Step |
Hyp |
Ref |
Expression |
1 |
|
stoweidlem2.1 |
⊢ Ⅎ 𝑡 𝜑 |
2 |
|
stoweidlem2.2 |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) · ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ) |
3 |
|
stoweidlem2.3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( 𝑡 ∈ 𝑇 ↦ 𝑥 ) ∈ 𝐴 ) |
4 |
|
stoweidlem2.4 |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) → 𝑓 : 𝑇 ⟶ ℝ ) |
5 |
|
stoweidlem2.5 |
⊢ ( 𝜑 → 𝐸 ∈ ℝ ) |
6 |
|
stoweidlem2.6 |
⊢ ( 𝜑 → 𝐹 ∈ 𝐴 ) |
7 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ) → 𝑡 ∈ 𝑇 ) |
8 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ) → 𝐸 ∈ ℝ ) |
9 |
|
eqidd |
⊢ ( 𝑠 = 𝑡 → 𝐸 = 𝐸 ) |
10 |
9
|
cbvmptv |
⊢ ( 𝑠 ∈ 𝑇 ↦ 𝐸 ) = ( 𝑡 ∈ 𝑇 ↦ 𝐸 ) |
11 |
10
|
fvmpt2 |
⊢ ( ( 𝑡 ∈ 𝑇 ∧ 𝐸 ∈ ℝ ) → ( ( 𝑠 ∈ 𝑇 ↦ 𝐸 ) ‘ 𝑡 ) = 𝐸 ) |
12 |
7 8 11
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ) → ( ( 𝑠 ∈ 𝑇 ↦ 𝐸 ) ‘ 𝑡 ) = 𝐸 ) |
13 |
12
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ) → 𝐸 = ( ( 𝑠 ∈ 𝑇 ↦ 𝐸 ) ‘ 𝑡 ) ) |
14 |
13
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ) → ( 𝐸 · ( 𝐹 ‘ 𝑡 ) ) = ( ( ( 𝑠 ∈ 𝑇 ↦ 𝐸 ) ‘ 𝑡 ) · ( 𝐹 ‘ 𝑡 ) ) ) |
15 |
1 14
|
mpteq2da |
⊢ ( 𝜑 → ( 𝑡 ∈ 𝑇 ↦ ( 𝐸 · ( 𝐹 ‘ 𝑡 ) ) ) = ( 𝑡 ∈ 𝑇 ↦ ( ( ( 𝑠 ∈ 𝑇 ↦ 𝐸 ) ‘ 𝑡 ) · ( 𝐹 ‘ 𝑡 ) ) ) ) |
16 |
|
id |
⊢ ( 𝑥 = 𝐸 → 𝑥 = 𝐸 ) |
17 |
16
|
mpteq2dv |
⊢ ( 𝑥 = 𝐸 → ( 𝑡 ∈ 𝑇 ↦ 𝑥 ) = ( 𝑡 ∈ 𝑇 ↦ 𝐸 ) ) |
18 |
17
|
eleq1d |
⊢ ( 𝑥 = 𝐸 → ( ( 𝑡 ∈ 𝑇 ↦ 𝑥 ) ∈ 𝐴 ↔ ( 𝑡 ∈ 𝑇 ↦ 𝐸 ) ∈ 𝐴 ) ) |
19 |
18
|
imbi2d |
⊢ ( 𝑥 = 𝐸 → ( ( 𝜑 → ( 𝑡 ∈ 𝑇 ↦ 𝑥 ) ∈ 𝐴 ) ↔ ( 𝜑 → ( 𝑡 ∈ 𝑇 ↦ 𝐸 ) ∈ 𝐴 ) ) ) |
20 |
3
|
expcom |
⊢ ( 𝑥 ∈ ℝ → ( 𝜑 → ( 𝑡 ∈ 𝑇 ↦ 𝑥 ) ∈ 𝐴 ) ) |
21 |
19 20
|
vtoclga |
⊢ ( 𝐸 ∈ ℝ → ( 𝜑 → ( 𝑡 ∈ 𝑇 ↦ 𝐸 ) ∈ 𝐴 ) ) |
22 |
5 21
|
mpcom |
⊢ ( 𝜑 → ( 𝑡 ∈ 𝑇 ↦ 𝐸 ) ∈ 𝐴 ) |
23 |
10 22
|
eqeltrid |
⊢ ( 𝜑 → ( 𝑠 ∈ 𝑇 ↦ 𝐸 ) ∈ 𝐴 ) |
24 |
|
fveq1 |
⊢ ( 𝑓 = ( 𝑠 ∈ 𝑇 ↦ 𝐸 ) → ( 𝑓 ‘ 𝑡 ) = ( ( 𝑠 ∈ 𝑇 ↦ 𝐸 ) ‘ 𝑡 ) ) |
25 |
24
|
oveq1d |
⊢ ( 𝑓 = ( 𝑠 ∈ 𝑇 ↦ 𝐸 ) → ( ( 𝑓 ‘ 𝑡 ) · ( 𝐹 ‘ 𝑡 ) ) = ( ( ( 𝑠 ∈ 𝑇 ↦ 𝐸 ) ‘ 𝑡 ) · ( 𝐹 ‘ 𝑡 ) ) ) |
26 |
25
|
mpteq2dv |
⊢ ( 𝑓 = ( 𝑠 ∈ 𝑇 ↦ 𝐸 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) · ( 𝐹 ‘ 𝑡 ) ) ) = ( 𝑡 ∈ 𝑇 ↦ ( ( ( 𝑠 ∈ 𝑇 ↦ 𝐸 ) ‘ 𝑡 ) · ( 𝐹 ‘ 𝑡 ) ) ) ) |
27 |
26
|
eleq1d |
