Step |
Hyp |
Ref |
Expression |
1 |
|
subrgascl.p |
⊢ 𝑃 = ( 𝐼 mPoly 𝑅 ) |
2 |
|
subrgascl.a |
⊢ 𝐴 = ( algSc ‘ 𝑃 ) |
3 |
|
subrgascl.h |
⊢ 𝐻 = ( 𝑅 ↾s 𝑇 ) |
4 |
|
subrgascl.u |
⊢ 𝑈 = ( 𝐼 mPoly 𝐻 ) |
5 |
|
subrgascl.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
6 |
|
subrgascl.r |
⊢ ( 𝜑 → 𝑇 ∈ ( SubRing ‘ 𝑅 ) ) |
7 |
|
subrgasclcl.b |
⊢ 𝐵 = ( Base ‘ 𝑈 ) |
8 |
|
subrgasclcl.k |
⊢ 𝐾 = ( Base ‘ 𝑅 ) |
9 |
|
subrgasclcl.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐾 ) |
10 |
|
iftrue |
⊢ ( 𝑥 = ( 𝐼 × { 0 } ) → if ( 𝑥 = ( 𝐼 × { 0 } ) , 𝑋 , ( 0g ‘ 𝑅 ) ) = 𝑋 ) |
11 |
10
|
eleq1d |
⊢ ( 𝑥 = ( 𝐼 × { 0 } ) → ( if ( 𝑥 = ( 𝐼 × { 0 } ) , 𝑋 , ( 0g ‘ 𝑅 ) ) ∈ ( Base ‘ 𝐻 ) ↔ 𝑋 ∈ ( Base ‘ 𝐻 ) ) ) |
12 |
|
eqid |
⊢ ( 𝐼 mPwSer 𝐻 ) = ( 𝐼 mPwSer 𝐻 ) |
13 |
|
eqid |
⊢ ( Base ‘ 𝐻 ) = ( Base ‘ 𝐻 ) |
14 |
|
eqid |
⊢ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
15 |
|
eqid |
⊢ ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) = ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) |
16 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
17 |
|
subrgrcl |
⊢ ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → 𝑅 ∈ Ring ) |
18 |
6 17
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
19 |
1 14 16 8 2 5 18 9
|
mplascl |
⊢ ( 𝜑 → ( 𝐴 ‘ 𝑋 ) = ( 𝑥 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 𝑋 , ( 0g ‘ 𝑅 ) ) ) ) |
20 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑋 ) ∈ 𝐵 ) → ( 𝐴 ‘ 𝑋 ) = ( 𝑥 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 𝑋 , ( 0g ‘ 𝑅 ) ) ) ) |
21 |
3
|
subrgring |
⊢ ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → 𝐻 ∈ Ring ) |
22 |
6 21
|
syl |
⊢ ( 𝜑 → 𝐻 ∈ Ring ) |
23 |
12 4 7 5 22
|
mplsubrg |
⊢ ( 𝜑 → 𝐵 ∈ ( SubRing ‘ ( 𝐼 mPwSer 𝐻 ) ) ) |
24 |
15
|
subrgss |
⊢ ( 𝐵 ∈ ( SubRing ‘ ( 𝐼 mPwSer 𝐻 ) ) → 𝐵 ⊆ ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ) |
25 |
23 24
|
syl |
⊢ ( 𝜑 → 𝐵 ⊆ ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ) |
26 |
25
|
sselda |
⊢ ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑋 ) ∈ 𝐵 ) → ( 𝐴 ‘ 𝑋 ) ∈ ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ) |
27 |
20 26
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑋 ) ∈ 𝐵 ) → ( 𝑥 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 𝑋 , ( 0g ‘ 𝑅 ) ) ) ∈ ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ) |
28 |
12 13 14 15 27
|
psrelbas |
⊢ ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑋 ) ∈ 𝐵 ) → ( 𝑥 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 𝑋 , ( 0g ‘ 𝑅 ) ) ) : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝐻 ) ) |
29 |
|
eqid |
⊢ ( 𝑥 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 𝑋 , ( 0g ‘ 𝑅 ) ) ) = ( 𝑥 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 𝑋 , ( 0g ‘ 𝑅 ) ) ) |
30 |
29
|
fmpt |
⊢ ( ∀ 𝑥 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } if ( 𝑥 = ( 𝐼 × { 0 } ) , 𝑋 , ( 0g ‘ 𝑅 ) ) ∈ ( Base ‘ 𝐻 ) ↔ ( 𝑥 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 𝑋 , ( 0g ‘ 𝑅 ) ) ) : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝐻 ) ) |
31 |
28 30
|
sylibr |
⊢ ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑋 ) ∈ 𝐵 ) → ∀ 𝑥 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } if ( 𝑥 = ( 𝐼 × { 0 } ) , 𝑋 , ( 0g ‘ 𝑅 ) ) ∈ ( Base ‘ 𝐻 ) ) |
32 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑋 ) ∈ 𝐵 ) → 𝐼 ∈ 𝑊 ) |
33 |
14
|
psrbag0 |
⊢ ( 𝐼 ∈ 𝑊 → ( 𝐼 × { 0 } ) ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) |
34 |
32 33
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑋 ) ∈ 𝐵 ) → ( 𝐼 × { 0 } ) ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) |
35 |
11 31 34
|
rspcdva |
⊢ ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑋 ) ∈ 𝐵 ) → 𝑋 ∈ ( Base ‘ 𝐻 ) ) |
36 |
3
|
subrgbas |
⊢ ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → 𝑇 = ( Base ‘ 𝐻 ) ) |
37 |
6 36
|
syl |
⊢ ( 𝜑 → 𝑇 = ( Base ‘ 𝐻 ) ) |
38 |
37
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑋 ) ∈ 𝐵 ) → 𝑇 = ( Base ‘ 𝐻 ) ) |
39 |
35 38
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ ( 𝐴 ‘ 𝑋 ) ∈ 𝐵 ) → 𝑋 ∈ 𝑇 ) |
40 |
|
eqid |
⊢ ( algSc ‘ 𝑈 ) = ( algSc ‘ 𝑈 ) |
41 |
1 2 3 4 5 6 40
|
subrgascl |
⊢ ( 𝜑 → ( algSc ‘ 𝑈 ) = ( 𝐴 ↾ 𝑇 ) ) |
42 |
41
|
fveq1d |
⊢ ( 𝜑 → ( ( algSc ‘ 𝑈 ) ‘ 𝑋 ) = ( ( 𝐴 ↾ 𝑇 ) ‘ 𝑋 ) ) |
43 |
|
fvres |
⊢ ( 𝑋 ∈ 𝑇 → ( ( 𝐴 ↾ 𝑇 ) ‘ 𝑋 ) = ( 𝐴 ‘ 𝑋 ) ) |
44 |
42 43
|
sylan9eq |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑇 ) → ( ( algSc ‘ 𝑈 ) ‘ 𝑋 ) = ( 𝐴 ‘ 𝑋 ) ) |
45 |
|
eqid |
⊢ ( Scalar ‘ 𝑈 ) = ( Scalar ‘ 𝑈 ) |
46 |
4
|
mplring |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝐻 ∈ Ring ) → 𝑈 ∈ Ring ) |
47 |
4
|
mpllmod |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝐻 ∈ Ring ) → 𝑈 ∈ LMod ) |
48 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑈 ) ) = ( Base ‘ ( Scalar ‘ 𝑈 ) ) |
49 |
40 45 46 47 48 7
|
asclf |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝐻 ∈ Ring ) → ( algSc ‘ 𝑈 ) : ( Base ‘ ( Scalar ‘ 𝑈 ) ) ⟶ 𝐵 ) |
50 |
5 22 49
|
syl2anc |
⊢ ( 𝜑 → ( algSc ‘ 𝑈 ) : ( Base ‘ ( Scalar ‘ 𝑈 ) ) ⟶ 𝐵 ) |
51 |
50
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑇 ) → ( algSc ‘ 𝑈 ) : ( Base ‘ ( Scalar ‘ 𝑈 ) ) ⟶ 𝐵 ) |
52 |
4 5 22
|
mplsca |
⊢ ( 𝜑 → 𝐻 = ( Scalar ‘ 𝑈 ) ) |
53 |
52
|
fveq2d |
⊢ ( 𝜑 → ( Base ‘ 𝐻 ) = ( Base ‘ ( Scalar ‘ 𝑈 ) ) ) |
54 |
37 53
|
eqtrd |
⊢ ( 𝜑 → 𝑇 = ( Base ‘ ( Scalar ‘ 𝑈 ) ) ) |
55 |
54
|
eleq2d |
⊢ ( 𝜑 → ( 𝑋 ∈ 𝑇 ↔ 𝑋 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ) ) |
56 |
55
|
biimpa |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑇 ) → 𝑋 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ) |
57 |
51 56
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑇 ) → ( ( algSc ‘ 𝑈 ) ‘ 𝑋 ) ∈ 𝐵 ) |
58 |
44 57
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝑇 ) → ( 𝐴 ‘ 𝑋 ) ∈ 𝐵 ) |
59 |
39 58
|
impbida |
⊢ ( 𝜑 → ( ( 𝐴 ‘ 𝑋 ) ∈ 𝐵 ↔ 𝑋 ∈ 𝑇 ) ) |