Step |
Hyp |
Ref |
Expression |
1 |
|
eldprdi.0 |
⊢ 0 = ( 0g ‘ 𝐺 ) |
2 |
|
eldprdi.w |
⊢ 𝑊 = { ℎ ∈ X 𝑖 ∈ 𝐼 ( 𝑆 ‘ 𝑖 ) ∣ ℎ finSupp 0 } |
3 |
|
eldprdi.1 |
⊢ ( 𝜑 → 𝐺 dom DProd 𝑆 ) |
4 |
|
eldprdi.2 |
⊢ ( 𝜑 → dom 𝑆 = 𝐼 ) |
5 |
|
eldprdi.3 |
⊢ ( 𝜑 → 𝐹 ∈ 𝑊 ) |
6 |
|
dprdfinv.b |
⊢ 𝑁 = ( invg ‘ 𝐺 ) |
7 |
|
dprdgrp |
⊢ ( 𝐺 dom DProd 𝑆 → 𝐺 ∈ Grp ) |
8 |
3 7
|
syl |
⊢ ( 𝜑 → 𝐺 ∈ Grp ) |
9 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
10 |
9 6
|
grpinvf |
⊢ ( 𝐺 ∈ Grp → 𝑁 : ( Base ‘ 𝐺 ) ⟶ ( Base ‘ 𝐺 ) ) |
11 |
8 10
|
syl |
⊢ ( 𝜑 → 𝑁 : ( Base ‘ 𝐺 ) ⟶ ( Base ‘ 𝐺 ) ) |
12 |
2 3 4 5 9
|
dprdff |
⊢ ( 𝜑 → 𝐹 : 𝐼 ⟶ ( Base ‘ 𝐺 ) ) |
13 |
|
fcompt |
⊢ ( ( 𝑁 : ( Base ‘ 𝐺 ) ⟶ ( Base ‘ 𝐺 ) ∧ 𝐹 : 𝐼 ⟶ ( Base ‘ 𝐺 ) ) → ( 𝑁 ∘ 𝐹 ) = ( 𝑥 ∈ 𝐼 ↦ ( 𝑁 ‘ ( 𝐹 ‘ 𝑥 ) ) ) ) |
14 |
11 12 13
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ∘ 𝐹 ) = ( 𝑥 ∈ 𝐼 ↦ ( 𝑁 ‘ ( 𝐹 ‘ 𝑥 ) ) ) ) |
15 |
3 4
|
dprdf2 |
⊢ ( 𝜑 → 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ) |
16 |
15
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝑆 ‘ 𝑥 ) ∈ ( SubGrp ‘ 𝐺 ) ) |
17 |
2 3 4 5
|
dprdfcl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( 𝑆 ‘ 𝑥 ) ) |
18 |
6
|
subginvcl |
⊢ ( ( ( 𝑆 ‘ 𝑥 ) ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐹 ‘ 𝑥 ) ∈ ( 𝑆 ‘ 𝑥 ) ) → ( 𝑁 ‘ ( 𝐹 ‘ 𝑥 ) ) ∈ ( 𝑆 ‘ 𝑥 ) ) |
19 |
16 17 18
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝑁 ‘ ( 𝐹 ‘ 𝑥 ) ) ∈ ( 𝑆 ‘ 𝑥 ) ) |
20 |
3 4
|
dprddomcld |
⊢ ( 𝜑 → 𝐼 ∈ V ) |
21 |
20
|
mptexd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐼 ↦ ( 𝑁 ‘ ( 𝐹 ‘ 𝑥 ) ) ) ∈ V ) |
22 |
|
funmpt |
⊢ Fun ( 𝑥 ∈ 𝐼 ↦ ( 𝑁 ‘ ( 𝐹 ‘ 𝑥 ) ) ) |
23 |
22
|
a1i |
⊢ ( 𝜑 → Fun ( 𝑥 ∈ 𝐼 ↦ ( 𝑁 ‘ ( 𝐹 ‘ 𝑥 ) ) ) ) |
24 |
2 3 4 5
|
dprdffsupp |
⊢ ( 𝜑 → 𝐹 finSupp 0 ) |
25 |
|
ssidd |
