| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | eqid 2736 | . . 3
⊢
(od‘𝑃) =
(od‘𝑃) | 
| 2 |  | eqid 2736 | . . 3
⊢
(1r‘𝑃) = (1r‘𝑃) | 
| 3 |  | eqid 2736 | . . 3
⊢
(chr‘𝑃) =
(chr‘𝑃) | 
| 4 | 1, 2, 3 | chrval 21539 | . 2
⊢
((od‘𝑃)‘(1r‘𝑃)) = (chr‘𝑃) | 
| 5 |  | eqid 2736 | . . . . . . . . . 10
⊢
(od‘𝑅) =
(od‘𝑅) | 
| 6 |  | eqid 2736 | . . . . . . . . . 10
⊢
(1r‘𝑅) = (1r‘𝑅) | 
| 7 |  | eqid 2736 | . . . . . . . . . 10
⊢
(chr‘𝑅) =
(chr‘𝑅) | 
| 8 | 5, 6, 7 | chrval 21539 | . . . . . . . . 9
⊢
((od‘𝑅)‘(1r‘𝑅)) = (chr‘𝑅) | 
| 9 | 8 | eqcomi 2745 | . . . . . . . 8
⊢
(chr‘𝑅) =
((od‘𝑅)‘(1r‘𝑅)) | 
| 10 |  | id 22 | . . . . . . . . . 10
⊢ (𝑅 ∈ CRing → 𝑅 ∈ CRing) | 
| 11 | 10 | crnggrpd 20245 | . . . . . . . . 9
⊢ (𝑅 ∈ CRing → 𝑅 ∈ Grp) | 
| 12 |  | crngring 20243 | . . . . . . . . . 10
⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | 
| 13 |  | eqid 2736 | . . . . . . . . . . 11
⊢
(Base‘𝑅) =
(Base‘𝑅) | 
| 14 | 13, 6 | ringidcl 20263 | . . . . . . . . . 10
⊢ (𝑅 ∈ Ring →
(1r‘𝑅)
∈ (Base‘𝑅)) | 
| 15 | 12, 14 | syl 17 | . . . . . . . . 9
⊢ (𝑅 ∈ CRing →
(1r‘𝑅)
∈ (Base‘𝑅)) | 
| 16 | 7 | chrcl 21540 | . . . . . . . . . 10
⊢ (𝑅 ∈ Ring →
(chr‘𝑅) ∈
ℕ0) | 
| 17 | 12, 16 | syl 17 | . . . . . . . . 9
⊢ (𝑅 ∈ CRing →
(chr‘𝑅) ∈
ℕ0) | 
| 18 |  | eqid 2736 | . . . . . . . . . 10
⊢
(.g‘𝑅) = (.g‘𝑅) | 
| 19 |  | eqid 2736 | . . . . . . . . . 10
⊢
(0g‘𝑅) = (0g‘𝑅) | 
| 20 | 13, 5, 18, 19 | odeq 19569 | . . . . . . . . 9
⊢ ((𝑅 ∈ Grp ∧
(1r‘𝑅)
∈ (Base‘𝑅) ∧
(chr‘𝑅) ∈
ℕ0) → ((chr‘𝑅) = ((od‘𝑅)‘(1r‘𝑅)) ↔ ∀𝑛 ∈ ℕ0
((chr‘𝑅) ∥
𝑛 ↔ (𝑛(.g‘𝑅)(1r‘𝑅)) = (0g‘𝑅)))) | 
| 21 | 11, 15, 17, 20 | syl3anc 1372 | . . . . . . . 8
⊢ (𝑅 ∈ CRing →
((chr‘𝑅) =
((od‘𝑅)‘(1r‘𝑅)) ↔ ∀𝑛 ∈ ℕ0
((chr‘𝑅) ∥
𝑛 ↔ (𝑛(.g‘𝑅)(1r‘𝑅)) = (0g‘𝑅)))) | 
| 22 | 9, 21 | mpbii 233 | . . . . . . 7
⊢ (𝑅 ∈ CRing →
∀𝑛 ∈
ℕ0 ((chr‘𝑅) ∥ 𝑛 ↔ (𝑛(.g‘𝑅)(1r‘𝑅)) = (0g‘𝑅))) | 
| 23 | 22 | r19.21bi 3250 | . . . . . 6
⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ ℕ0)
→ ((chr‘𝑅)
∥ 𝑛 ↔ (𝑛(.g‘𝑅)(1r‘𝑅)) = (0g‘𝑅))) | 
| 24 |  | ply1chr.1 | . . . . . . 7
⊢ 𝑃 = (Poly1‘𝑅) | 
| 25 |  | eqid 2736 | . . . . . . 7
⊢
(algSc‘𝑃) =
(algSc‘𝑃) | 
| 26 | 12 | adantr 480 | . . . . . . 7
⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ ℕ0)
→ 𝑅 ∈
Ring) | 
| 27 | 11 | grpmndd 18965 | . . . . . . . . 9
⊢ (𝑅 ∈ CRing → 𝑅 ∈ Mnd) | 
| 28 | 27 | adantr 480 | . . . . . . . 8
⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ ℕ0)
→ 𝑅 ∈
Mnd) | 
| 29 |  | simpr 484 | . . . . . . . 8
⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ ℕ0)
