Step |
Hyp |
Ref |
Expression |
1 |
|
subrgpropd.1 |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐾 ) ) |
2 |
|
subrgpropd.2 |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐿 ) ) |
3 |
|
subrgpropd.3 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ) |
4 |
|
subrgpropd.4 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( .r ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ) |
5 |
1 2 3 4
|
ringpropd |
⊢ ( 𝜑 → ( 𝐾 ∈ Ring ↔ 𝐿 ∈ Ring ) ) |
6 |
1
|
ineq2d |
⊢ ( 𝜑 → ( 𝑠 ∩ 𝐵 ) = ( 𝑠 ∩ ( Base ‘ 𝐾 ) ) ) |
7 |
|
eqid |
⊢ ( 𝐾 ↾s 𝑠 ) = ( 𝐾 ↾s 𝑠 ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
9 |
7 8
|
ressbas |
⊢ ( 𝑠 ∈ V → ( 𝑠 ∩ ( Base ‘ 𝐾 ) ) = ( Base ‘ ( 𝐾 ↾s 𝑠 ) ) ) |
10 |
9
|
elv |
⊢ ( 𝑠 ∩ ( Base ‘ 𝐾 ) ) = ( Base ‘ ( 𝐾 ↾s 𝑠 ) ) |
11 |
6 10
|
eqtrdi |
⊢ ( 𝜑 → ( 𝑠 ∩ 𝐵 ) = ( Base ‘ ( 𝐾 ↾s 𝑠 ) ) ) |
12 |
2
|
ineq2d |
⊢ ( 𝜑 → ( 𝑠 ∩ 𝐵 ) = ( 𝑠 ∩ ( Base ‘ 𝐿 ) ) ) |
13 |
|
eqid |
⊢ ( 𝐿 ↾s 𝑠 ) = ( 𝐿 ↾s 𝑠 ) |
14 |
|
eqid |
⊢ ( Base ‘ 𝐿 ) = ( Base ‘ 𝐿 ) |
15 |
13 14
|
ressbas |
⊢ ( 𝑠 ∈ V → ( 𝑠 ∩ ( Base ‘ 𝐿 ) ) = ( Base ‘ ( 𝐿 ↾s 𝑠 ) ) ) |
16 |
15
|
elv |
⊢ ( 𝑠 ∩ ( Base ‘ 𝐿 ) ) = ( Base ‘ ( 𝐿 ↾s 𝑠 ) ) |
17 |
12 16
|
eqtrdi |
⊢ ( 𝜑 → ( 𝑠 ∩ 𝐵 ) = ( Base ‘ ( 𝐿 ↾s 𝑠 ) ) ) |
18 |
|
elinel2 |
⊢ ( 𝑥 ∈ ( 𝑠 ∩ 𝐵 ) → 𝑥 ∈ 𝐵 ) |
19 |
|
elinel2 |
⊢ ( 𝑦 ∈ ( 𝑠 ∩ 𝐵 ) → 𝑦 ∈ 𝐵 ) |
20 |
18 19
|
anim12i |
⊢ ( ( 𝑥 ∈ ( 𝑠 ∩ 𝐵 ) ∧ 𝑦 ∈ ( 𝑠 ∩ 𝐵 ) ) → ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) |
21 |
|
eqid |
⊢ ( +g ‘ 𝐾 ) = ( +g ‘ 𝐾 ) |
22 |
7 21
|
ressplusg |
⊢ ( 𝑠 ∈ V → ( +g ‘ 𝐾 ) = ( +g ‘ ( 𝐾 ↾s 𝑠 ) ) ) |
23 |
22
|
elv |
⊢ ( +g ‘ 𝐾 ) = ( +g ‘ ( 𝐾 ↾s 𝑠 ) ) |
24 |
23
|
oveqi |
⊢ ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( +g ‘ ( 𝐾 ↾s 𝑠 ) ) 𝑦 ) |
25 |
|
eqid |
⊢ ( +g ‘ 𝐿 ) = ( +g ‘ 𝐿 ) |
26 |
13 25
|
ressplusg |
⊢ ( 𝑠 ∈ V → ( +g ‘ 𝐿 ) = ( +g ‘ ( 𝐿 ↾s 𝑠 ) ) ) |
27 |
26
|
elv |
⊢ ( +g ‘ 𝐿 ) = ( +g ‘ ( 𝐿 ↾s 𝑠 ) ) |
28 |
27
|
oveqi |
⊢ ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 𝑥 ( +g ‘ ( 𝐿 ↾s 𝑠 ) ) 𝑦 ) |
29 |
3 24 28
|
3eqtr3g |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ ( 𝐾 ↾s 𝑠 ) ) 𝑦 ) = ( 𝑥 ( +g ‘ ( 𝐿 ↾s 𝑠 ) ) 𝑦 ) ) |
30 |
20 29
|
sylan2 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑠 ∩ 𝐵 ) ∧ 𝑦 ∈ ( 𝑠 ∩ 𝐵 ) ) ) → ( 𝑥 ( +g ‘ ( 𝐾 ↾s 𝑠 ) ) 𝑦 ) = ( 𝑥 ( +g ‘ ( 𝐿 ↾s 𝑠 ) ) 𝑦 ) ) |
