Step |
Hyp |
Ref |
Expression |
1 |
|
submrcl |
⊢ ( 𝐴 ∈ ( SubMnd ‘ 𝑀 ) → 𝑀 ∈ Mnd ) |
2 |
|
ssinss1 |
⊢ ( 𝐴 ⊆ ( Base ‘ 𝑀 ) → ( 𝐴 ∩ 𝐵 ) ⊆ ( Base ‘ 𝑀 ) ) |
3 |
2
|
3ad2ant1 |
⊢ ( ( 𝐴 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐴 ∧ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐴 ) → ( 𝐴 ∩ 𝐵 ) ⊆ ( Base ‘ 𝑀 ) ) |
4 |
3
|
ad2antrl |
⊢ ( ( 𝑀 ∈ Mnd ∧ ( ( 𝐴 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐴 ∧ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐴 ) ∧ ( 𝐵 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ) ) ) → ( 𝐴 ∩ 𝐵 ) ⊆ ( Base ‘ 𝑀 ) ) |
5 |
|
elin |
⊢ ( ( 0g ‘ 𝑀 ) ∈ ( 𝐴 ∩ 𝐵 ) ↔ ( ( 0g ‘ 𝑀 ) ∈ 𝐴 ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 ) ) |
6 |
5
|
simplbi2com |
⊢ ( ( 0g ‘ 𝑀 ) ∈ 𝐵 → ( ( 0g ‘ 𝑀 ) ∈ 𝐴 → ( 0g ‘ 𝑀 ) ∈ ( 𝐴 ∩ 𝐵 ) ) ) |
7 |
6
|
3ad2ant2 |
⊢ ( ( 𝐵 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ) → ( ( 0g ‘ 𝑀 ) ∈ 𝐴 → ( 0g ‘ 𝑀 ) ∈ ( 𝐴 ∩ 𝐵 ) ) ) |
8 |
7
|
com12 |
⊢ ( ( 0g ‘ 𝑀 ) ∈ 𝐴 → ( ( 𝐵 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ) → ( 0g ‘ 𝑀 ) ∈ ( 𝐴 ∩ 𝐵 ) ) ) |
9 |
8
|
3ad2ant2 |
⊢ ( ( 𝐴 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐴 ∧ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐴 ) → ( ( 𝐵 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ) → ( 0g ‘ 𝑀 ) ∈ ( 𝐴 ∩ 𝐵 ) ) ) |
10 |
9
|
imp |
⊢ ( ( ( 𝐴 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐴 ∧ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐴 ) ∧ ( 𝐵 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ) ) → ( 0g ‘ 𝑀 ) ∈ ( 𝐴 ∩ 𝐵 ) ) |
11 |
10
|
adantl |
⊢ ( ( 𝑀 ∈ Mnd ∧ ( ( 𝐴 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐴 ∧ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐴 ) ∧ ( 𝐵 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ) ) ) → ( 0g ‘ 𝑀 ) ∈ ( 𝐴 ∩ 𝐵 ) ) |
12 |
|
elin |
⊢ ( 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ) |
13 |
|
elin |
⊢ ( 𝑦 ∈ ( 𝐴 ∩ 𝐵 ) ↔ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) |
14 |
12 13
|
anbi12i |
⊢ ( ( 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) ∧ 𝑦 ∈ ( 𝐴 ∩ 𝐵 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) ) |
15 |
|
oveq1 |
⊢ ( 𝑎 = 𝑥 → ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) = ( 𝑥 ( +g ‘ 𝑀 ) 𝑏 ) ) |
16 |
15
|
eleq1d |
⊢ ( 𝑎 = 𝑥 → ( ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐴 ↔ ( 𝑥 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐴 ) ) |
17 |
|
oveq2 |
⊢ ( 𝑏 = 𝑦 → ( 𝑥 ( +g ‘ 𝑀 ) 𝑏 ) = ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ) |
18 |
17
|
eleq1d |
⊢ ( 𝑏 = 𝑦 → ( ( 𝑥 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐴 ↔ ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝐴 ) ) |
19 |
|
simpl |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ 𝐴 ) |
20 |
19
|
adantr |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑥 ∈ 𝐴 ) |
21 |
|
eqidd |
⊢ ( ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑎 = 𝑥 ) → 𝐴 = 𝐴 ) |
22 |
|
simpl |
⊢ ( ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → 𝑦 ∈ 𝐴 ) |
23 |
22
|
adantl |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑦 ∈ 𝐴 ) |
24 |
16 18 20 21 23
|
rspc2vd |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐴 → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝐴 ) ) |
25 |
24
|
com12 |
⊢ ( ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐴 → ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝐴 ) ) |
