| Step | Hyp | Ref
| Expression |
| 1 | | rloc1r.i |
. 2
⊢ 𝐼 = [〈 1 , 1 〉] ∼ |
| 2 | | eqid 2737 |
. . . . 5
⊢
(Base‘𝑅) =
(Base‘𝑅) |
| 3 | | eqid 2737 |
. . . . 5
⊢
(.r‘𝑅) = (.r‘𝑅) |
| 4 | | eqid 2737 |
. . . . 5
⊢
(+g‘𝑅) = (+g‘𝑅) |
| 5 | | rloc0g.3 |
. . . . 5
⊢ 𝐿 = (𝑅 RLocal 𝑆) |
| 6 | | rloc0g.4 |
. . . . 5
⊢ ∼ =
(𝑅 ~RL
𝑆) |
| 7 | | rloc0g.5 |
. . . . 5
⊢ (𝜑 → 𝑅 ∈ CRing) |
| 8 | | rloc0g.6 |
. . . . 5
⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅))) |
| 9 | 2, 3, 4, 5, 6, 7, 8 | rloccring 33274 |
. . . 4
⊢ (𝜑 → 𝐿 ∈ CRing) |
| 10 | 9 | crngringd 20243 |
. . 3
⊢ (𝜑 → 𝐿 ∈ Ring) |
| 11 | | eqid 2737 |
. . . . . . . . . 10
⊢
(mulGrp‘𝑅) =
(mulGrp‘𝑅) |
| 12 | 11, 2 | mgpbas 20142 |
. . . . . . . . 9
⊢
(Base‘𝑅) =
(Base‘(mulGrp‘𝑅)) |
| 13 | 12 | submss 18822 |
. . . . . . . 8
⊢ (𝑆 ∈
(SubMnd‘(mulGrp‘𝑅)) → 𝑆 ⊆ (Base‘𝑅)) |
| 14 | 8, 13 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑅)) |
| 15 | | rloc0g.2 |
. . . . . . . . . 10
⊢ 1 =
(1r‘𝑅) |
| 16 | 11, 15 | ringidval 20180 |
. . . . . . . . 9
⊢ 1 =
(0g‘(mulGrp‘𝑅)) |
| 17 | 16 | subm0cl 18824 |
. . . . . . . 8
⊢ (𝑆 ∈
(SubMnd‘(mulGrp‘𝑅)) → 1 ∈ 𝑆) |
| 18 | 8, 17 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 1 ∈ 𝑆) |
| 19 | 14, 18 | sseldd 3984 |
. . . . . 6
⊢ (𝜑 → 1 ∈ (Base‘𝑅)) |
| 20 | 19, 18 | opelxpd 5724 |
. . . . 5
⊢ (𝜑 → 〈 1 , 1 〉 ∈
((Base‘𝑅) ×
𝑆)) |
| 21 | 6 | ovexi 7465 |
. . . . . 6
⊢ ∼ ∈
V |
| 22 | 21 | ecelqsi 8813 |
. . . . 5
⊢ (〈
1 , 1 〉 ∈
((Base‘𝑅) ×
𝑆) → [〈 1 , 1 〉] ∼ ∈
(((Base‘𝑅) ×
𝑆) / ∼
)) |
| 23 | 20, 22 | syl 17 |
. . . 4
⊢ (𝜑 → [〈 1 , 1 〉] ∼ ∈
(((Base‘𝑅) ×
𝑆) / ∼
)) |
| 24 | | rloc0g.1 |
. . . . 5
⊢ 0 =
(0g‘𝑅) |
| 25 | | eqid 2737 |
. . . . 5
⊢
(-g‘𝑅) = (-g‘𝑅) |
| 26 | | eqid 2737 |
. . . . 5
⊢
((Base‘𝑅)
× 𝑆) =
((Base‘𝑅) ×
𝑆) |
| 27 | 2, 24, 3, 25, 26, 5, 6, 7, 14 | rlocbas 33271 |
. . . 4
⊢ (𝜑 → (((Base‘𝑅) × 𝑆) / ∼ ) =
(Base‘𝐿)) |
| 28 | 23, 27 | eleqtrd 2843 |
. . 3
⊢ (𝜑 → [〈 1 , 1 〉] ∼ ∈
(Base‘𝐿)) |
| 29 | 7 | ad4antr 732 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) → 𝑅 ∈ CRing) |
| 30 | 8 | ad4antr 732 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) → 𝑆 ∈
(SubMnd‘(mulGrp‘𝑅))) |
| 31 | 19 | ad4antr 732 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) → 1 ∈
(Base‘𝑅)) |
| 32 | | simpllr 776 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) → 𝑎 ∈ (Base‘𝑅)) |