→ 𝑛 ∈
ℕ0) | 
| 30 | 15 | adantr 480 | . . . . . . . 8
⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ ℕ0)
→ (1r‘𝑅) ∈ (Base‘𝑅)) | 
| 31 | 13, 18, 28, 29, 30 | mulgnn0cld 19114 | . . . . . . 7
⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ ℕ0)
→ (𝑛(.g‘𝑅)(1r‘𝑅)) ∈ (Base‘𝑅)) | 
| 32 |  | simpl 482 | . . . . . . . 8
⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ ℕ0)
→ 𝑅 ∈
CRing) | 
| 33 | 13, 19 | ring0cl 20265 | . . . . . . . 8
⊢ (𝑅 ∈ Ring →
(0g‘𝑅)
∈ (Base‘𝑅)) | 
| 34 | 32, 12, 33 | 3syl 18 | . . . . . . 7
⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ ℕ0)
→ (0g‘𝑅) ∈ (Base‘𝑅)) | 
| 35 | 24, 13, 25, 26, 31, 34 | ply1scleq 22310 | . . . . . 6
⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ ℕ0)
→ (((algSc‘𝑃)‘(𝑛(.g‘𝑅)(1r‘𝑅))) = ((algSc‘𝑃)‘(0g‘𝑅)) ↔ (𝑛(.g‘𝑅)(1r‘𝑅)) = (0g‘𝑅))) | 
| 36 | 24 | ply1sca 22255 | . . . . . . . . . . . 12
⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃)) | 
| 37 | 36 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ ℕ0)
→ 𝑅 =
(Scalar‘𝑃)) | 
| 38 | 37 | fveq2d 6909 | . . . . . . . . . 10
⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ ℕ0)
→ (.g‘𝑅) = (.g‘(Scalar‘𝑃))) | 
| 39 | 38 | oveqd 7449 | . . . . . . . . 9
⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ ℕ0)
→ (𝑛(.g‘𝑅)(1r‘𝑅)) = (𝑛(.g‘(Scalar‘𝑃))(1r‘𝑅))) | 
| 40 | 39 | fveq2d 6909 | . . . . . . . 8
⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ ℕ0)
→ ((algSc‘𝑃)‘(𝑛(.g‘𝑅)(1r‘𝑅))) = ((algSc‘𝑃)‘(𝑛(.g‘(Scalar‘𝑃))(1r‘𝑅)))) | 
| 41 | 24 | ply1assa 22202 | . . . . . . . . . 10
⊢ (𝑅 ∈ CRing → 𝑃 ∈ AssAlg) | 
| 42 | 41 | adantr 480 | . . . . . . . . 9
⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ ℕ0)
→ 𝑃 ∈
AssAlg) | 
| 43 | 37 | fveq2d 6909 | . . . . . . . . . 10
⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ ℕ0)
→ (Base‘𝑅) =
(Base‘(Scalar‘𝑃))) | 
| 44 | 30, 43 | eleqtrd 2842 | . . . . . . . . 9
⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ ℕ0)
→ (1r‘𝑅) ∈ (Base‘(Scalar‘𝑃))) | 
| 45 |  | eqid 2736 | . . . . . . . . . 10
⊢
(Scalar‘𝑃) =
(Scalar‘𝑃) | 
| 46 |  | eqid 2736 | . . . . . . . . . 10
⊢
(Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃)) | 
| 47 |  | eqid 2736 | . . . . . . . . . 10
⊢
(.g‘𝑃) = (.g‘𝑃) | 
| 48 |  | eqid 2736 | . . . . . . . . . 10
⊢
(.g‘(Scalar‘𝑃)) =
(.g‘(Scalar‘𝑃)) | 
| 49 | 25, 45, 46, 47, 48 | asclmulg 21923 | . . . . . . . . 9
⊢ ((𝑃 ∈ AssAlg ∧ 𝑛 ∈ ℕ0
∧ (1r‘𝑅) ∈ (Base‘(Scalar‘𝑃))) → ((algSc‘𝑃)‘(𝑛(.g‘(Scalar‘𝑃))(1r‘𝑅))) = (𝑛(.g‘𝑃)((algSc‘𝑃)‘(1r‘𝑅)))) | 