26 |
25
|
3ad2ant3 |
⊢ ( ( 𝐴 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐴 ∧ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐴 ) → ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝐴 ) ) |
27 |
26
|
ad2antrl |
⊢ ( ( 𝑀 ∈ Mnd ∧ ( ( 𝐴 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐴 ∧ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐴 ) ∧ ( 𝐵 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ) ) ) → ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝐴 ) ) |
28 |
27
|
imp |
⊢ ( ( ( 𝑀 ∈ Mnd ∧ ( ( 𝐴 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐴 ∧ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐴 ) ∧ ( 𝐵 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ) ) ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) ) → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝐴 ) |
29 |
15
|
eleq1d |
⊢ ( 𝑎 = 𝑥 → ( ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ↔ ( 𝑥 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ) ) |
30 |
17
|
eleq1d |
⊢ ( 𝑏 = 𝑦 → ( ( 𝑥 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ↔ ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝐵 ) ) |
31 |
|
simpr |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ 𝐵 ) |
32 |
31
|
adantr |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑥 ∈ 𝐵 ) |
33 |
|
eqidd |
⊢ ( ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝑎 = 𝑥 ) → 𝐵 = 𝐵 ) |
34 |
|
simpr |
⊢ ( ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → 𝑦 ∈ 𝐵 ) |
35 |
34
|
adantl |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → 𝑦 ∈ 𝐵 ) |
36 |
29 30 32 33 35
|
rspc2vd |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝐵 ) ) |
37 |
36
|
com12 |
⊢ ( ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 → ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝐵 ) ) |
38 |
37
|
3ad2ant3 |
⊢ ( ( 𝐵 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ) → ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝐵 ) ) |
39 |
38
|
adantl |
⊢ ( ( ( 𝐴 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐴 ∧ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐴 ) ∧ ( 𝐵 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ) ) → ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝐵 ) ) |
40 |
39
|
adantl |
⊢ ( ( 𝑀 ∈ Mnd ∧ ( ( 𝐴 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐴 ∧ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐴 ) ∧ ( 𝐵 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ) ) ) → ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝐵 ) ) |
41 |
40
|
imp |
⊢ ( ( ( 𝑀 ∈ Mnd ∧ ( ( 𝐴 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐴 ∧ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐴 ) ∧ ( 𝐵 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ) ) ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) ) → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝐵 ) |
42 |
28 41
|
elind |
⊢ ( ( ( 𝑀 ∈ Mnd ∧ ( ( 𝐴 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐴 ∧ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐴 ) ∧ ( 𝐵 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ) ) ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) ) → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ ( 𝐴 ∩ 𝐵 ) ) |
43 |
42
|
ex |