| 33 | 30, 17 | syl 17 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) → 1 ∈ 𝑆) |
| 34 | | simplr 769 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) → 𝑏 ∈ 𝑆) |
| 35 | | eqid 2737 |
. . . . . . . . 9
⊢
(.r‘𝐿) = (.r‘𝐿) |
| 36 | 2, 3, 4, 5, 6, 29,
30, 31, 32, 33, 34, 35 | rlocmulval 33273 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) → ([〈
1 , 1 〉] ∼
(.r‘𝐿)[〈𝑎, 𝑏〉] ∼ ) = [〈( 1
(.r‘𝑅)𝑎), ( 1 (.r‘𝑅)𝑏)〉] ∼ ) |
| 37 | 29 | crngringd 20243 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) → 𝑅 ∈ Ring) |
| 38 | 2, 3, 15, 37, 32 | ringlidmd 20269 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) → ( 1
(.r‘𝑅)𝑎) = 𝑎) |
| 39 | 30, 13 | syl 17 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) → 𝑆 ⊆ (Base‘𝑅)) |
| 40 | 39, 34 | sseldd 3984 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) → 𝑏 ∈ (Base‘𝑅)) |
| 41 | 2, 3, 15, 37, 40 | ringlidmd 20269 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) → ( 1
(.r‘𝑅)𝑏) = 𝑏) |
| 42 | 38, 41 | opeq12d 4881 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) → 〈(
1
(.r‘𝑅)𝑎), ( 1 (.r‘𝑅)𝑏)〉 = 〈𝑎, 𝑏〉) |
| 43 | 42 | eceq1d 8785 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) → [〈(
1
(.r‘𝑅)𝑎), ( 1 (.r‘𝑅)𝑏)〉] ∼ = [〈𝑎, 𝑏〉] ∼ ) |
| 44 | 36, 43 | eqtrd 2777 |
. . . . . . 7
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) → ([〈
1 , 1 〉] ∼
(.r‘𝐿)[〈𝑎, 𝑏〉] ∼ ) = [〈𝑎, 𝑏〉] ∼ ) |
| 45 | | simpr 484 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) → 𝑥 = [〈𝑎, 𝑏〉] ∼ ) |
| 46 | 45 | oveq2d 7447 |
. . . . . . 7
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) → ([〈
1 , 1 〉] ∼
(.r‘𝐿)𝑥) = ([〈 1 , 1 〉] ∼
(.r‘𝐿)[〈𝑎, 𝑏〉] ∼ )) |
| 47 | 44, 46, 45 | 3eqtr4d 2787 |
. . . . . 6
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) → ([〈
1 , 1 〉] ∼
(.r‘𝐿)𝑥) = 𝑥) |
| 48 | 27 | eqcomd 2743 |
. . . . . . . . 9
⊢ (𝜑 → (Base‘𝐿) = (((Base‘𝑅) × 𝑆) / ∼ )) |
| 49 | 48 | eleq2d 2827 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (Base‘𝐿) ↔ 𝑥 ∈ (((Base‘𝑅) × 𝑆) / ∼
))) |
| 50 | 49 | biimpa 476 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) → 𝑥 ∈ (((Base‘𝑅) × 𝑆) / ∼ )) |
| 51 | 50 | elrlocbasi 33270 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) → ∃𝑎 ∈ (Base‘𝑅)∃𝑏 ∈ 𝑆 𝑥 = [〈𝑎, 𝑏〉] ∼ ) |
| 52 | 47, 51 | r19.29vva 3216 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) → ([〈 1 , 1 〉] ∼
(.r‘𝐿)𝑥) = 𝑥) |
