Step | Hyp | Ref
| Expression |
1 | | issrg.g |
. . . . . 6
⊢ 𝐺 = (mulGrp‘𝑅) |
2 | 1 | eleq1i 2831 |
. . . . 5
⊢ (𝐺 ∈ Mnd ↔
(mulGrp‘𝑅) ∈
Mnd) |
3 | 2 | bicomi 223 |
. . . 4
⊢
((mulGrp‘𝑅)
∈ Mnd ↔ 𝐺 ∈
Mnd) |
4 | | issrg.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝑅) |
5 | 4 | fvexi 6785 |
. . . . 5
⊢ 𝐵 ∈ V |
6 | | issrg.p |
. . . . . 6
⊢ + =
(+g‘𝑅) |
7 | 6 | fvexi 6785 |
. . . . 5
⊢ + ∈
V |
8 | | issrg.t |
. . . . . . . 8
⊢ · =
(.r‘𝑅) |
9 | 8 | fvexi 6785 |
. . . . . . 7
⊢ · ∈
V |
10 | 9 | a1i 11 |
. . . . . 6
⊢ ((𝑏 = 𝐵 ∧ 𝑝 = + ) → · ∈
V) |
11 | | issrg.0 |
. . . . . . . . 9
⊢ 0 =
(0g‘𝑅) |
12 | 11 | fvexi 6785 |
. . . . . . . 8
⊢ 0 ∈
V |
13 | 12 | a1i 11 |
. . . . . . 7
⊢ (((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) → 0 ∈
V) |
14 | | simplll 772 |
. . . . . . . 8
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → 𝑏 = 𝐵) |
15 | | simplr 766 |
. . . . . . . . . . . . . 14
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → 𝑡 = · ) |
16 | | eqidd 2741 |
. . . . . . . . . . . . . 14
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → 𝑥 = 𝑥) |
17 | | simpllr 773 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → 𝑝 = + ) |
18 | 17 | oveqd 7288 |
. . . . . . . . . . . . . 14
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (𝑦𝑝𝑧) = (𝑦 + 𝑧)) |
19 | 15, 16, 18 | oveq123d 7292 |
. . . . . . . . . . . . 13
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (𝑥𝑡(𝑦𝑝𝑧)) = (𝑥 · (𝑦 + 𝑧))) |
20 | 15 | oveqd 7288 |
. . . . . . . . . . . . . 14
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (𝑥𝑡𝑦) = (𝑥 · 𝑦)) |
21 | 15 | oveqd 7288 |
. . . . . . . . . . . . . 14
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (𝑥𝑡𝑧) = (𝑥 · 𝑧)) |
22 | 17, 20, 21 | oveq123d 7292 |
. . . . . . . . . . . . 13
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))) |
23 | 19, 22 | eqeq12d 2756 |
. . . . . . . . . . . 12
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ↔ (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))) |
24 | 17 | oveqd 7288 |
. . . . . . . . . . . . . 14
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (𝑥𝑝𝑦) = (𝑥 + 𝑦)) |
25 | | eqidd 2741 |
. . . . . . . . . . . . . 14
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → 𝑧 = 𝑧) |
26 | 15, 24, 25 | oveq123d 7292 |
. . . . . . . . . . . . 13
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥 + 𝑦) · 𝑧)) |
27 | 15 | oveqd 7288 |
. . . . . . . . . . . . . 14
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (𝑦𝑡𝑧) = (𝑦 · 𝑧)) |
28 | 17, 21, 27 | oveq123d 7292 |
. . . . . . . . . . . . 13
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧)) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) |
29 | 26, 28 | eqeq12d 2756 |
. . . . . . . . . . . 12
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧)) ↔ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))) |
30 | 23, 29 | anbi12d 631 |
. . . . . . . . . . 11
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ↔ ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))) |
31 | 14, 30 | raleqbidv 3335 |
. . . . . . . . . 10
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ↔ ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))) |
32 | 14, 31 | raleqbidv 3335 |
. . . . . . . . 9
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))) |
33 | | simpr 485 |
. . . . . . . . . . . 12
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → 𝑛 = 0 ) |
34 | 15, 33, 16 | oveq123d 7292 |
. . . . . . . . . . 11
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (𝑛𝑡𝑥) = ( 0 · 𝑥)) |
35 | 34, 33 | eqeq12d 2756 |
. . . . . . . . . 10
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → ((𝑛𝑡𝑥) = 𝑛 ↔ ( 0 · 𝑥) = 0 )) |
36 | 15, 16, 33 | oveq123d 7292 |
. . . . . . . . . . 11
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (𝑥𝑡𝑛) = (𝑥 · 0 )) |
37 | 36, 33 | eqeq12d 2756 |
. . . . . . . . . 10
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → ((𝑥𝑡𝑛) = 𝑛 ↔ (𝑥 · 0 ) = 0 )) |
38 | 35, 37 | anbi12d 631 |
. . . . . . . . 9