⊢ ( ( 𝑀 ∈ Mnd ∧ ( ( 𝐴 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐴 ∧ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐴 ) ∧ ( 𝐵 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ) ) ) → ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ ( 𝐴 ∩ 𝐵 ) ) ) |
44 |
14 43
|
syl5bi |
⊢ ( ( 𝑀 ∈ Mnd ∧ ( ( 𝐴 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐴 ∧ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐴 ) ∧ ( 𝐵 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ) ) ) → ( ( 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) ∧ 𝑦 ∈ ( 𝐴 ∩ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ ( 𝐴 ∩ 𝐵 ) ) ) |
45 |
44
|
ralrimivv |
⊢ ( ( 𝑀 ∈ Mnd ∧ ( ( 𝐴 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐴 ∧ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐴 ) ∧ ( 𝐵 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ) ) ) → ∀ 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) ∀ 𝑦 ∈ ( 𝐴 ∩ 𝐵 ) ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ ( 𝐴 ∩ 𝐵 ) ) |
46 |
4 11 45
|
3jca |
⊢ ( ( 𝑀 ∈ Mnd ∧ ( ( 𝐴 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐴 ∧ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐴 ) ∧ ( 𝐵 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ) ) ) → ( ( 𝐴 ∩ 𝐵 ) ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ ( 𝐴 ∩ 𝐵 ) ∧ ∀ 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) ∀ 𝑦 ∈ ( 𝐴 ∩ 𝐵 ) ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ ( 𝐴 ∩ 𝐵 ) ) ) |
47 |
46
|
ex |
⊢ ( 𝑀 ∈ Mnd → ( ( ( 𝐴 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐴 ∧ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐴 ) ∧ ( 𝐵 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ) ) → ( ( 𝐴 ∩ 𝐵 ) ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ ( 𝐴 ∩ 𝐵 ) ∧ ∀ 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) ∀ 𝑦 ∈ ( 𝐴 ∩ 𝐵 ) ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ ( 𝐴 ∩ 𝐵 ) ) ) ) |
48 |
|
eqid |
⊢ ( Base ‘ 𝑀 ) = ( Base ‘ 𝑀 ) |
49 |
|
eqid |
⊢ ( 0g ‘ 𝑀 ) = ( 0g ‘ 𝑀 ) |
50 |
|
eqid |
⊢ ( +g ‘ 𝑀 ) = ( +g ‘ 𝑀 ) |
51 |
48 49 50
|
issubm |
⊢ ( 𝑀 ∈ Mnd → ( 𝐴 ∈ ( SubMnd ‘ 𝑀 ) ↔ ( 𝐴 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐴 ∧ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐴 ) ) ) |
52 |
48 49 50
|
issubm |
⊢ ( 𝑀 ∈ Mnd → ( 𝐵 ∈ ( SubMnd ‘ 𝑀 ) ↔ ( 𝐵 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ) ) ) |
53 |
51 52
|
anbi12d |
⊢ ( 𝑀 ∈ Mnd → ( ( 𝐴 ∈ ( SubMnd ‘ 𝑀 ) ∧ 𝐵 ∈ ( SubMnd ‘ 𝑀 ) ) ↔ ( ( 𝐴 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐴 ∧ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐴 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐴 ) ∧ ( 𝐵 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ) ) ) ) |
54 |
48 49 50
|
issubm |
⊢ ( 𝑀 ∈ Mnd → ( ( 𝐴 ∩ 𝐵 ) ∈ ( SubMnd ‘ 𝑀 ) ↔ ( ( 𝐴 ∩ 𝐵 ) ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ ( 𝐴 ∩ 𝐵 ) ∧ ∀ 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) ∀ 𝑦 ∈ ( 𝐴 ∩ 𝐵 ) ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ ( 𝐴 ∩ 𝐵 ) ) ) ) |
55 |
47 53 54
|
3imtr4d |
⊢ ( 𝑀 ∈ Mnd → ( ( 𝐴 ∈ ( SubMnd ‘ 𝑀 ) ∧ 𝐵 ∈ ( SubMnd ‘ 𝑀 ) ) → ( 𝐴 ∩ 𝐵 ) ∈ ( SubMnd ‘ 𝑀 ) ) ) |
56 |
55
|
expd |
⊢ ( 𝑀 ∈ Mnd → ( 𝐴 ∈ ( SubMnd ‘ 𝑀 ) → ( 𝐵 ∈ ( SubMnd ‘ 𝑀 ) → ( 𝐴 ∩ 𝐵 ) ∈ ( SubMnd ‘ 𝑀 ) ) ) ) |
57 |
1 56
|
mpcom |
⊢ ( 𝐴 ∈ ( SubMnd ‘ 𝑀 ) → ( 𝐵 ∈ ( SubMnd ‘ 𝑀 ) → ( 𝐴 ∩ 𝐵 ) ∈ ( SubMnd ‘ 𝑀 ) ) ) |
58 |
57
|
imp |
⊢ ( ( 𝐴 ∈ ( SubMnd ‘ 𝑀 ) ∧ 𝐵 ∈ ( SubMnd ‘ 𝑀 ) ) → ( 𝐴 ∩ 𝐵 ) ∈ ( SubMnd ‘ 𝑀 ) ) |