31 |
|
eqid |
⊢ ( .r ‘ 𝐾 ) = ( .r ‘ 𝐾 ) |
32 |
7 31
|
ressmulr |
⊢ ( 𝑠 ∈ V → ( .r ‘ 𝐾 ) = ( .r ‘ ( 𝐾 ↾s 𝑠 ) ) ) |
33 |
32
|
elv |
⊢ ( .r ‘ 𝐾 ) = ( .r ‘ ( 𝐾 ↾s 𝑠 ) ) |
34 |
33
|
oveqi |
⊢ ( 𝑥 ( .r ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( .r ‘ ( 𝐾 ↾s 𝑠 ) ) 𝑦 ) |
35 |
|
eqid |
⊢ ( .r ‘ 𝐿 ) = ( .r ‘ 𝐿 ) |
36 |
13 35
|
ressmulr |
⊢ ( 𝑠 ∈ V → ( .r ‘ 𝐿 ) = ( .r ‘ ( 𝐿 ↾s 𝑠 ) ) ) |
37 |
36
|
elv |
⊢ ( .r ‘ 𝐿 ) = ( .r ‘ ( 𝐿 ↾s 𝑠 ) ) |
38 |
37
|
oveqi |
⊢ ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = ( 𝑥 ( .r ‘ ( 𝐿 ↾s 𝑠 ) ) 𝑦 ) |
39 |
4 34 38
|
3eqtr3g |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( .r ‘ ( 𝐾 ↾s 𝑠 ) ) 𝑦 ) = ( 𝑥 ( .r ‘ ( 𝐿 ↾s 𝑠 ) ) 𝑦 ) ) |
40 |
20 39
|
sylan2 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑠 ∩ 𝐵 ) ∧ 𝑦 ∈ ( 𝑠 ∩ 𝐵 ) ) ) → ( 𝑥 ( .r ‘ ( 𝐾 ↾s 𝑠 ) ) 𝑦 ) = ( 𝑥 ( .r ‘ ( 𝐿 ↾s 𝑠 ) ) 𝑦 ) ) |
41 |
11 17 30 40
|
ringpropd |
⊢ ( 𝜑 → ( ( 𝐾 ↾s 𝑠 ) ∈ Ring ↔ ( 𝐿 ↾s 𝑠 ) ∈ Ring ) ) |
42 |
5 41
|
anbi12d |
⊢ ( 𝜑 → ( ( 𝐾 ∈ Ring ∧ ( 𝐾 ↾s 𝑠 ) ∈ Ring ) ↔ ( 𝐿 ∈ Ring ∧ ( 𝐿 ↾s 𝑠 ) ∈ Ring ) ) ) |
43 |
1 2
|
eqtr3d |
⊢ ( 𝜑 → ( Base ‘ 𝐾 ) = ( Base ‘ 𝐿 ) ) |
44 |
43
|
sseq2d |
⊢ ( 𝜑 → ( 𝑠 ⊆ ( Base ‘ 𝐾 ) ↔ 𝑠 ⊆ ( Base ‘ 𝐿 ) ) ) |
45 |
1 2 4
|
rngidpropd |
⊢ ( 𝜑 → ( 1r ‘ 𝐾 ) = ( 1r ‘ 𝐿 ) ) |
46 |
45
|
eleq1d |
⊢ ( 𝜑 → ( ( 1r ‘ 𝐾 ) ∈ 𝑠 ↔ ( 1r ‘ 𝐿 ) ∈ 𝑠 ) ) |
47 |
44 46
|
anbi12d |
⊢ ( 𝜑 → ( ( 𝑠 ⊆ ( Base ‘ 𝐾 ) ∧ ( 1r ‘ 𝐾 ) ∈ 𝑠 ) ↔ ( 𝑠 ⊆ ( Base ‘ 𝐿 ) ∧ ( 1r ‘ 𝐿 ) ∈ 𝑠 ) ) ) |
48 |
42 47
|
anbi12d |
⊢ ( 𝜑 → ( ( ( 𝐾 ∈ Ring ∧ ( 𝐾 ↾s 𝑠 ) ∈ Ring ) ∧ ( 𝑠 ⊆ ( Base ‘ 𝐾 ) ∧ ( 1r ‘ 𝐾 ) ∈ 𝑠 ) ) ↔ ( ( 𝐿 ∈ Ring ∧ ( 𝐿 ↾s 𝑠 ) ∈ Ring ) ∧ ( 𝑠 ⊆ ( Base ‘ 𝐿 ) ∧ ( 1r ‘ 𝐿 ) ∈ 𝑠 ) ) ) ) |
49 |
|
eqid |
⊢ ( 1r ‘ 𝐾 ) = ( 1r ‘ 𝐾 ) |
50 |
8 49
|
issubrg |
⊢ ( 𝑠 ∈ ( SubRing ‘ 𝐾 ) ↔ ( ( 𝐾 ∈ Ring ∧ ( 𝐾 ↾s 𝑠 ) ∈ Ring ) ∧ ( 𝑠 ⊆ ( Base ‘ 𝐾 ) ∧ ( 1r ‘ 𝐾 ) ∈ 𝑠 ) ) ) |
51 |
|
eqid |
⊢ ( 1r ‘ 𝐿 ) = ( 1r ‘ 𝐿 ) |
52 |
14 51
|
issubrg |
⊢ ( 𝑠 ∈ ( SubRing ‘ 𝐿 ) ↔ ( ( 𝐿 ∈ Ring ∧ ( 𝐿 ↾s 𝑠 ) ∈ Ring ) ∧ ( 𝑠 ⊆ ( Base ‘ 𝐿 ) ∧ ( 1r ‘ 𝐿 ) ∈ 𝑠 ) ) ) |
53 |
48 50 52
|
3bitr4g |
⊢ ( 𝜑 → ( 𝑠 ∈ ( SubRing ‘ 𝐾 ) ↔ 𝑠 ∈ ( SubRing ‘ 𝐿 ) ) ) |
54 |
53
|
eqrdv |
⊢ ( 𝜑 → ( SubRing ‘ 𝐾 ) = ( SubRing ‘ 𝐿 ) ) |