Step |
Hyp |
Ref |
Expression |
1 |
|
climneg.1 |
⊢ Ⅎ 𝑘 𝜑 |
2 |
|
climneg.2 |
⊢ Ⅎ 𝑘 𝐹 |
3 |
|
climneg.3 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
4 |
|
climneg.4 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
5 |
|
climneg.5 |
⊢ ( 𝜑 → 𝐹 ⇝ 𝐴 ) |
6 |
|
climneg.6 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
7 |
|
nfmpt1 |
⊢ Ⅎ 𝑘 ( 𝑘 ∈ 𝑍 ↦ - 1 ) |
8 |
|
nfmpt1 |
⊢ Ⅎ 𝑘 ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) |
9 |
3
|
fvexi |
⊢ 𝑍 ∈ V |
10 |
9
|
mptex |
⊢ ( 𝑘 ∈ 𝑍 ↦ - 1 ) ∈ V |
11 |
10
|
a1i |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑍 ↦ - 1 ) ∈ V ) |
12 |
|
1cnd |
⊢ ( 𝜑 → 1 ∈ ℂ ) |
13 |
12
|
negcld |
⊢ ( 𝜑 → - 1 ∈ ℂ ) |
14 |
|
eqidd |
⊢ ( 𝑗 ∈ 𝑍 → ( 𝑘 ∈ 𝑍 ↦ - 1 ) = ( 𝑘 ∈ 𝑍 ↦ - 1 ) ) |
15 |
|
eqidd |
⊢ ( ( 𝑗 ∈ 𝑍 ∧ 𝑘 = 𝑗 ) → - 1 = - 1 ) |
16 |
|
id |
⊢ ( 𝑗 ∈ 𝑍 → 𝑗 ∈ 𝑍 ) |
17 |
|
1cnd |
⊢ ( 𝑗 ∈ 𝑍 → 1 ∈ ℂ ) |
18 |
17
|
negcld |
⊢ ( 𝑗 ∈ 𝑍 → - 1 ∈ ℂ ) |
19 |
14 15 16 18
|
fvmptd |
⊢ ( 𝑗 ∈ 𝑍 → ( ( 𝑘 ∈ 𝑍 ↦ - 1 ) ‘ 𝑗 ) = - 1 ) |
20 |
19
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( ( 𝑘 ∈ 𝑍 ↦ - 1 ) ‘ 𝑗 ) = - 1 ) |
21 |
3 4 11 13 20
|
climconst |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑍 ↦ - 1 ) ⇝ - 1 ) |
22 |
9
|
mptex |
⊢ ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) ∈ V |
23 |
22
|
a1i |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) ∈ V ) |
24 |
|
neg1cn |
⊢ - 1 ∈ ℂ |
25 |
|
eqid |
⊢ ( 𝑘 ∈ 𝑍 ↦ - 1 ) = ( 𝑘 ∈ 𝑍 ↦ - 1 ) |
26 |
25
|
fvmpt2 |
⊢ ( ( 𝑘 ∈ 𝑍 ∧ - 1 ∈ ℂ ) → ( ( 𝑘 ∈ 𝑍 ↦ - 1 ) ‘ 𝑘 ) = - 1 ) |
27 |
24 26
|
mpan2 |
⊢ ( 𝑘 ∈ 𝑍 → ( ( 𝑘 ∈ 𝑍 ↦ - 1 ) ‘ 𝑘 ) = - 1 ) |
28 |
27 24
|
eqeltrdi |
⊢ ( 𝑘 ∈ 𝑍 → ( ( 𝑘 ∈ 𝑍 ↦ - 1 ) ‘ 𝑘 ) ∈ ℂ ) |
29 |
28
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝑘 ∈ 𝑍 ↦ - 1 ) ‘ 𝑘 ) ∈ ℂ ) |
30 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 𝑘 ∈ 𝑍 ) |
31 |
6
|
negcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → - ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
32 |
|
eqid |
⊢ ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) = ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) |
33 |
32
|
fvmpt2 |
⊢ ( ( 𝑘 ∈ 𝑍 ∧ - ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) → ( ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) ‘ 𝑘 ) = - ( 𝐹 ‘ 𝑘 ) ) |
34 |
30 31 33
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) ‘ 𝑘 ) = - ( 𝐹 ‘ 𝑘 ) ) |
35 |
6
|
mulm1d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( - 1 · ( 𝐹 ‘ 𝑘 ) ) = - ( 𝐹 ‘ 𝑘 ) ) |
36 |
27
|
eqcomd |
⊢ ( 𝑘 ∈ 𝑍 → - 1 = ( ( 𝑘 ∈ 𝑍 ↦ - 1 ) ‘ 𝑘 ) ) |
37 |
36
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → - 1 = ( ( 𝑘 ∈ 𝑍 ↦ - 1 ) ‘ 𝑘 ) ) |
38 |
37
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( - 1 · ( 𝐹 ‘ 𝑘 ) ) = ( ( ( 𝑘 ∈ 𝑍 ↦ - 1 ) ‘ 𝑘 ) · ( 𝐹 ‘ 𝑘 ) ) ) |
39 |
34 35 38
|
3eqtr2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) ‘ 𝑘 ) = ( ( ( 𝑘 ∈ 𝑍 ↦ - 1 ) ‘ 𝑘 ) · ( 𝐹 ‘ 𝑘 ) ) ) |
40 |
1 7 2 8 3 4 21 23 5 29 6 39
|
climmulf |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) ⇝ ( - 1 · 𝐴 ) ) |
41 |
|
climcl |
⊢ ( 𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ ) |
42 |
5 41
|
syl |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
43 |
42
|
mulm1d |
⊢ ( 𝜑 → ( - 1 · 𝐴 ) = - 𝐴 ) |
44 |
40 43
|
breqtrd |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑍 ↦ - ( 𝐹 ‘ 𝑘 ) ) ⇝ - 𝐴 ) |