Step |
Hyp |
Ref |
Expression |
1 |
|
nn0gsumfz.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
nn0gsumfz.0 |
⊢ 0 = ( 0g ‘ 𝐺 ) |
3 |
|
nn0gsumfz.g |
⊢ ( 𝜑 → 𝐺 ∈ CMnd ) |
4 |
|
nn0gsumfz.f |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝐵 ↑m ℕ0 ) ) |
5 |
|
fsfnn0gsumfsffz.s |
⊢ ( 𝜑 → 𝑆 ∈ ℕ0 ) |
6 |
|
fsfnn0gsumfsffz.h |
⊢ 𝐻 = ( 𝐹 ↾ ( 0 ... 𝑆 ) ) |
7 |
6
|
oveq2i |
⊢ ( 𝐺 Σg 𝐻 ) = ( 𝐺 Σg ( 𝐹 ↾ ( 0 ... 𝑆 ) ) ) |
8 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑆 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 0 ) ) → 𝐺 ∈ CMnd ) |
9 |
|
nn0ex |
⊢ ℕ0 ∈ V |
10 |
9
|
a1i |
⊢ ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑆 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 0 ) ) → ℕ0 ∈ V ) |
11 |
|
elmapi |
⊢ ( 𝐹 ∈ ( 𝐵 ↑m ℕ0 ) → 𝐹 : ℕ0 ⟶ 𝐵 ) |
12 |
4 11
|
syl |
⊢ ( 𝜑 → 𝐹 : ℕ0 ⟶ 𝐵 ) |
13 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑆 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 0 ) ) → 𝐹 : ℕ0 ⟶ 𝐵 ) |
14 |
2
|
fvexi |
⊢ 0 ∈ V |
15 |
14
|
a1i |
⊢ ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑆 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 0 ) ) → 0 ∈ V ) |
16 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑆 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 0 ) ) → 𝐹 ∈ ( 𝐵 ↑m ℕ0 ) ) |
17 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑆 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 0 ) ) → 𝑆 ∈ ℕ0 ) |
18 |
|
simpr |
⊢ ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑆 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 0 ) ) → ∀ 𝑥 ∈ ℕ0 ( 𝑆 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 0 ) ) |
19 |
15 16 17 18
|
suppssfz |
⊢ ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑆 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 0 ) ) → ( 𝐹 supp 0 ) ⊆ ( 0 ... 𝑆 ) ) |
20 |
|
elmapfun |
⊢ ( 𝐹 ∈ ( 𝐵 ↑m ℕ0 ) → Fun 𝐹 ) |
21 |
4 20
|
syl |
⊢ ( 𝜑 → Fun 𝐹 ) |
22 |
14
|
a1i |
⊢ ( 𝜑 → 0 ∈ V ) |
23 |
4 21 22
|
3jca |
⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝐵 ↑m ℕ0 ) ∧ Fun 𝐹 ∧ 0 ∈ V ) ) |
24 |
|
fzfid |
⊢ ( 𝜑 → ( 0 ... 𝑆 ) ∈ Fin ) |
25 |
24
|
anim1i |
⊢ ( ( 𝜑 ∧ ( 𝐹 supp 0 ) ⊆ ( 0 ... 𝑆 ) ) → ( ( 0 ... 𝑆 ) ∈ Fin ∧ ( 𝐹 supp 0 ) ⊆ ( 0 ... 𝑆 ) ) ) |
26 |
|
suppssfifsupp |
⊢ ( ( ( 𝐹 ∈ ( 𝐵 ↑m ℕ0 ) ∧ Fun 𝐹 ∧ 0 ∈ V ) ∧ ( ( 0 ... 𝑆 ) ∈ Fin ∧ ( 𝐹 supp 0 ) ⊆ ( 0 ... 𝑆 ) ) ) → 𝐹 finSupp 0 ) |
27 |
23 25 26
|
syl2an2r |
⊢ ( ( 𝜑 ∧ ( 𝐹 supp 0 ) ⊆ ( 0 ... 𝑆 ) ) → 𝐹 finSupp 0 ) |
28 |
19 27
|
syldan |
⊢ ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑆 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 0 ) ) → 𝐹 finSupp 0 ) |
29 |
1 2 8 10 13 19 28
|
gsumres |
⊢ ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑆 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 0 ) ) → ( 𝐺 Σg ( 𝐹 ↾ ( 0 ... 𝑆 ) ) ) = ( 𝐺 Σg 𝐹 ) ) |
30 |
7 29
|
eqtr2id |
⊢ ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑆 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 0 ) ) → ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝐻 ) ) |
31 |
30
|
ex |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℕ0 ( 𝑆 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 0 ) → ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝐻 ) ) ) |