Step |
Hyp |
Ref |
Expression |
1 |
|
stoweidlem8.1 |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) + ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ) |
2 |
|
stoweidlem8.2 |
⊢ Ⅎ 𝑡 𝐹 |
3 |
|
stoweidlem8.3 |
⊢ Ⅎ 𝑡 𝐺 |
4 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ) → 𝐺 ∈ 𝐴 ) |
5 |
|
eleq1 |
⊢ ( 𝑔 = 𝐺 → ( 𝑔 ∈ 𝐴 ↔ 𝐺 ∈ 𝐴 ) ) |
6 |
5
|
3anbi3d |
⊢ ( 𝑔 = 𝐺 → ( ( 𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) ↔ ( 𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ) ) ) |
7 |
3
|
nfeq2 |
⊢ Ⅎ 𝑡 𝑔 = 𝐺 |
8 |
|
fveq1 |
⊢ ( 𝑔 = 𝐺 → ( 𝑔 ‘ 𝑡 ) = ( 𝐺 ‘ 𝑡 ) ) |
9 |
8
|
oveq2d |
⊢ ( 𝑔 = 𝐺 → ( ( 𝐹 ‘ 𝑡 ) + ( 𝑔 ‘ 𝑡 ) ) = ( ( 𝐹 ‘ 𝑡 ) + ( 𝐺 ‘ 𝑡 ) ) ) |
10 |
9
|
adantr |
⊢ ( ( 𝑔 = 𝐺 ∧ 𝑡 ∈ 𝑇 ) → ( ( 𝐹 ‘ 𝑡 ) + ( 𝑔 ‘ 𝑡 ) ) = ( ( 𝐹 ‘ 𝑡 ) + ( 𝐺 ‘ 𝑡 ) ) ) |
11 |
7 10
|
mpteq2da |
⊢ ( 𝑔 = 𝐺 → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝐹 ‘ 𝑡 ) + ( 𝑔 ‘ 𝑡 ) ) ) = ( 𝑡 ∈ 𝑇 ↦ ( ( 𝐹 ‘ 𝑡 ) + ( 𝐺 ‘ 𝑡 ) ) ) ) |
12 |
11
|
eleq1d |
⊢ ( 𝑔 = 𝐺 → ( ( 𝑡 ∈ 𝑇 ↦ ( ( 𝐹 ‘ 𝑡 ) + ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ↔ ( 𝑡 ∈ 𝑇 ↦ ( ( 𝐹 ‘ 𝑡 ) + ( 𝐺 ‘ 𝑡 ) ) ) ∈ 𝐴 ) ) |
13 |
6 12
|
imbi12d |
⊢ ( 𝑔 = 𝐺 → ( ( ( 𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝐹 ‘ 𝑡 ) + ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ) ↔ ( ( 𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝐹 ‘ 𝑡 ) + ( 𝐺 ‘ 𝑡 ) ) ) ∈ 𝐴 ) ) ) |
14 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) → 𝐹 ∈ 𝐴 ) |
15 |
|
eleq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 ∈ 𝐴 ↔ 𝐹 ∈ 𝐴 ) ) |
16 |
15
|
3anbi2d |
⊢ ( 𝑓 = 𝐹 → ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) ↔ ( 𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) ) ) |
17 |
2
|
nfeq2 |
⊢ Ⅎ 𝑡 𝑓 = 𝐹 |
18 |
|
fveq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑡 ) = ( 𝐹 ‘ 𝑡 ) ) |
19 |
18
|
oveq1d |
⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ 𝑡 ) + ( 𝑔 ‘ 𝑡 ) ) = ( ( 𝐹 ‘ 𝑡 ) + ( 𝑔 ‘ 𝑡 ) ) ) |
20 |
19
|
adantr |
⊢ ( ( 𝑓 = 𝐹 ∧ 𝑡 ∈ 𝑇 ) → ( ( 𝑓 ‘ 𝑡 ) + ( 𝑔 ‘ 𝑡 ) ) = ( ( 𝐹 ‘ 𝑡 ) + ( 𝑔 ‘ 𝑡 ) ) ) |
21 |
17 20
|
mpteq2da |
⊢ ( 𝑓 = 𝐹 → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) + ( 𝑔 ‘ 𝑡 ) ) ) = ( 𝑡 ∈ 𝑇 ↦ ( ( 𝐹 ‘ 𝑡 ) + ( 𝑔 ‘ 𝑡 ) ) ) ) |
22 |
21
|
eleq1d |
⊢ ( 𝑓 = 𝐹 → ( ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) + ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ↔ ( 𝑡 ∈ 𝑇 ↦ ( ( 𝐹 ‘ 𝑡 ) + ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ) ) |
23 |
16 22
|
imbi12d |
⊢ ( 𝑓 = 𝐹 → ( ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) + ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ) ↔ ( ( 𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝐹 ‘ 𝑡 ) + ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ) ) ) |
24 |
23 1
|
vtoclg |
⊢ ( 𝐹 ∈ 𝐴 → ( ( 𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝐹 ‘ 𝑡 ) + ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ) ) |
25 |
14 24
|
mpcom |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝐹 ‘ 𝑡 ) + ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ) |
26 |
13 25
|
vtoclg |
⊢ ( 𝐺 ∈ 𝐴 → ( ( 𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝐹 ‘ 𝑡 ) + ( 𝐺 ‘ 𝑡 ) ) ) ∈ 𝐴 ) ) |
27 |
4 26
|
mpcom |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝐹 ‘ 𝑡 ) + ( 𝐺 ‘ 𝑡 ) ) ) ∈ 𝐴 ) |