Step |
Hyp |
Ref |
Expression |
1 |
|
psrplusgpropd.b1 |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑅 ) ) |
2 |
|
psrplusgpropd.b2 |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑆 ) ) |
3 |
|
psrplusgpropd.p |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) |
4 |
|
simpl1 |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) ∧ 𝑑 ∈ { 𝑐 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑐 “ ℕ ) ∈ Fin } ) → 𝜑 ) |
5 |
|
eqid |
⊢ ( 𝐼 mPwSer 𝑅 ) = ( 𝐼 mPwSer 𝑅 ) |
6 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
7 |
|
eqid |
⊢ { 𝑐 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑐 “ ℕ ) ∈ Fin } = { 𝑐 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑐 “ ℕ ) ∈ Fin } |
8 |
|
eqid |
⊢ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) |
9 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) → 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) |
10 |
5 6 7 8 9
|
psrelbas |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) → 𝑎 : { 𝑐 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑐 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) ) |
11 |
10
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) ∧ 𝑑 ∈ { 𝑐 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑐 “ ℕ ) ∈ Fin } ) → ( 𝑎 ‘ 𝑑 ) ∈ ( Base ‘ 𝑅 ) ) |
12 |
4 1
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) ∧ 𝑑 ∈ { 𝑐 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑐 “ ℕ ) ∈ Fin } ) → 𝐵 = ( Base ‘ 𝑅 ) ) |
13 |
11 12
|
eleqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) ∧ 𝑑 ∈ { 𝑐 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑐 “ ℕ ) ∈ Fin } ) → ( 𝑎 ‘ 𝑑 ) ∈ 𝐵 ) |
14 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) → 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) |
15 |
5 6 7 8 14
|
psrelbas |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) → 𝑏 : { 𝑐 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑐 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) ) |
16 |
15
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) ∧ 𝑑 ∈ { 𝑐 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑐 “ ℕ ) ∈ Fin } ) → ( 𝑏 ‘ 𝑑 ) ∈ ( Base ‘ 𝑅 ) ) |
17 |
16 12
|
eleqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) ∧ 𝑑 ∈ { 𝑐 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑐 “ ℕ ) ∈ Fin } ) → ( 𝑏 ‘ 𝑑 ) ∈ 𝐵 ) |
18 |
3
|
oveqrspc2v |
⊢ ( ( 𝜑 ∧ ( ( 𝑎 ‘ 𝑑 ) ∈ 𝐵 ∧ ( 𝑏 ‘ 𝑑 ) ∈ 𝐵 ) ) → ( ( 𝑎 ‘ 𝑑 ) ( +g ‘ 𝑅 ) ( 𝑏 ‘ 𝑑 ) ) = ( ( 𝑎 ‘ 𝑑 ) ( +g ‘ 𝑆 ) ( 𝑏 ‘ 𝑑 ) ) ) |
19 |
4 13 17 18
|
syl12anc |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) ∧ 𝑑 ∈ { 𝑐 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑐 “ ℕ ) ∈ Fin } ) → ( ( 𝑎 ‘ 𝑑 ) ( +g ‘ 𝑅 ) ( 𝑏 ‘ 𝑑 ) ) = ( ( 𝑎 ‘ 𝑑 ) ( +g ‘ 𝑆 ) ( 𝑏 ‘ 𝑑 ) ) ) |
20 |
19
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) → ( 𝑑 ∈ { 𝑐 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑐 “ ℕ ) ∈ Fin } ↦ ( ( 𝑎 ‘ 𝑑 ) ( +g ‘ 𝑅 ) ( 𝑏 ‘ 𝑑 ) ) ) = ( 𝑑 ∈ { 𝑐 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑐 “ ℕ ) ∈ Fin } ↦ ( ( 𝑎 ‘ 𝑑 ) ( +g ‘ 𝑆 ) ( 𝑏 ‘ 𝑑 ) ) ) ) |
21 |
10
|
ffnd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) → 𝑎 Fn { 𝑐 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑐 “ ℕ ) ∈ Fin } ) |
22 |
15
|
ffnd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) → 𝑏 Fn { 𝑐 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑐 “ ℕ ) ∈ Fin } ) |
23 |
|
ovex |
⊢ ( ℕ0 ↑m 𝐼 ) ∈ V |
24 |
23
|
rabex |
⊢ { 𝑐 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑐 “ ℕ ) ∈ Fin } ∈ V |
25 |
24
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) → { 𝑐 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑐 “ ℕ ) ∈ Fin } ∈ V ) |
26 |
|
inidm |
⊢ ( { 𝑐 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑐 “ ℕ ) ∈ Fin } ∩ { 𝑐 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑐 “ ℕ ) ∈ Fin } ) = { 𝑐 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑐 “ ℕ ) ∈ Fin } |
27 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) ∧ 𝑑 ∈ { 𝑐 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑐 “ ℕ ) ∈ Fin } ) → ( 𝑎 ‘ 𝑑 ) = ( 𝑎 ‘ 𝑑 ) ) |