| 50 | 42, 29, 44, 49 | syl3anc 1372 | . . . . . . . 8
⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ ℕ0)
→ ((algSc‘𝑃)‘(𝑛(.g‘(Scalar‘𝑃))(1r‘𝑅))) = (𝑛(.g‘𝑃)((algSc‘𝑃)‘(1r‘𝑅)))) | 
| 51 | 40, 50 | eqtrd 2776 | . . . . . . 7
⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ ℕ0)
→ ((algSc‘𝑃)‘(𝑛(.g‘𝑅)(1r‘𝑅))) = (𝑛(.g‘𝑃)((algSc‘𝑃)‘(1r‘𝑅)))) | 
| 52 |  | eqid 2736 | . . . . . . . . 9
⊢
(0g‘𝑃) = (0g‘𝑃) | 
| 53 | 24, 25, 19, 52 | ply1scl0 22294 | . . . . . . . 8
⊢ (𝑅 ∈ Ring →
((algSc‘𝑃)‘(0g‘𝑅)) = (0g‘𝑃)) | 
| 54 | 32, 12, 53 | 3syl 18 | . . . . . . 7
⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ ℕ0)
→ ((algSc‘𝑃)‘(0g‘𝑅)) = (0g‘𝑃)) | 
| 55 | 51, 54 | eqeq12d 2752 | . . . . . 6
⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ ℕ0)
→ (((algSc‘𝑃)‘(𝑛(.g‘𝑅)(1r‘𝑅))) = ((algSc‘𝑃)‘(0g‘𝑅)) ↔ (𝑛(.g‘𝑃)((algSc‘𝑃)‘(1r‘𝑅))) = (0g‘𝑃))) | 
| 56 | 23, 35, 55 | 3bitr2d 307 | . . . . 5
⊢ ((𝑅 ∈ CRing ∧ 𝑛 ∈ ℕ0)
→ ((chr‘𝑅)
∥ 𝑛 ↔ (𝑛(.g‘𝑃)((algSc‘𝑃)‘(1r‘𝑅))) = (0g‘𝑃))) | 
| 57 | 56 | ralrimiva 3145 | . . . 4
⊢ (𝑅 ∈ CRing →
∀𝑛 ∈
ℕ0 ((chr‘𝑅) ∥ 𝑛 ↔ (𝑛(.g‘𝑃)((algSc‘𝑃)‘(1r‘𝑅))) = (0g‘𝑃))) | 
| 58 | 24 | ply1crng 22201 | . . . . . 6
⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing) | 
| 59 | 58 | crnggrpd 20245 | . . . . 5
⊢ (𝑅 ∈ CRing → 𝑃 ∈ Grp) | 
| 60 |  | eqid 2736 | . . . . . . 7
⊢
(Base‘𝑃) =
(Base‘𝑃) | 
| 61 | 24, 25, 13, 60 | ply1sclcl 22290 | . . . . . 6
⊢ ((𝑅 ∈ Ring ∧
(1r‘𝑅)
∈ (Base‘𝑅))
→ ((algSc‘𝑃)‘(1r‘𝑅)) ∈ (Base‘𝑃)) | 
| 62 | 12, 15, 61 | syl2anc 584 | . . . . 5
⊢ (𝑅 ∈ CRing →
((algSc‘𝑃)‘(1r‘𝑅)) ∈ (Base‘𝑃)) | 
| 63 | 60, 1, 47, 52 | odeq 19569 | . . . . 5
⊢ ((𝑃 ∈ Grp ∧
((algSc‘𝑃)‘(1r‘𝑅)) ∈ (Base‘𝑃) ∧ (chr‘𝑅) ∈ ℕ0)
→ ((chr‘𝑅) =
((od‘𝑃)‘((algSc‘𝑃)‘(1r‘𝑅))) ↔ ∀𝑛 ∈ ℕ0
((chr‘𝑅) ∥
𝑛 ↔ (𝑛(.g‘𝑃)((algSc‘𝑃)‘(1r‘𝑅))) = (0g‘𝑃)))) | 
| 64 | 59, 62, 17, 63 | syl3anc 1372 | . . . 4
⊢ (𝑅 ∈ CRing →
((chr‘𝑅) =
((od‘𝑃)‘((algSc‘𝑃)‘(1r‘𝑅))) ↔ ∀𝑛 ∈ ℕ0
((chr‘𝑅) ∥
𝑛 ↔ (𝑛(.g‘𝑃)((algSc‘𝑃)‘(1r‘𝑅))) = (0g‘𝑃)))) | 
| 65 | 57, 64 | mpbird 257 | . . 3
⊢ (𝑅 ∈ CRing →
(chr‘𝑅) =
((od‘𝑃)‘((algSc‘𝑃)‘(1r‘𝑅)))) | 
| 66 | 24, 25, 6, 2 | ply1scl1 22297 | . . . . 5
⊢ (𝑅 ∈ Ring →
((algSc‘𝑃)‘(1r‘𝑅)) = (1r‘𝑃)) | 
| 67 | 66 | fveq2d 6909 | . . . 4
⊢ (𝑅 ∈ Ring →
((od‘𝑃)‘((algSc‘𝑃)‘(1r‘𝑅))) = ((od‘𝑃)‘(1r‘𝑃))) | 
| 68 | 12, 67 | syl 17 | . . 3
⊢ (𝑅 ∈ CRing →
((od‘𝑃)‘((algSc‘𝑃)‘(1r‘𝑅))) = ((od‘𝑃)‘(1r‘𝑃))) | 
| 69 | 65, 68 | eqtr2d 2777 | . 2
⊢ (𝑅 ∈ CRing →
((od‘𝑃)‘(1r‘𝑃)) = (chr‘𝑅)) | 
| 70 | 4, 69 | eqtr3id 2790 | 1
⊢ (𝑅 ∈ CRing →
(chr‘𝑃) =
(chr‘𝑅)) |