| 53 | 2, 3, 4, 5, 6, 29,
30, 32, 31, 34, 33, 35 | rlocmulval 33273 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) →
([〈𝑎, 𝑏〉] ∼
(.r‘𝐿)[〈 1 , 1 〉] ∼ ) = [〈(𝑎(.r‘𝑅) 1 ), (𝑏(.r‘𝑅) 1 )〉] ∼
) |
| 54 | 2, 3, 15, 37, 32 | ringridmd 20270 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) → (𝑎(.r‘𝑅) 1 ) = 𝑎) |
| 55 | 2, 3, 15, 37, 40 | ringridmd 20270 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) → (𝑏(.r‘𝑅) 1 ) = 𝑏) |
| 56 | 54, 55 | opeq12d 4881 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) →
〈(𝑎(.r‘𝑅) 1 ), (𝑏(.r‘𝑅) 1 )〉 = 〈𝑎, 𝑏〉) |
| 57 | 56 | eceq1d 8785 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) →
[〈(𝑎(.r‘𝑅) 1 ), (𝑏(.r‘𝑅) 1 )〉] ∼ =
[〈𝑎, 𝑏〉] ∼ ) |
| 58 | 53, 57 | eqtrd 2777 |
. . . . . . 7
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) →
([〈𝑎, 𝑏〉] ∼
(.r‘𝐿)[〈 1 , 1 〉] ∼ ) = [〈𝑎, 𝑏〉] ∼ ) |
| 59 | 45 | oveq1d 7446 |
. . . . . . 7
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) → (𝑥(.r‘𝐿)[〈 1 , 1 〉] ∼ ) = ([〈𝑎, 𝑏〉] ∼
(.r‘𝐿)[〈 1 , 1 〉] ∼ )) |
| 60 | 58, 59, 45 | 3eqtr4d 2787 |
. . . . . 6
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [〈𝑎, 𝑏〉] ∼ ) → (𝑥(.r‘𝐿)[〈 1 , 1 〉] ∼ ) = 𝑥) |
| 61 | 60, 51 | r19.29vva 3216 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) → (𝑥(.r‘𝐿)[〈 1 , 1 〉] ∼ ) = 𝑥) |
| 62 | 52, 61 | jca 511 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) → (([〈 1 , 1 〉] ∼
(.r‘𝐿)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐿)[〈 1 , 1 〉] ∼ ) = 𝑥)) |
| 63 | 62 | ralrimiva 3146 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐿)(([〈 1 , 1 〉] ∼
(.r‘𝐿)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐿)[〈 1 , 1 〉] ∼ ) = 𝑥)) |
| 64 | | eqid 2737 |
. . . . 5
⊢
(Base‘𝐿) =
(Base‘𝐿) |
| 65 | | eqid 2737 |
. . . . 5
⊢
(1r‘𝐿) = (1r‘𝐿) |
| 66 | 64, 35, 65 | isringid 20268 |
. . . 4
⊢ (𝐿 ∈ Ring → (([〈
1 , 1 〉] ∼ ∈
(Base‘𝐿) ∧
∀𝑥 ∈
(Base‘𝐿)(([〈
1 , 1 〉] ∼
(.r‘𝐿)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐿)[〈 1 , 1 〉] ∼ ) = 𝑥)) ↔
(1r‘𝐿) =
[〈 1
, 1
〉] ∼ )) |
| 67 | 66 | biimpa 476 |
. . 3
⊢ ((𝐿 ∈ Ring ∧ ([〈
1 , 1 〉] ∼ ∈
(Base‘𝐿) ∧
∀𝑥 ∈
(Base‘𝐿)(([〈
1 , 1 〉] ∼
(.r‘𝐿)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐿)[〈 1 , 1 〉] ∼ ) = 𝑥))) →
(1r‘𝐿) =
[〈 1
, 1
〉] ∼ ) |
| 68 | 10, 28, 63, 67 | syl12anc 837 |
. 2
⊢ (𝜑 → (1r‘𝐿) = [〈 1 , 1 〉] ∼ ) |
| 69 | 1, 68 | eqtr4id 2796 |
1
⊢ (𝜑 → 𝐼 = (1r‘𝐿)) |