Step |
Hyp |
Ref |
Expression |
1 |
|
rhmpsr1.p |
⊢ 𝑃 = ( PwSer1 ‘ 𝑅 ) |
2 |
|
rhmpsr1.q |
⊢ 𝑄 = ( PwSer1 ‘ 𝑆 ) |
3 |
|
rhmpsr1.b |
⊢ 𝐵 = ( Base ‘ 𝑃 ) |
4 |
|
rhmpsr1.f |
⊢ 𝐹 = ( 𝑝 ∈ 𝐵 ↦ ( 𝐻 ∘ 𝑝 ) ) |
5 |
|
rhmpsr1.h |
⊢ ( 𝜑 → 𝐻 ∈ ( 𝑅 RingHom 𝑆 ) ) |
6 |
|
eqid |
⊢ ( 1o mPwSer 𝑅 ) = ( 1o mPwSer 𝑅 ) |
7 |
|
eqid |
⊢ ( 1o mPwSer 𝑆 ) = ( 1o mPwSer 𝑆 ) |
8 |
1 3 6
|
psr1bas2 |
⊢ 𝐵 = ( Base ‘ ( 1o mPwSer 𝑅 ) ) |
9 |
|
1oex |
⊢ 1o ∈ V |
10 |
9
|
a1i |
⊢ ( 𝜑 → 1o ∈ V ) |
11 |
6 7 8 4 10 5
|
rhmpsr |
⊢ ( 𝜑 → 𝐹 ∈ ( ( 1o mPwSer 𝑅 ) RingHom ( 1o mPwSer 𝑆 ) ) ) |
12 |
|
eqid |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 ) |
13 |
1 12 6
|
psr1bas2 |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ ( 1o mPwSer 𝑅 ) ) |
14 |
13
|
a1i |
⊢ ( 𝜑 → ( Base ‘ 𝑃 ) = ( Base ‘ ( 1o mPwSer 𝑅 ) ) ) |
15 |
|
eqid |
⊢ ( Base ‘ 𝑄 ) = ( Base ‘ 𝑄 ) |
16 |
2 15 7
|
psr1bas2 |
⊢ ( Base ‘ 𝑄 ) = ( Base ‘ ( 1o mPwSer 𝑆 ) ) |
17 |
16
|
a1i |
⊢ ( 𝜑 → ( Base ‘ 𝑄 ) = ( Base ‘ ( 1o mPwSer 𝑆 ) ) ) |
18 |
|
eqidd |
⊢ ( 𝜑 → ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 ) ) |
19 |
|
eqidd |
⊢ ( 𝜑 → ( Base ‘ 𝑄 ) = ( Base ‘ 𝑄 ) ) |
20 |
|
eqid |
⊢ ( +g ‘ 𝑃 ) = ( +g ‘ 𝑃 ) |
21 |
1 6 20
|
psr1plusg |
⊢ ( +g ‘ 𝑃 ) = ( +g ‘ ( 1o mPwSer 𝑅 ) ) |
22 |
21
|
eqcomi |
⊢ ( +g ‘ ( 1o mPwSer 𝑅 ) ) = ( +g ‘ 𝑃 ) |
23 |
22
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑃 ) ∧ 𝑦 ∈ ( Base ‘ 𝑃 ) ) ) → ( +g ‘ ( 1o mPwSer 𝑅 ) ) = ( +g ‘ 𝑃 ) ) |
24 |
23
|
oveqd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑃 ) ∧ 𝑦 ∈ ( Base ‘ 𝑃 ) ) ) → ( 𝑥 ( +g ‘ ( 1o mPwSer 𝑅 ) ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) ) |
25 |
|
eqid |
⊢ ( +g ‘ 𝑄 ) = ( +g ‘ 𝑄 ) |
26 |
2 7 25
|
psr1plusg |
⊢ ( +g ‘ 𝑄 ) = ( +g ‘ ( 1o mPwSer 𝑆 ) ) |
27 |
26
|
eqcomi |
⊢ ( +g ‘ ( 1o mPwSer 𝑆 ) ) = ( +g ‘ 𝑄 ) |
28 |
27
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑄 ) ∧ 𝑦 ∈ ( Base ‘ 𝑄 ) ) ) → ( +g ‘ ( 1o mPwSer 𝑆 ) ) = ( +g ‘ 𝑄 ) ) |
29 |
28
|
oveqd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑄 ) ∧ 𝑦 ∈ ( Base ‘ 𝑄 ) ) ) → ( 𝑥 ( +g ‘ ( 1o mPwSer 𝑆 ) ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝑄 ) 𝑦 ) ) |
30 |
|
eqid |
⊢ ( .r ‘ 𝑃 ) = ( .r ‘ 𝑃 ) |
31 |
1 6 30
|
psr1mulr |
⊢ ( .r ‘ 𝑃 ) = ( .r ‘ ( 1o mPwSer 𝑅 ) ) |
32 |
31
|
eqcomi |
⊢ ( .r ‘ ( 1o mPwSer 𝑅 ) ) = ( .r ‘ 𝑃 ) |
33 |
32
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑃 ) ∧ 𝑦 ∈ ( Base ‘ 𝑃 ) ) ) → ( .r ‘ ( 1o mPwSer 𝑅 ) ) = ( .r ‘ 𝑃 ) ) |
34 |
33
|
oveqd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑃 ) ∧ 𝑦 ∈ ( Base ‘ 𝑃 ) ) ) → ( 𝑥 ( .r ‘ ( 1o mPwSer 𝑅 ) ) 𝑦 ) = ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) ) |
35 |
|
eqid |
⊢ ( .r ‘ 𝑄 ) = ( .r ‘ 𝑄 ) |
36 |
2 7 35
|
psr1mulr |
⊢ ( .r ‘ 𝑄 ) = ( .r ‘ ( 1o mPwSer 𝑆 ) ) |
37 |
36
|
eqcomi |
⊢ ( .r ‘ ( 1o mPwSer 𝑆 ) ) = ( .r ‘ 𝑄 ) |
38 |
37
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑄 ) ∧ 𝑦 ∈ ( Base ‘ 𝑄 ) ) ) → ( .r ‘ ( 1o mPwSer 𝑆 ) ) = ( .r ‘ 𝑄 ) ) |
39 |
38
|
oveqd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑄 ) ∧ 𝑦 ∈ ( Base ‘ 𝑄 ) ) ) → ( 𝑥 ( .r ‘ ( 1o mPwSer 𝑆 ) ) 𝑦 ) = ( 𝑥 ( .r ‘ 𝑄 ) 𝑦 ) ) |
40 |
14 17 18 19 24 29 34 39
|
rhmpropd |
⊢ ( 𝜑 → ( ( 1o mPwSer 𝑅 ) RingHom ( 1o mPwSer 𝑆 ) ) = ( 𝑃 RingHom 𝑄 ) ) |
41 |
11 40
|
eleqtrd |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝑃 RingHom 𝑄 ) ) |