⊢ ( 𝜑 → ( 𝐹 supp 0 ) ⊆ ( 𝐹 supp 0 ) ) |
26 |
1
|
fvexi |
⊢ 0 ∈ V |
27 |
26
|
a1i |
⊢ ( 𝜑 → 0 ∈ V ) |
28 |
12 25 20 27
|
suppssr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐼 ∖ ( 𝐹 supp 0 ) ) ) → ( 𝐹 ‘ 𝑥 ) = 0 ) |
29 |
28
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐼 ∖ ( 𝐹 supp 0 ) ) ) → ( 𝑁 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝑁 ‘ 0 ) ) |
30 |
1 6
|
grpinvid |
⊢ ( 𝐺 ∈ Grp → ( 𝑁 ‘ 0 ) = 0 ) |
31 |
8 30
|
syl |
⊢ ( 𝜑 → ( 𝑁 ‘ 0 ) = 0 ) |
32 |
31
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐼 ∖ ( 𝐹 supp 0 ) ) ) → ( 𝑁 ‘ 0 ) = 0 ) |
33 |
29 32
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐼 ∖ ( 𝐹 supp 0 ) ) ) → ( 𝑁 ‘ ( 𝐹 ‘ 𝑥 ) ) = 0 ) |
34 |
33 20
|
suppss2 |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐼 ↦ ( 𝑁 ‘ ( 𝐹 ‘ 𝑥 ) ) ) supp 0 ) ⊆ ( 𝐹 supp 0 ) ) |
35 |
|
fsuppsssupp |
⊢ ( ( ( ( 𝑥 ∈ 𝐼 ↦ ( 𝑁 ‘ ( 𝐹 ‘ 𝑥 ) ) ) ∈ V ∧ Fun ( 𝑥 ∈ 𝐼 ↦ ( 𝑁 ‘ ( 𝐹 ‘ 𝑥 ) ) ) ) ∧ ( 𝐹 finSupp 0 ∧ ( ( 𝑥 ∈ 𝐼 ↦ ( 𝑁 ‘ ( 𝐹 ‘ 𝑥 ) ) ) supp 0 ) ⊆ ( 𝐹 supp 0 ) ) ) → ( 𝑥 ∈ 𝐼 ↦ ( 𝑁 ‘ ( 𝐹 ‘ 𝑥 ) ) ) finSupp 0 ) |
36 |
21 23 24 34 35
|
syl22anc |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐼 ↦ ( 𝑁 ‘ ( 𝐹 ‘ 𝑥 ) ) ) finSupp 0 ) |
37 |
2 3 4 19 36
|
dprdwd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐼 ↦ ( 𝑁 ‘ ( 𝐹 ‘ 𝑥 ) ) ) ∈ 𝑊 ) |
38 |
14 37
|
eqeltrd |
⊢ ( 𝜑 → ( 𝑁 ∘ 𝐹 ) ∈ 𝑊 ) |
39 |
|
eqid |
⊢ ( Cntz ‘ 𝐺 ) = ( Cntz ‘ 𝐺 ) |
40 |
2 3 4 5 39
|
dprdfcntz |
⊢ ( 𝜑 → ran 𝐹 ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ran 𝐹 ) ) |
41 |
9 1 39 6 8 20 12 40 24
|
gsumzinv |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑁 ∘ 𝐹 ) ) = ( 𝑁 ‘ ( 𝐺 Σg 𝐹 ) ) ) |
42 |
38 41
|
jca |
⊢ ( 𝜑 → ( ( 𝑁 ∘ 𝐹 ) ∈ 𝑊 ∧ ( 𝐺 Σg ( 𝑁 ∘ 𝐹 ) ) = ( 𝑁 ‘ ( 𝐺 Σg 𝐹 ) ) ) ) |