Step |
Hyp |
Ref |
Expression |
1 |
|
resspsr.s |
⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 ) |
2 |
|
resspsr.h |
⊢ 𝐻 = ( 𝑅 ↾s 𝑇 ) |
3 |
|
resspsr.u |
⊢ 𝑈 = ( 𝐼 mPwSer 𝐻 ) |
4 |
|
resspsr.b |
⊢ 𝐵 = ( Base ‘ 𝑈 ) |
5 |
|
resspsr.p |
⊢ 𝑃 = ( 𝑆 ↾s 𝐵 ) |
6 |
|
resspsr.2 |
⊢ ( 𝜑 → 𝑇 ∈ ( SubRing ‘ 𝑅 ) ) |
7 |
|
eqid |
⊢ ( +g ‘ 𝐻 ) = ( +g ‘ 𝐻 ) |
8 |
|
eqid |
⊢ ( +g ‘ 𝑈 ) = ( +g ‘ 𝑈 ) |
9 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 ) |
10 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 ) |
11 |
3 4 7 8 9 10
|
psradd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( +g ‘ 𝑈 ) 𝑌 ) = ( 𝑋 ∘f ( +g ‘ 𝐻 ) 𝑌 ) ) |
12 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
13 |
|
eqid |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) |
14 |
|
eqid |
⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 ) |
15 |
|
fvex |
⊢ ( Base ‘ 𝑅 ) ∈ V |
16 |
2
|
subrgbas |
⊢ ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → 𝑇 = ( Base ‘ 𝐻 ) ) |
17 |
6 16
|
syl |
⊢ ( 𝜑 → 𝑇 = ( Base ‘ 𝐻 ) ) |
18 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
19 |
18
|
subrgss |
⊢ ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → 𝑇 ⊆ ( Base ‘ 𝑅 ) ) |
20 |
6 19
|
syl |
⊢ ( 𝜑 → 𝑇 ⊆ ( Base ‘ 𝑅 ) ) |
21 |
17 20
|
eqsstrrd |
⊢ ( 𝜑 → ( Base ‘ 𝐻 ) ⊆ ( Base ‘ 𝑅 ) ) |
22 |
|
mapss |
⊢ ( ( ( Base ‘ 𝑅 ) ∈ V ∧ ( Base ‘ 𝐻 ) ⊆ ( Base ‘ 𝑅 ) ) → ( ( Base ‘ 𝐻 ) ↑m { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ⊆ ( ( Base ‘ 𝑅 ) ↑m { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ) |
23 |
15 21 22
|
sylancr |
⊢ ( 𝜑 → ( ( Base ‘ 𝐻 ) ↑m { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ⊆ ( ( Base ‘ 𝑅 ) ↑m { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ) |
24 |
23
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( Base ‘ 𝐻 ) ↑m { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ⊆ ( ( Base ‘ 𝑅 ) ↑m { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ) |
25 |
|
eqid |
⊢ ( Base ‘ 𝐻 ) = ( Base ‘ 𝐻 ) |
26 |
|
eqid |
⊢ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
27 |
|
reldmpsr |
⊢ Rel dom mPwSer |
28 |
27 3 4
|
elbasov |
⊢ ( 𝑋 ∈ 𝐵 → ( 𝐼 ∈ V ∧ 𝐻 ∈ V ) ) |
29 |
28
|
ad2antrl |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐼 ∈ V ∧ 𝐻 ∈ V ) ) |
30 |
29
|
simpld |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐼 ∈ V ) |
31 |
3 25 26 4 30
|
psrbas |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐵 = ( ( Base ‘ 𝐻 ) ↑m { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ) |
32 |
1 18 26 12 30
|
psrbas |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( Base ‘ 𝑆 ) = ( ( Base ‘ 𝑅 ) ↑m { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ) |
33 |
24 31 32
|
3sstr4d |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐵 ⊆ ( Base ‘ 𝑆 ) ) |
34 |
33 9
|
sseldd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑋 ∈ ( Base ‘ 𝑆 ) ) |
35 |
33 10
|
sseldd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑌 ∈ ( Base ‘ 𝑆 ) ) |
36 |
1 12 13 14 34 35
|
psradd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( +g ‘ 𝑆 ) 𝑌 ) = ( 𝑋 ∘f ( +g ‘ 𝑅 ) 𝑌 ) ) |
37 |
2 13
|
ressplusg |
⊢ ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → ( +g ‘ 𝑅 ) = ( +g ‘ 𝐻 ) ) |
38 |
6 37
|
syl |
⊢ ( 𝜑 → ( +g ‘ 𝑅 ) = ( +g ‘ 𝐻 ) ) |
39 |
38
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( +g ‘ 𝑅 ) = ( +g ‘ 𝐻 ) ) |
40 |
|
ofeq |
⊢ ( ( +g ‘ 𝑅 ) = ( +g ‘ 𝐻 ) → ∘f ( +g ‘ 𝑅 ) = ∘f ( +g ‘ 𝐻 ) ) |
41 |
39 40
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ∘f ( +g ‘ 𝑅 ) = ∘f ( +g ‘ 𝐻 ) ) |
42 |
41
|
oveqd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ∘f ( +g ‘ 𝑅 ) 𝑌 ) = ( 𝑋 ∘f ( +g ‘ 𝐻 ) 𝑌 ) ) |
43 |
36 42
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( +g ‘ 𝑆 ) 𝑌 ) = ( 𝑋 ∘f ( +g ‘ 𝐻 ) 𝑌 ) ) |
44 |
4
|
fvexi |
⊢ 𝐵 ∈ V |
45 |
5 14
|
ressplusg |
⊢ ( 𝐵 ∈ V → ( +g ‘ 𝑆 ) = ( +g ‘ 𝑃 ) ) |
46 |
44 45
|
mp1i |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( +g ‘ 𝑆 ) = ( +g ‘ 𝑃 ) ) |
47 |
46
|
oveqd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( +g ‘ 𝑆 ) 𝑌 ) = ( 𝑋 ( +g ‘ 𝑃 ) 𝑌 ) ) |
48 |
11 43 47
|
3eqtr2d |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( +g ‘ 𝑈 ) 𝑌 ) = ( 𝑋 ( +g ‘ 𝑃 ) 𝑌 ) ) |