28 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) ∧ 𝑑 ∈ { 𝑐 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑐 “ ℕ ) ∈ Fin } ) → ( 𝑏 ‘ 𝑑 ) = ( 𝑏 ‘ 𝑑 ) ) |
29 |
21 22 25 25 26 27 28
|
offval |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) → ( 𝑎 ∘f ( +g ‘ 𝑅 ) 𝑏 ) = ( 𝑑 ∈ { 𝑐 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑐 “ ℕ ) ∈ Fin } ↦ ( ( 𝑎 ‘ 𝑑 ) ( +g ‘ 𝑅 ) ( 𝑏 ‘ 𝑑 ) ) ) ) |
30 |
21 22 25 25 26 27 28
|
offval |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) → ( 𝑎 ∘f ( +g ‘ 𝑆 ) 𝑏 ) = ( 𝑑 ∈ { 𝑐 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑐 “ ℕ ) ∈ Fin } ↦ ( ( 𝑎 ‘ 𝑑 ) ( +g ‘ 𝑆 ) ( 𝑏 ‘ 𝑑 ) ) ) ) |
31 |
20 29 30
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) → ( 𝑎 ∘f ( +g ‘ 𝑅 ) 𝑏 ) = ( 𝑎 ∘f ( +g ‘ 𝑆 ) 𝑏 ) ) |
32 |
31
|
mpoeq3dva |
⊢ ( 𝜑 → ( 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) , 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ↦ ( 𝑎 ∘f ( +g ‘ 𝑅 ) 𝑏 ) ) = ( 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) , 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ↦ ( 𝑎 ∘f ( +g ‘ 𝑆 ) 𝑏 ) ) ) |
33 |
1 2
|
eqtr3d |
⊢ ( 𝜑 → ( Base ‘ 𝑅 ) = ( Base ‘ 𝑆 ) ) |
34 |
33
|
psrbaspropd |
⊢ ( 𝜑 → ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( Base ‘ ( 𝐼 mPwSer 𝑆 ) ) ) |
35 |
|
mpoeq12 |
⊢ ( ( ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( Base ‘ ( 𝐼 mPwSer 𝑆 ) ) ∧ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( Base ‘ ( 𝐼 mPwSer 𝑆 ) ) ) → ( 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) , 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ↦ ( 𝑎 ∘f ( +g ‘ 𝑆 ) 𝑏 ) ) = ( 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑆 ) ) , 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑆 ) ) ↦ ( 𝑎 ∘f ( +g ‘ 𝑆 ) 𝑏 ) ) ) |
36 |
34 34 35
|
syl2anc |
⊢ ( 𝜑 → ( 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) , 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ↦ ( 𝑎 ∘f ( +g ‘ 𝑆 ) 𝑏 ) ) = ( 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑆 ) ) , 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑆 ) ) ↦ ( 𝑎 ∘f ( +g ‘ 𝑆 ) 𝑏 ) ) ) |
37 |
32 36
|
eqtrd |
⊢ ( 𝜑 → ( 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) , 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ↦ ( 𝑎 ∘f ( +g ‘ 𝑅 ) 𝑏 ) ) = ( 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑆 ) ) , 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑆 ) ) ↦ ( 𝑎 ∘f ( +g ‘ 𝑆 ) 𝑏 ) ) ) |
38 |
|
ofmres |
⊢ ( ∘f ( +g ‘ 𝑅 ) ↾ ( ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) × ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) ) = ( 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) , 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ↦ ( 𝑎 ∘f ( +g ‘ 𝑅 ) 𝑏 ) ) |
39 |
|
ofmres |
⊢ ( ∘f ( +g ‘ 𝑆 ) ↾ ( ( Base ‘ ( 𝐼 mPwSer 𝑆 ) ) × ( Base ‘ ( 𝐼 mPwSer 𝑆 ) ) ) ) = ( 𝑎 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑆 ) ) , 𝑏 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑆 ) ) ↦ ( 𝑎 ∘f ( +g ‘ 𝑆 ) 𝑏 ) ) |
40 |
37 38 39
|
3eqtr4g |
⊢ ( 𝜑 → ( ∘f ( +g ‘ 𝑅 ) ↾ ( ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) × ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) ) = ( ∘f ( +g ‘ 𝑆 ) ↾ ( ( Base ‘ ( 𝐼 mPwSer 𝑆 ) ) × ( Base ‘ ( 𝐼 mPwSer 𝑆 ) ) ) ) ) |
41 |
|
eqid |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) |
42 |
|
eqid |
⊢ ( +g ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( +g ‘ ( 𝐼 mPwSer 𝑅 ) ) |
43 |
5 8 41 42
|
psrplusg |
⊢ ( +g ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( ∘f ( +g ‘ 𝑅 ) ↾ ( ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) × ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) ) |
44 |
|
eqid |
⊢ ( 𝐼 mPwSer 𝑆 ) = ( 𝐼 mPwSer 𝑆 ) |
45 |
|
eqid |
⊢ ( Base ‘ ( 𝐼 mPwSer 𝑆 ) ) = ( Base ‘ ( 𝐼 mPwSer 𝑆 ) ) |
46 |
|
eqid |
⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 ) |
47 |
|
eqid |
⊢ ( +g ‘ ( 𝐼 mPwSer 𝑆 ) ) = ( +g ‘ ( 𝐼 mPwSer 𝑆 ) ) |
48 |
44 45 46 47
|
psrplusg |
⊢ ( +g ‘ ( 𝐼 mPwSer 𝑆 ) ) = ( ∘f ( +g ‘ 𝑆 ) ↾ ( ( Base ‘ ( 𝐼 mPwSer 𝑆 ) ) × ( Base ‘ ( 𝐼 mPwSer 𝑆 ) ) ) ) |
49 |
40 43 48
|
3eqtr4g |
⊢ ( 𝜑 → ( +g ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( +g ‘ ( 𝐼 mPwSer 𝑆 ) ) ) |