⊢ ( 𝑓 = ( 𝑠 ∈ 𝑇 ↦ 𝐸 ) → ( ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) · ( 𝐹 ‘ 𝑡 ) ) ) ∈ 𝐴 ↔ ( 𝑡 ∈ 𝑇 ↦ ( ( ( 𝑠 ∈ 𝑇 ↦ 𝐸 ) ‘ 𝑡 ) · ( 𝐹 ‘ 𝑡 ) ) ) ∈ 𝐴 ) ) |
28 |
27
|
imbi2d |
⊢ ( 𝑓 = ( 𝑠 ∈ 𝑇 ↦ 𝐸 ) → ( ( 𝜑 → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) · ( 𝐹 ‘ 𝑡 ) ) ) ∈ 𝐴 ) ↔ ( 𝜑 → ( 𝑡 ∈ 𝑇 ↦ ( ( ( 𝑠 ∈ 𝑇 ↦ 𝐸 ) ‘ 𝑡 ) · ( 𝐹 ‘ 𝑡 ) ) ) ∈ 𝐴 ) ) ) |
29 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) → 𝐹 ∈ 𝐴 ) |
30 |
|
fveq1 |
⊢ ( 𝑔 = 𝐹 → ( 𝑔 ‘ 𝑡 ) = ( 𝐹 ‘ 𝑡 ) ) |
31 |
30
|
oveq2d |
⊢ ( 𝑔 = 𝐹 → ( ( 𝑓 ‘ 𝑡 ) · ( 𝑔 ‘ 𝑡 ) ) = ( ( 𝑓 ‘ 𝑡 ) · ( 𝐹 ‘ 𝑡 ) ) ) |
32 |
31
|
mpteq2dv |
⊢ ( 𝑔 = 𝐹 → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) · ( 𝑔 ‘ 𝑡 ) ) ) = ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) · ( 𝐹 ‘ 𝑡 ) ) ) ) |
33 |
32
|
eleq1d |
⊢ ( 𝑔 = 𝐹 → ( ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) · ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ↔ ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) · ( 𝐹 ‘ 𝑡 ) ) ) ∈ 𝐴 ) ) |
34 |
33
|
imbi2d |
⊢ ( 𝑔 = 𝐹 → ( ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) · ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ) ↔ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) · ( 𝐹 ‘ 𝑡 ) ) ) ∈ 𝐴 ) ) ) |
35 |
2
|
3comr |
⊢ ( ( 𝑔 ∈ 𝐴 ∧ 𝜑 ∧ 𝑓 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) · ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ) |
36 |
35
|
3expib |
⊢ ( 𝑔 ∈ 𝐴 → ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) · ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ) ) |
37 |
34 36
|
vtoclga |
⊢ ( 𝐹 ∈ 𝐴 → ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) · ( 𝐹 ‘ 𝑡 ) ) ) ∈ 𝐴 ) ) |
38 |
29 37
|
mpcom |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) · ( 𝐹 ‘ 𝑡 ) ) ) ∈ 𝐴 ) |
39 |
38
|
expcom |
⊢ ( 𝑓 ∈ 𝐴 → ( 𝜑 → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) · ( 𝐹 ‘ 𝑡 ) ) ) ∈ 𝐴 ) ) |
40 |
28 39
|
vtoclga |
⊢ ( ( 𝑠 ∈ 𝑇 ↦ 𝐸 ) ∈ 𝐴 → ( 𝜑 → ( 𝑡 ∈ 𝑇 ↦ ( ( ( 𝑠 ∈ 𝑇 ↦ 𝐸 ) ‘ 𝑡 ) · ( 𝐹 ‘ 𝑡 ) ) ) ∈ 𝐴 ) ) |
41 |
23 40
|
mpcom |
⊢ ( 𝜑 → ( 𝑡 ∈ 𝑇 ↦ ( ( ( 𝑠 ∈ 𝑇 ↦ 𝐸 ) ‘ 𝑡 ) · ( 𝐹 ‘ 𝑡 ) ) ) ∈ 𝐴 ) |
42 |
15 41
|
eqeltrd |
⊢ ( 𝜑 → ( 𝑡 ∈ 𝑇 ↦ ( 𝐸 · ( 𝐹 ‘ 𝑡 ) ) ) ∈ 𝐴 ) |