Step |
Hyp |
Ref |
Expression |
1 |
|
rhmsscmap.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑉 ) |
2 |
|
rhmsscmap.r |
⊢ ( 𝜑 → 𝑅 = ( Ring ∩ 𝑈 ) ) |
3 |
|
inss2 |
⊢ ( Ring ∩ 𝑈 ) ⊆ 𝑈 |
4 |
2 3
|
eqsstrdi |
⊢ ( 𝜑 → 𝑅 ⊆ 𝑈 ) |
5 |
|
eqid |
⊢ ( Base ‘ 𝑎 ) = ( Base ‘ 𝑎 ) |
6 |
|
eqid |
⊢ ( Base ‘ 𝑏 ) = ( Base ‘ 𝑏 ) |
7 |
5 6
|
rhmf |
⊢ ( ℎ ∈ ( 𝑎 RingHom 𝑏 ) → ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) |
8 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) ∧ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) → ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) |
9 |
|
fvex |
⊢ ( Base ‘ 𝑏 ) ∈ V |
10 |
|
fvex |
⊢ ( Base ‘ 𝑎 ) ∈ V |
11 |
9 10
|
pm3.2i |
⊢ ( ( Base ‘ 𝑏 ) ∈ V ∧ ( Base ‘ 𝑎 ) ∈ V ) |
12 |
|
elmapg |
⊢ ( ( ( Base ‘ 𝑏 ) ∈ V ∧ ( Base ‘ 𝑎 ) ∈ V ) → ( ℎ ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ↔ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) ) |
13 |
11 12
|
mp1i |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) ∧ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) → ( ℎ ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ↔ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) ) |
14 |
8 13
|
mpbird |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) ∧ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) → ℎ ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ) |
15 |
14
|
ex |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) → ℎ ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ) ) |
16 |
7 15
|
syl5 |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( ℎ ∈ ( 𝑎 RingHom 𝑏 ) → ℎ ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ) ) |
17 |
16
|
ssrdv |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( 𝑎 RingHom 𝑏 ) ⊆ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ) |
18 |
|
ovres |
⊢ ( ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) → ( 𝑎 ( RingHom ↾ ( 𝑅 × 𝑅 ) ) 𝑏 ) = ( 𝑎 RingHom 𝑏 ) ) |
19 |
18
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( 𝑎 ( RingHom ↾ ( 𝑅 × 𝑅 ) ) 𝑏 ) = ( 𝑎 RingHom 𝑏 ) ) |
20 |
|
eqidd |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ) |
21 |
|
fveq2 |
⊢ ( 𝑦 = 𝑏 → ( Base ‘ 𝑦 ) = ( Base ‘ 𝑏 ) ) |
22 |
|
fveq2 |
⊢ ( 𝑥 = 𝑎 → ( Base ‘ 𝑥 ) = ( Base ‘ 𝑎 ) ) |
23 |
21 22
|
oveqan12rd |
⊢ ( ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) → ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) = ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ) |
24 |
23
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) ∧ ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) ) → ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) = ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ) |
25 |
4
|
sseld |
⊢ ( 𝜑 → ( 𝑎 ∈ 𝑅 → 𝑎 ∈ 𝑈 ) ) |
26 |
25
|
com12 |
⊢ ( 𝑎 ∈ 𝑅 → ( 𝜑 → 𝑎 ∈ 𝑈 ) ) |
27 |
26
|
adantr |
⊢ ( ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) → ( 𝜑 → 𝑎 ∈ 𝑈 ) ) |
28 |
27
|
impcom |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → 𝑎 ∈ 𝑈 ) |
29 |
4
|
sseld |
⊢ ( 𝜑 → ( 𝑏 ∈ 𝑅 → 𝑏 ∈ 𝑈 ) ) |
30 |
29
|
com12 |
⊢ ( 𝑏 ∈ 𝑅 → ( 𝜑 → 𝑏 ∈ 𝑈 ) ) |
31 |
30
|
adantl |
⊢ ( ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) → ( 𝜑 → 𝑏 ∈ 𝑈 ) ) |
32 |
31
|
impcom |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → 𝑏 ∈ 𝑈 ) |
33 |
|
ovexd |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ∈ V ) |
34 |
20 24 28 32 33
|
ovmpod |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( 𝑎 ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) 𝑏 ) = ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ) |
35 |
17 19 34
|
3sstr4d |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( 𝑎 ( RingHom ↾ ( 𝑅 × 𝑅 ) ) 𝑏 ) ⊆ ( 𝑎 ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) 𝑏 ) ) |
36 |
35
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝑅 ∀ 𝑏 ∈ 𝑅 ( 𝑎 ( RingHom ↾ ( 𝑅 × 𝑅 ) ) 𝑏 ) ⊆ ( 𝑎 ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) 𝑏 ) ) |
37 |
|
rhmfn |
⊢ RingHom Fn ( Ring × Ring ) |
38 |
37
|
a1i |
⊢ ( 𝜑 → RingHom Fn ( Ring × Ring ) ) |
39 |
|
inss1 |
⊢ ( Ring ∩ 𝑈 ) ⊆ Ring |
40 |
2 39
|
eqsstrdi |
⊢ ( 𝜑 → 𝑅 ⊆ Ring ) |
41 |
|
xpss12 |
⊢ ( ( 𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring ) → ( 𝑅 × 𝑅 ) ⊆ ( Ring × Ring ) ) |
42 |
40 40 41
|
syl2anc |
⊢ ( 𝜑 → ( 𝑅 × 𝑅 ) ⊆ ( Ring × Ring ) ) |
43 |
|
fnssres |
⊢ ( ( RingHom Fn ( Ring × Ring ) ∧ ( 𝑅 × 𝑅 ) ⊆ ( Ring × Ring ) ) → ( RingHom ↾ ( 𝑅 × 𝑅 ) ) Fn ( 𝑅 × 𝑅 ) ) |
44 |
38 42 43
|
syl2anc |
⊢ ( 𝜑 → ( RingHom ↾ ( 𝑅 × 𝑅 ) ) Fn ( 𝑅 × 𝑅 ) ) |
45 |
|
eqid |
⊢ ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) |
46 |
|
ovex |
⊢ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ∈ V |
47 |
45 46
|
fnmpoi |
⊢ ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) Fn ( 𝑈 × 𝑈 ) |
48 |
47
|
a1i |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) Fn ( 𝑈 × 𝑈 ) ) |
49 |
|
elex |
⊢ ( 𝑈 ∈ 𝑉 → 𝑈 ∈ V ) |
50 |
1 49
|
syl |
⊢ ( 𝜑 → 𝑈 ∈ V ) |
51 |
44 48 50
|
isssc |
⊢ ( 𝜑 → ( ( RingHom ↾ ( 𝑅 × 𝑅 ) ) ⊆cat ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ↔ ( 𝑅 ⊆ 𝑈 ∧ ∀ 𝑎 ∈ 𝑅 ∀ 𝑏 ∈ 𝑅 ( 𝑎 ( RingHom ↾ ( 𝑅 × 𝑅 ) ) 𝑏 ) ⊆ ( 𝑎 ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) 𝑏 ) ) ) ) |
52 |
4 36 51
|
mpbir2and |
⊢ ( 𝜑 → ( RingHom ↾ ( 𝑅 × 𝑅 ) ) ⊆cat ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ) |