Step |
Hyp |
Ref |
Expression |
1 |
|
subrgply1.s |
⊢ 𝑆 = ( Poly1 ‘ 𝑅 ) |
2 |
|
subrgply1.h |
⊢ 𝐻 = ( 𝑅 ↾s 𝑇 ) |
3 |
|
subrgply1.u |
⊢ 𝑈 = ( Poly1 ‘ 𝐻 ) |
4 |
|
subrgply1.b |
⊢ 𝐵 = ( Base ‘ 𝑈 ) |
5 |
|
gsumply1subr.s |
⊢ ( 𝜑 → 𝑇 ∈ ( SubRing ‘ 𝑅 ) ) |
6 |
|
gsumply1subr.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
7 |
|
gsumply1subr.f |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
8 |
1 2 3 4
|
subrgply1 |
⊢ ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → 𝐵 ∈ ( SubRing ‘ 𝑆 ) ) |
9 |
|
subrgsubg |
⊢ ( 𝐵 ∈ ( SubRing ‘ 𝑆 ) → 𝐵 ∈ ( SubGrp ‘ 𝑆 ) ) |
10 |
|
subgsubm |
⊢ ( 𝐵 ∈ ( SubGrp ‘ 𝑆 ) → 𝐵 ∈ ( SubMnd ‘ 𝑆 ) ) |
11 |
9 10
|
syl |
⊢ ( 𝐵 ∈ ( SubRing ‘ 𝑆 ) → 𝐵 ∈ ( SubMnd ‘ 𝑆 ) ) |
12 |
5 8 11
|
3syl |
⊢ ( 𝜑 → 𝐵 ∈ ( SubMnd ‘ 𝑆 ) ) |
13 |
|
eqid |
⊢ ( 𝑆 ↾s 𝐵 ) = ( 𝑆 ↾s 𝐵 ) |
14 |
6 12 7 13
|
gsumsubm |
⊢ ( 𝜑 → ( 𝑆 Σg 𝐹 ) = ( ( 𝑆 ↾s 𝐵 ) Σg 𝐹 ) ) |
15 |
|
fex |
⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐴 ∈ 𝑉 ) → 𝐹 ∈ V ) |
16 |
7 6 15
|
syl2anc |
⊢ ( 𝜑 → 𝐹 ∈ V ) |
17 |
|
ovexd |
⊢ ( 𝜑 → ( 𝑆 ↾s 𝐵 ) ∈ V ) |
18 |
3
|
fvexi |
⊢ 𝑈 ∈ V |
19 |
18
|
a1i |
⊢ ( 𝜑 → 𝑈 ∈ V ) |
20 |
|
eqid |
⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 ) |
21 |
4
|
oveq2i |
⊢ ( 𝑆 ↾s 𝐵 ) = ( 𝑆 ↾s ( Base ‘ 𝑈 ) ) |
22 |
1 2 3 20 5 21
|
ressply1bas |
⊢ ( 𝜑 → ( Base ‘ 𝑈 ) = ( Base ‘ ( 𝑆 ↾s 𝐵 ) ) ) |
23 |
22
|
eqcomd |
⊢ ( 𝜑 → ( Base ‘ ( 𝑆 ↾s 𝐵 ) ) = ( Base ‘ 𝑈 ) ) |
24 |
13
|
subrgring |
⊢ ( 𝐵 ∈ ( SubRing ‘ 𝑆 ) → ( 𝑆 ↾s 𝐵 ) ∈ Ring ) |
25 |
8 24
|
syl |
⊢ ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → ( 𝑆 ↾s 𝐵 ) ∈ Ring ) |
26 |
|
ringmgm |
⊢ ( ( 𝑆 ↾s 𝐵 ) ∈ Ring → ( 𝑆 ↾s 𝐵 ) ∈ Mgm ) |
27 |
5 25 26
|
3syl |