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛) ↔ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))) |
39 | 32, 38 | anbi12d 631 |
. . . . . . . 8
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → ((∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 )))) |
40 | 14, 39 | raleqbidv 3335 |
. . . . . . 7
⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 )))) |
41 | 13, 40 | sbcied 3765 |
. . . . . 6
⊢ (((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) → ([
0 / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 )))) |
42 | 10, 41 | sbcied 3765 |
. . . . 5
⊢ ((𝑏 = 𝐵 ∧ 𝑝 = + ) → ([ · /
𝑡][ 0 / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 )))) |
43 | 5, 7, 42 | sbc2ie 3804 |
. . . 4
⊢
([𝐵 / 𝑏][ + / 𝑝][ · /
𝑡][ 0 / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))) |
44 | 3, 43 | anbi12i 627 |
. . 3
⊢
(((mulGrp‘𝑅)
∈ Mnd ∧ [𝐵
/ 𝑏][ + / 𝑝][ · /
𝑡][ 0 / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛))) ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 )))) |
45 | 44 | anbi2i 623 |
. 2
⊢ ((𝑅 ∈ CMnd ∧
((mulGrp‘𝑅) ∈
Mnd ∧ [𝐵 / 𝑏][ + / 𝑝][ · /
𝑡][ 0 / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))) ↔ (𝑅 ∈ CMnd ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))))) |
46 | | fveq2 6771 |
. . . . 5
⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅)) |
47 | 46 | eleq1d 2825 |
. . . 4
⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ Mnd ↔ (mulGrp‘𝑅) ∈ Mnd)) |
48 | | fveq2 6771 |
. . . . . 6
⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) |
49 | 48, 4 | eqtr4di 2798 |
. . . . 5
⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵) |
50 | | fveq2 6771 |
. . . . . . 7
⊢ (𝑟 = 𝑅 → (+g‘𝑟) = (+g‘𝑅)) |
51 | 50, 6 | eqtr4di 2798 |
. . . . . 6
⊢ (𝑟 = 𝑅 → (+g‘𝑟) = + ) |
52 | | fveq2 6771 |
. . . . . . . 8
⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅)) |
53 | 52, 8 | eqtr4di 2798 |
. . . . . . 7
⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · ) |
54 | | fveq2 6771 |
. . . . . . . . 9
⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅)) |
55 | 54, 11 | eqtr4di 2798 |
. . . . . . . 8
⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 ) |
56 | 55 | sbceq1d 3725 |
. . . . . . 7
⊢ (𝑟 = 𝑅 → ([(0g‘𝑟) / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ [ 0 / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))) |
57 | 53, 56 | sbceqbid 3727 |
. . . . . 6
⊢ (𝑟 = 𝑅 → ([(.r‘𝑟) / 𝑡][(0g‘𝑟) / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ [ · / 𝑡][ 0 / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))) |
58 | 51, 57 | sbceqbid 3727 |
. . . . 5
⊢ (𝑟 = 𝑅 → ([(+g‘𝑟) / 𝑝][(.r‘𝑟) / 𝑡][(0g‘𝑟) / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ [ + / 𝑝][ · / 𝑡][ 0 / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))) |
59 | 49, 58 | sbceqbid 3727 |
. . . 4
⊢ (𝑟 = 𝑅 → ([(Base‘𝑟) / 𝑏][(+g‘𝑟) / 𝑝][(.r‘𝑟) / 𝑡][(0g‘𝑟) / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ [𝐵 / 𝑏][ + / 𝑝][ · / 𝑡][ 0 / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))) |
60 | 47, 59 | anbi12d 631 |
. . 3
⊢ (𝑟 = 𝑅 → (((mulGrp‘𝑟) ∈ Mnd ∧ [(Base‘𝑟) / 𝑏][(+g‘𝑟) / 𝑝][(.r‘𝑟) / 𝑡][(0g‘𝑟) / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛))) ↔ ((mulGrp‘𝑅) ∈ Mnd ∧ [𝐵 / 𝑏][ + / 𝑝][ · / 𝑡][ 0 / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛))))) |
61 | | df-srg 19740 |
. . 3
⊢ SRing =
{𝑟 ∈ CMnd ∣
((mulGrp‘𝑟) ∈
Mnd ∧ [(Base‘𝑟) / 𝑏][(+g‘𝑟) / 𝑝][(.r‘𝑟) / 𝑡][(0g‘𝑟) / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))} |
62 | 60, 61 | elrab2 3629 |
. 2
⊢ (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧
((mulGrp‘𝑅) ∈
Mnd ∧ [𝐵 / 𝑏][ + / 𝑝][ · /
𝑡][ 0 / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛))))) |
63 | | 3anass 1094 |
. 2
⊢ ((𝑅 ∈ CMnd ∧ 𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))) ↔ (𝑅 ∈ CMnd ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))))) |
64 | 45, 62, 63 | 3bitr4i 303 |
1
⊢ (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ 𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 )))) |