Step |
Hyp |
Ref |
Expression |
1 |
|
mulc1cncfg.1 |
⊢ Ⅎ 𝑥 𝐹 |
2 |
|
mulc1cncfg.2 |
⊢ Ⅎ 𝑥 𝜑 |
3 |
|
mulc1cncfg.3 |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝐴 –cn→ ℂ ) ) |
4 |
|
mulc1cncfg.4 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
5 |
|
eqid |
⊢ ( 𝑥 ∈ ℂ ↦ ( 𝐵 · 𝑥 ) ) = ( 𝑥 ∈ ℂ ↦ ( 𝐵 · 𝑥 ) ) |
6 |
5
|
mulc1cncf |
⊢ ( 𝐵 ∈ ℂ → ( 𝑥 ∈ ℂ ↦ ( 𝐵 · 𝑥 ) ) ∈ ( ℂ –cn→ ℂ ) ) |
7 |
4 6
|
syl |
⊢ ( 𝜑 → ( 𝑥 ∈ ℂ ↦ ( 𝐵 · 𝑥 ) ) ∈ ( ℂ –cn→ ℂ ) ) |
8 |
|
cncff |
⊢ ( ( 𝑥 ∈ ℂ ↦ ( 𝐵 · 𝑥 ) ) ∈ ( ℂ –cn→ ℂ ) → ( 𝑥 ∈ ℂ ↦ ( 𝐵 · 𝑥 ) ) : ℂ ⟶ ℂ ) |
9 |
7 8
|
syl |
⊢ ( 𝜑 → ( 𝑥 ∈ ℂ ↦ ( 𝐵 · 𝑥 ) ) : ℂ ⟶ ℂ ) |
10 |
|
cncff |
⊢ ( 𝐹 ∈ ( 𝐴 –cn→ ℂ ) → 𝐹 : 𝐴 ⟶ ℂ ) |
11 |
3 10
|
syl |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ℂ ) |
12 |
|
fcompt |
⊢ ( ( ( 𝑥 ∈ ℂ ↦ ( 𝐵 · 𝑥 ) ) : ℂ ⟶ ℂ ∧ 𝐹 : 𝐴 ⟶ ℂ ) → ( ( 𝑥 ∈ ℂ ↦ ( 𝐵 · 𝑥 ) ) ∘ 𝐹 ) = ( 𝑡 ∈ 𝐴 ↦ ( ( 𝑥 ∈ ℂ ↦ ( 𝐵 · 𝑥 ) ) ‘ ( 𝐹 ‘ 𝑡 ) ) ) ) |
13 |
9 11 12
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ℂ ↦ ( 𝐵 · 𝑥 ) ) ∘ 𝐹 ) = ( 𝑡 ∈ 𝐴 ↦ ( ( 𝑥 ∈ ℂ ↦ ( 𝐵 · 𝑥 ) ) ‘ ( 𝐹 ‘ 𝑡 ) ) ) ) |
14 |
11
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑡 ) ∈ ℂ ) |
15 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) |
16 |
15 14
|
mulcld |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝐴 ) → ( 𝐵 · ( 𝐹 ‘ 𝑡 ) ) ∈ ℂ ) |
17 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑡 |
18 |
1 17
|
nffv |
⊢ Ⅎ 𝑥 ( 𝐹 ‘ 𝑡 ) |
19 |
|
nfcv |
⊢ Ⅎ 𝑥 𝐵 |
20 |
|
nfcv |
⊢ Ⅎ 𝑥 · |
21 |
19 20 18
|
nfov |
⊢ Ⅎ 𝑥 ( 𝐵 · ( 𝐹 ‘ 𝑡 ) ) |
22 |
|
oveq2 |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑡 ) → ( 𝐵 · 𝑥 ) = ( 𝐵 · ( 𝐹 ‘ 𝑡 ) ) ) |
23 |
18 21 22 5
|
fvmptf |
⊢ ( ( ( 𝐹 ‘ 𝑡 ) ∈ ℂ ∧ ( 𝐵 · ( 𝐹 ‘ 𝑡 ) ) ∈ ℂ ) → ( ( 𝑥 ∈ ℂ ↦ ( 𝐵 · 𝑥 ) ) ‘ ( 𝐹 ‘ 𝑡 ) ) = ( 𝐵 · ( 𝐹 ‘ 𝑡 ) ) ) |
24 |
14 16 23
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝐴 ) → ( ( 𝑥 ∈ ℂ ↦ ( 𝐵 · 𝑥 ) ) ‘ ( 𝐹 ‘ 𝑡 ) ) = ( 𝐵 · ( 𝐹 ‘ 𝑡 ) ) ) |
25 |
24
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑡 ∈ 𝐴 ↦ ( ( 𝑥 ∈ ℂ ↦ ( 𝐵 · 𝑥 ) ) ‘ ( 𝐹 ‘ 𝑡 ) ) ) = ( 𝑡 ∈ 𝐴 ↦ ( 𝐵 · ( 𝐹 ‘ 𝑡 ) ) ) ) |
26 |
|
nfcv |
⊢ Ⅎ 𝑡 𝐵 |
27 |
|
nfcv |
⊢ Ⅎ 𝑡 · |
28 |
|
nfcv |
⊢ Ⅎ 𝑡 ( 𝐹 ‘ 𝑥 ) |
29 |
26 27 28
|
nfov |
⊢ Ⅎ 𝑡 ( 𝐵 · ( 𝐹 ‘ 𝑥 ) ) |
30 |
|
fveq2 |
⊢ ( 𝑡 = 𝑥 → ( 𝐹 ‘ 𝑡 ) = ( 𝐹 ‘ 𝑥 ) ) |
31 |
30
|
oveq2d |
⊢ ( 𝑡 = 𝑥 → ( 𝐵 · ( 𝐹 ‘ 𝑡 ) ) = ( 𝐵 · ( 𝐹 ‘ 𝑥 ) ) ) |
32 |
21 29 31
|
cbvmpt |
⊢ ( 𝑡 ∈ 𝐴 ↦ ( 𝐵 · ( 𝐹 ‘ 𝑡 ) ) ) = ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 · ( 𝐹 ‘ 𝑥 ) ) ) |
33 |
25 32
|
eqtrdi |
⊢ ( 𝜑 → ( 𝑡 ∈ 𝐴 ↦ ( ( 𝑥 ∈ ℂ ↦ ( 𝐵 · 𝑥 ) ) ‘ ( 𝐹 ‘ 𝑡 ) ) ) = ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 · ( 𝐹 ‘ 𝑥 ) ) ) ) |
34 |
13 33
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ℂ ↦ ( 𝐵 · 𝑥 ) ) ∘ 𝐹 ) = ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 · ( 𝐹 ‘ 𝑥 ) ) ) ) |
35 |
3 7
|
cncfco |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ℂ ↦ ( 𝐵 · 𝑥 ) ) ∘ 𝐹 ) ∈ ( 𝐴 –cn→ ℂ ) ) |
36 |
34 35
|
eqeltrrd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 · ( 𝐹 ‘ 𝑥 ) ) ) ∈ ( 𝐴 –cn→ ℂ ) ) |