Step |
Hyp |
Ref |
Expression |
1 |
|
gsummptres.0 |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
gsummptres.1 |
⊢ 0 = ( 0g ‘ 𝐺 ) |
3 |
|
gsummptres.2 |
⊢ ( 𝜑 → 𝐺 ∈ CMnd ) |
4 |
|
gsummptres.3 |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
5 |
|
gsummptres.4 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ 𝐵 ) |
6 |
|
gsummptres.5 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∖ 𝐷 ) ) → 𝐶 = 0 ) |
7 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
8 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) |
9 |
2
|
fvexi |
⊢ 0 ∈ V |
10 |
9
|
a1i |
⊢ ( 𝜑 → 0 ∈ V ) |
11 |
8 4 5 10
|
fsuppmptdm |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) finSupp 0 ) |
12 |
|
inindif |
⊢ ( ( 𝐴 ∩ 𝐷 ) ∩ ( 𝐴 ∖ 𝐷 ) ) = ∅ |
13 |
12
|
a1i |
⊢ ( 𝜑 → ( ( 𝐴 ∩ 𝐷 ) ∩ ( 𝐴 ∖ 𝐷 ) ) = ∅ ) |
14 |
|
inundif |
⊢ ( ( 𝐴 ∩ 𝐷 ) ∪ ( 𝐴 ∖ 𝐷 ) ) = 𝐴 |
15 |
14
|
eqcomi |
⊢ 𝐴 = ( ( 𝐴 ∩ 𝐷 ) ∪ ( 𝐴 ∖ 𝐷 ) ) |
16 |
15
|
a1i |
⊢ ( 𝜑 → 𝐴 = ( ( 𝐴 ∩ 𝐷 ) ∪ ( 𝐴 ∖ 𝐷 ) ) ) |
17 |
1 2 7 3 4 5 11 13 16
|
gsumsplit2 |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ) = ( ( 𝐺 Σg ( 𝑥 ∈ ( 𝐴 ∩ 𝐷 ) ↦ 𝐶 ) ) ( +g ‘ 𝐺 ) ( 𝐺 Σg ( 𝑥 ∈ ( 𝐴 ∖ 𝐷 ) ↦ 𝐶 ) ) ) ) |
18 |
6
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐴 ∖ 𝐷 ) ↦ 𝐶 ) = ( 𝑥 ∈ ( 𝐴 ∖ 𝐷 ) ↦ 0 ) ) |
19 |
18
|
oveq2d |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑥 ∈ ( 𝐴 ∖ 𝐷 ) ↦ 𝐶 ) ) = ( 𝐺 Σg ( 𝑥 ∈ ( 𝐴 ∖ 𝐷 ) ↦ 0 ) ) ) |
20 |
|
cmnmnd |
⊢ ( 𝐺 ∈ CMnd → 𝐺 ∈ Mnd ) |
21 |
3 20
|
syl |
⊢ ( 𝜑 → 𝐺 ∈ Mnd ) |
22 |
|
diffi |
⊢ ( 𝐴 ∈ Fin → ( 𝐴 ∖ 𝐷 ) ∈ Fin ) |
23 |
4 22
|
syl |
⊢ ( 𝜑 → ( 𝐴 ∖ 𝐷 ) ∈ Fin ) |
24 |
2
|
gsumz |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝐴 ∖ 𝐷 ) ∈ Fin ) → ( 𝐺 Σg ( 𝑥 ∈ ( 𝐴 ∖ 𝐷 ) ↦ 0 ) ) = 0 ) |
25 |
21 23 24
|
syl2anc |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑥 ∈ ( 𝐴 ∖ 𝐷 ) ↦ 0 ) ) = 0 ) |
26 |
19 25
|
eqtrd |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑥 ∈ ( 𝐴 ∖ 𝐷 ) ↦ 𝐶 ) ) = 0 ) |
27 |
26
|
oveq2d |
⊢ ( 𝜑 → ( ( 𝐺 Σg ( 𝑥 ∈ ( 𝐴 ∩ 𝐷 ) ↦ 𝐶 ) ) ( +g ‘ 𝐺 ) ( 𝐺 Σg ( 𝑥 ∈ ( 𝐴 ∖ 𝐷 ) ↦ 𝐶 ) ) ) = ( ( 𝐺 Σg ( 𝑥 ∈ ( 𝐴 ∩ 𝐷 ) ↦ 𝐶 ) ) ( +g ‘ 𝐺 ) 0 ) ) |
28 |
|
infi |
⊢ ( 𝐴 ∈ Fin → ( 𝐴 ∩ 𝐷 ) ∈ Fin ) |
29 |
4 28
|
syl |
⊢ ( 𝜑 → ( 𝐴 ∩ 𝐷 ) ∈ Fin ) |
30 |
|
inss1 |
⊢ ( 𝐴 ∩ 𝐷 ) ⊆ 𝐴 |
31 |
30
|
sseli |
⊢ ( 𝑥 ∈ ( 𝐴 ∩ 𝐷 ) → 𝑥 ∈ 𝐴 ) |
32 |
31 5
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∩ 𝐷 ) ) → 𝐶 ∈ 𝐵 ) |
33 |
32
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ( 𝐴 ∩ 𝐷 ) 𝐶 ∈ 𝐵 ) |
34 |
1 3 29 33
|
gsummptcl |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑥 ∈ ( 𝐴 ∩ 𝐷 ) ↦ 𝐶 ) ) ∈ 𝐵 ) |
35 |
1 7 2
|
mndrid |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝐺 Σg ( 𝑥 ∈ ( 𝐴 ∩ 𝐷 ) ↦ 𝐶 ) ) ∈ 𝐵 ) → ( ( 𝐺 Σg ( 𝑥 ∈ ( 𝐴 ∩ 𝐷 ) ↦ 𝐶 ) ) ( +g ‘ 𝐺 ) 0 ) = ( 𝐺 Σg ( 𝑥 ∈ ( 𝐴 ∩ 𝐷 ) ↦ 𝐶 ) ) ) |
36 |
21 34 35
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐺 Σg ( 𝑥 ∈ ( 𝐴 ∩ 𝐷 ) ↦ 𝐶 ) ) ( +g ‘ 𝐺 ) 0 ) = ( 𝐺 Σg ( 𝑥 ∈ ( 𝐴 ∩ 𝐷 ) ↦ 𝐶 ) ) ) |
37 |
27 36
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝐺 Σg ( 𝑥 ∈ ( 𝐴 ∩ 𝐷 ) ↦ 𝐶 ) ) ( +g ‘ 𝐺 ) ( 𝐺 Σg ( 𝑥 ∈ ( 𝐴 ∖ 𝐷 ) ↦ 𝐶 ) ) ) = ( 𝐺 Σg ( 𝑥 ∈ ( 𝐴 ∩ 𝐷 ) ↦ 𝐶 ) ) ) |
38 |
17 37
|
eqtrd |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ) = ( 𝐺 Σg ( 𝑥 ∈ ( 𝐴 ∩ 𝐷 ) ↦ 𝐶 ) ) ) |