Step |
Hyp |
Ref |
Expression |
1 |
|
ressmpl.s |
⊢ 𝑆 = ( 𝐼 mPoly 𝑅 ) |
2 |
|
ressmpl.h |
⊢ 𝐻 = ( 𝑅 ↾s 𝑇 ) |
3 |
|
ressmpl.u |
⊢ 𝑈 = ( 𝐼 mPoly 𝐻 ) |
4 |
|
ressmpl.b |
⊢ 𝐵 = ( Base ‘ 𝑈 ) |
5 |
|
ressmpl.1 |
⊢ ( 𝜑 → 𝐼 ∈ 𝑉 ) |
6 |
|
ressmpl.2 |
⊢ ( 𝜑 → 𝑇 ∈ ( SubRing ‘ 𝑅 ) ) |
7 |
|
ressmplbas2.w |
⊢ 𝑊 = ( 𝐼 mPwSer 𝐻 ) |
8 |
|
ressmplbas2.c |
⊢ 𝐶 = ( Base ‘ 𝑊 ) |
9 |
|
ressmplbas2.k |
⊢ 𝐾 = ( Base ‘ 𝑆 ) |
10 |
|
eqid |
⊢ ( 𝐼 mPwSer 𝑅 ) = ( 𝐼 mPwSer 𝑅 ) |
11 |
10 2 7 8
|
subrgpsr |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑇 ∈ ( SubRing ‘ 𝑅 ) ) → 𝐶 ∈ ( SubRing ‘ ( 𝐼 mPwSer 𝑅 ) ) ) |
12 |
5 6 11
|
syl2anc |
⊢ ( 𝜑 → 𝐶 ∈ ( SubRing ‘ ( 𝐼 mPwSer 𝑅 ) ) ) |
13 |
|
eqid |
⊢ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) |
14 |
13
|
subrgss |
⊢ ( 𝐶 ∈ ( SubRing ‘ ( 𝐼 mPwSer 𝑅 ) ) → 𝐶 ⊆ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) |
15 |
12 14
|
syl |
⊢ ( 𝜑 → 𝐶 ⊆ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) |
16 |
|
df-ss |
⊢ ( 𝐶 ⊆ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ↔ ( 𝐶 ∩ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) = 𝐶 ) |
17 |
15 16
|
sylib |
⊢ ( 𝜑 → ( 𝐶 ∩ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) = 𝐶 ) |
18 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
19 |
2 18
|
subrg0 |
⊢ ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝐻 ) ) |
20 |
6 19
|
syl |
⊢ ( 𝜑 → ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝐻 ) ) |
21 |
20
|
breq2d |
⊢ ( 𝜑 → ( 𝑓 finSupp ( 0g ‘ 𝑅 ) ↔ 𝑓 finSupp ( 0g ‘ 𝐻 ) ) ) |
22 |
21
|
abbidv |
⊢ ( 𝜑 → { 𝑓 ∣ 𝑓 finSupp ( 0g ‘ 𝑅 ) } = { 𝑓 ∣ 𝑓 finSupp ( 0g ‘ 𝐻 ) } ) |
23 |
17 22
|
ineq12d |
⊢ ( 𝜑 → ( ( 𝐶 ∩ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) ∩ { 𝑓 ∣ 𝑓 finSupp ( 0g ‘ 𝑅 ) } ) = ( 𝐶 ∩ { 𝑓 ∣ 𝑓 finSupp ( 0g ‘ 𝐻 ) } ) ) |
24 |
23
|
eqcomd |
⊢ ( 𝜑 → ( 𝐶 ∩ { 𝑓 ∣ 𝑓 finSupp ( 0g ‘ 𝐻 ) } ) = ( ( 𝐶 ∩ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) ∩ { 𝑓 ∣ 𝑓 finSupp ( 0g ‘ 𝑅 ) } ) ) |
25 |
|
eqid |
⊢ ( 0g ‘ 𝐻 ) = ( 0g ‘ 𝐻 ) |
26 |
3 7 8 25 4
|
mplbas |
⊢ 𝐵 = { 𝑓 ∈ 𝐶 ∣ 𝑓 finSupp ( 0g ‘ 𝐻 ) } |
27 |
|
dfrab3 |
⊢ { 𝑓 ∈ 𝐶 ∣ 𝑓 finSupp ( 0g ‘ 𝐻 ) } = ( 𝐶 ∩ { 𝑓 ∣ 𝑓 finSupp ( 0g ‘ 𝐻 ) } ) |
28 |
26 27
|
eqtri |
⊢ 𝐵 = ( 𝐶 ∩ { 𝑓 ∣ 𝑓 finSupp ( 0g ‘ 𝐻 ) } ) |
29 |
1 10 13 18 9
|
mplbas |
⊢ 𝐾 = { 𝑓 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∣ 𝑓 finSupp ( 0g ‘ 𝑅 ) } |
30 |
|
dfrab3 |
⊢ { 𝑓 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∣ 𝑓 finSupp ( 0g ‘ 𝑅 ) } = ( ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∩ { 𝑓 ∣ 𝑓 finSupp ( 0g ‘ 𝑅 ) } ) |
31 |
29 30
|
eqtri |
⊢ 𝐾 = ( ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∩ { 𝑓 ∣ 𝑓 finSupp ( 0g ‘ 𝑅 ) } ) |
32 |
31
|
ineq2i |
⊢ ( 𝐶 ∩ 𝐾 ) = ( 𝐶 ∩ ( ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∩ { 𝑓 ∣ 𝑓 finSupp ( 0g ‘ 𝑅 ) } ) ) |
33 |
|
inass |
⊢ ( ( 𝐶 ∩ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) ∩ { 𝑓 ∣ 𝑓 finSupp ( 0g ‘ 𝑅 ) } ) = ( 𝐶 ∩ ( ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∩ { 𝑓 ∣ 𝑓 finSupp ( 0g ‘ 𝑅 ) } ) ) |
34 |
32 33
|
eqtr4i |
⊢ ( 𝐶 ∩ 𝐾 ) = ( ( 𝐶 ∩ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) ∩ { 𝑓 ∣ 𝑓 finSupp ( 0g ‘ 𝑅 ) } ) |
35 |
24 28 34
|
3eqtr4g |
⊢ ( 𝜑 → 𝐵 = ( 𝐶 ∩ 𝐾 ) ) |