⊢ ( 𝜑 → ( 𝑆 ↾s 𝐵 ) ∈ Mgm ) |
28 |
|
simpl |
⊢ ( ( 𝜑 ∧ ( 𝑠 ∈ ( Base ‘ ( 𝑆 ↾s 𝐵 ) ) ∧ 𝑡 ∈ ( Base ‘ ( 𝑆 ↾s 𝐵 ) ) ) ) → 𝜑 ) |
29 |
1 2 3 4 5 13
|
ressply1bas |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ ( 𝑆 ↾s 𝐵 ) ) ) |
30 |
29
|
eqcomd |
⊢ ( 𝜑 → ( Base ‘ ( 𝑆 ↾s 𝐵 ) ) = 𝐵 ) |
31 |
30
|
eleq2d |
⊢ ( 𝜑 → ( 𝑠 ∈ ( Base ‘ ( 𝑆 ↾s 𝐵 ) ) ↔ 𝑠 ∈ 𝐵 ) ) |
32 |
31
|
biimpcd |
⊢ ( 𝑠 ∈ ( Base ‘ ( 𝑆 ↾s 𝐵 ) ) → ( 𝜑 → 𝑠 ∈ 𝐵 ) ) |
33 |
32
|
adantr |
⊢ ( ( 𝑠 ∈ ( Base ‘ ( 𝑆 ↾s 𝐵 ) ) ∧ 𝑡 ∈ ( Base ‘ ( 𝑆 ↾s 𝐵 ) ) ) → ( 𝜑 → 𝑠 ∈ 𝐵 ) ) |
34 |
33
|
impcom |
⊢ ( ( 𝜑 ∧ ( 𝑠 ∈ ( Base ‘ ( 𝑆 ↾s 𝐵 ) ) ∧ 𝑡 ∈ ( Base ‘ ( 𝑆 ↾s 𝐵 ) ) ) ) → 𝑠 ∈ 𝐵 ) |
35 |
30
|
eleq2d |
⊢ ( 𝜑 → ( 𝑡 ∈ ( Base ‘ ( 𝑆 ↾s 𝐵 ) ) ↔ 𝑡 ∈ 𝐵 ) ) |
36 |
35
|
biimpcd |
⊢ ( 𝑡 ∈ ( Base ‘ ( 𝑆 ↾s 𝐵 ) ) → ( 𝜑 → 𝑡 ∈ 𝐵 ) ) |
37 |
36
|
adantl |
⊢ ( ( 𝑠 ∈ ( Base ‘ ( 𝑆 ↾s 𝐵 ) ) ∧ 𝑡 ∈ ( Base ‘ ( 𝑆 ↾s 𝐵 ) ) ) → ( 𝜑 → 𝑡 ∈ 𝐵 ) ) |
38 |
37
|
impcom |
⊢ ( ( 𝜑 ∧ ( 𝑠 ∈ ( Base ‘ ( 𝑆 ↾s 𝐵 ) ) ∧ 𝑡 ∈ ( Base ‘ ( 𝑆 ↾s 𝐵 ) ) ) ) → 𝑡 ∈ 𝐵 ) |
39 |
1 2 3 4 5 13
|
ressply1add |
⊢ ( ( 𝜑 ∧ ( 𝑠 ∈ 𝐵 ∧ 𝑡 ∈ 𝐵 ) ) → ( 𝑠 ( +g ‘ 𝑈 ) 𝑡 ) = ( 𝑠 ( +g ‘ ( 𝑆 ↾s 𝐵 ) ) 𝑡 ) ) |
40 |
28 34 38 39
|
syl12anc |
⊢ ( ( 𝜑 ∧ ( 𝑠 ∈ ( Base ‘ ( 𝑆 ↾s 𝐵 ) ) ∧ 𝑡 ∈ ( Base ‘ ( 𝑆 ↾s 𝐵 ) ) ) ) → ( 𝑠 ( +g ‘ 𝑈 ) 𝑡 ) = ( 𝑠 ( +g ‘ ( 𝑆 ↾s 𝐵 ) ) 𝑡 ) ) |
41 |
40
|
eqcomd |
⊢ ( ( 𝜑 ∧ ( 𝑠 ∈ ( Base ‘ ( 𝑆 ↾s 𝐵 ) ) ∧ 𝑡 ∈ ( Base ‘ ( 𝑆 ↾s 𝐵 ) ) ) ) → ( 𝑠 ( +g ‘ ( 𝑆 ↾s 𝐵 ) ) 𝑡 ) = ( 𝑠 ( +g ‘ 𝑈 ) 𝑡 ) ) |
42 |
7
|
ffund |
⊢ ( 𝜑 → Fun 𝐹 ) |
43 |
7
|
frnd |
⊢ ( 𝜑 → ran 𝐹 ⊆ 𝐵 ) |
44 |
43 29
|
sseqtrd |
⊢ ( 𝜑 → ran 𝐹 ⊆ ( Base ‘ ( 𝑆 ↾s 𝐵 ) ) ) |
45 |
16 17 19 23 27 41 42 44
|
gsummgmpropd |
⊢ ( 𝜑 → ( ( 𝑆 ↾s 𝐵 ) Σg 𝐹 ) = ( 𝑈 Σg 𝐹 ) ) |
46 |
14 45
|
eqtrd |
⊢ ( 𝜑 → ( 𝑆 Σg 𝐹 ) = ( 𝑈 Σg 𝐹 ) ) |