mulGrp‘𝑅))(0g‘𝑅)) = ((0g‘𝑅)(+g‘(mulGrp‘𝑅))(0g‘𝑅))) | 
| 33 |  | eqid 2736 | . . . . . . . . . . 11
⊢
(.r‘𝑅) = (.r‘𝑅) | 
| 34 | 18, 33 | mgpplusg 20142 | . . . . . . . . . 10
⊢
(.r‘𝑅) = (+g‘(mulGrp‘𝑅)) | 
| 35 | 34 | eqcomi 2745 | . . . . . . . . 9
⊢
(+g‘(mulGrp‘𝑅)) = (.r‘𝑅) | 
| 36 | 14, 35, 15 | ringrz 20292 | . . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧
(0g‘𝑅)
∈ (Base‘𝑅))
→ ((0g‘𝑅)(+g‘(mulGrp‘𝑅))(0g‘𝑅)) = (0g‘𝑅)) | 
| 37 | 11, 17, 36 | syl2anc 584 | . . . . . . 7
⊢ (𝜑 →
((0g‘𝑅)(+g‘(mulGrp‘𝑅))(0g‘𝑅)) = (0g‘𝑅)) | 
| 38 | 37 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) →
((0g‘𝑅)(+g‘(mulGrp‘𝑅))(0g‘𝑅)) = (0g‘𝑅)) | 
| 39 | 38 | adantr 480 | . . . . 5
⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑦 ↑
(0g‘𝑅)) =
(0g‘𝑅))
→ ((0g‘𝑅)(+g‘(mulGrp‘𝑅))(0g‘𝑅)) = (0g‘𝑅)) | 
| 40 | 32, 39 | eqtrd 2776 | . . . 4
⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑦 ↑
(0g‘𝑅)) =
(0g‘𝑅))
→ ((𝑦 ↑
(0g‘𝑅))(+g‘(mulGrp‘𝑅))(0g‘𝑅)) = (0g‘𝑅)) | 
| 41 | 30, 40 | eqtrd 2776 | . . 3
⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ (𝑦 ↑
(0g‘𝑅)) =
(0g‘𝑅))
→ ((𝑦 + 1) ↑
(0g‘𝑅)) =
(0g‘𝑅)) | 
| 42 | 4, 6, 8, 10, 25, 41 | nnindd 12287 | . 2
⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝑁 ↑
(0g‘𝑅)) =
(0g‘𝑅)) | 
| 43 | 2, 42 | syl 17 | 1
⊢ (𝜑 → (𝑁 ↑
(0g‘𝑅)) =
(0g‘𝑅)) |