Step |
Hyp |
Ref |
Expression |
1 |
|
musumsum.1 |
⊢ ( 𝑚 = 1 → 𝐵 = 𝐶 ) |
2 |
|
musumsum.2 |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
3 |
|
musumsum.3 |
⊢ ( 𝜑 → 𝐴 ⊆ ℕ ) |
4 |
|
musumsum.4 |
⊢ ( 𝜑 → 1 ∈ 𝐴 ) |
5 |
|
musumsum.5 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) |
6 |
3
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ 𝐴 ) → 𝑚 ∈ ℕ ) |
7 |
|
musum |
⊢ ( 𝑚 ∈ ℕ → Σ 𝑘 ∈ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚 } ( μ ‘ 𝑘 ) = if ( 𝑚 = 1 , 1 , 0 ) ) |
8 |
6 7
|
syl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ 𝐴 ) → Σ 𝑘 ∈ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚 } ( μ ‘ 𝑘 ) = if ( 𝑚 = 1 , 1 , 0 ) ) |
9 |
8
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ 𝐴 ) → ( Σ 𝑘 ∈ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚 } ( μ ‘ 𝑘 ) · 𝐵 ) = ( if ( 𝑚 = 1 , 1 , 0 ) · 𝐵 ) ) |
10 |
|
fzfid |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ 𝐴 ) → ( 1 ... 𝑚 ) ∈ Fin ) |
11 |
|
dvdsssfz1 |
⊢ ( 𝑚 ∈ ℕ → { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚 } ⊆ ( 1 ... 𝑚 ) ) |
12 |
6 11
|
syl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ 𝐴 ) → { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚 } ⊆ ( 1 ... 𝑚 ) ) |
13 |
10 12
|
ssfid |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ 𝐴 ) → { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚 } ∈ Fin ) |
14 |
|
elrabi |
⊢ ( 𝑘 ∈ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚 } → 𝑘 ∈ ℕ ) |
15 |
|
mucl |
⊢ ( 𝑘 ∈ ℕ → ( μ ‘ 𝑘 ) ∈ ℤ ) |
16 |
14 15
|
syl |
⊢ ( 𝑘 ∈ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚 } → ( μ ‘ 𝑘 ) ∈ ℤ ) |
17 |
16
|
zcnd |
⊢ ( 𝑘 ∈ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚 } → ( μ ‘ 𝑘 ) ∈ ℂ ) |
18 |
17
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ 𝐴 ) ∧ 𝑘 ∈ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚 } ) → ( μ ‘ 𝑘 ) ∈ ℂ ) |
19 |
13 5 18
|
fsummulc1 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ 𝐴 ) → ( Σ 𝑘 ∈ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚 } ( μ ‘ 𝑘 ) · 𝐵 ) = Σ 𝑘 ∈ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚 } ( ( μ ‘ 𝑘 ) · 𝐵 ) ) |
20 |
|
ovif |
⊢ ( if ( 𝑚 = 1 , 1 , 0 ) · 𝐵 ) = if ( 𝑚 = 1 , ( 1 · 𝐵 ) , ( 0 · 𝐵 ) ) |
21 |
|
velsn |
⊢ ( 𝑚 ∈ { 1 } ↔ 𝑚 = 1 ) |
22 |
21
|
bicomi |
⊢ ( 𝑚 = 1 ↔ 𝑚 ∈ { 1 } ) |
23 |
22
|
a1i |
⊢ ( 𝐵 ∈ ℂ → ( 𝑚 = 1 ↔ 𝑚 ∈ { 1 } ) ) |
24 |
|
mulid2 |
⊢ ( 𝐵 ∈ ℂ → ( 1 · 𝐵 ) = 𝐵 ) |
25 |
|
mul02 |
⊢ ( 𝐵 ∈ ℂ → ( 0 · 𝐵 ) = 0 ) |
26 |
23 24 25
|
ifbieq12d |
⊢ ( 𝐵 ∈ ℂ → if ( 𝑚 = 1 , ( 1 · 𝐵 ) , ( 0 · 𝐵 ) ) = if ( 𝑚 ∈ { 1 } , 𝐵 , 0 ) ) |
27 |
5 26
|
syl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ 𝐴 ) → if ( 𝑚 = 1 , ( 1 · 𝐵 ) , ( 0 · 𝐵 ) ) = if ( 𝑚 ∈ { 1 } , 𝐵 , 0 ) ) |
28 |
20 27
|
syl5eq |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ 𝐴 ) → ( if ( 𝑚 = 1 , 1 , 0 ) · 𝐵 ) = if ( 𝑚 ∈ { 1 } , 𝐵 , 0 ) ) |
29 |
9 19 28
|
3eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ 𝐴 ) → Σ 𝑘 ∈ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚 } ( ( μ ‘ 𝑘 ) · 𝐵 ) = if ( 𝑚 ∈ { 1 } , 𝐵 , 0 ) ) |
30 |
29
|
sumeq2dv |
⊢ ( 𝜑 → Σ 𝑚 ∈ 𝐴 Σ 𝑘 ∈ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚 } ( ( μ ‘ 𝑘 ) · 𝐵 ) = Σ 𝑚 ∈ 𝐴 if ( 𝑚 ∈ { 1 } , 𝐵 , 0 ) ) |
31 |
4
|
snssd |
⊢ ( 𝜑 → { 1 } ⊆ 𝐴 ) |
32 |
31
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ { 1 } ) → 𝑚 ∈ 𝐴 ) |
33 |
32 5
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ { 1 } ) → 𝐵 ∈ ℂ ) |
34 |
33
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑚 ∈ { 1 } 𝐵 ∈ ℂ ) |
35 |
2
|
olcd |
⊢ ( 𝜑 → ( 𝐴 ⊆ ( ℤ≥ ‘ 1 ) ∨ 𝐴 ∈ Fin ) ) |
36 |
|
sumss2 |
⊢ ( ( ( { 1 } ⊆ 𝐴 ∧ ∀ 𝑚 ∈ { 1 } 𝐵 ∈ ℂ ) ∧ ( 𝐴 ⊆ ( ℤ≥ ‘ 1 ) ∨ 𝐴 ∈ Fin ) ) → Σ 𝑚 ∈ { 1 } 𝐵 = Σ 𝑚 ∈ 𝐴 if ( 𝑚 ∈ { 1 } , 𝐵 , 0 ) ) |
37 |
31 34 35 36
|
syl21anc |
⊢ ( 𝜑 → Σ 𝑚 ∈ { 1 } 𝐵 = Σ 𝑚 ∈ 𝐴 if ( 𝑚 ∈ { 1 } , 𝐵 , 0 ) ) |
38 |
1
|
eleq1d |
⊢ ( 𝑚 = 1 → ( 𝐵 ∈ ℂ ↔ 𝐶 ∈ ℂ ) ) |
39 |
5
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑚 ∈ 𝐴 𝐵 ∈ ℂ ) |
40 |
38 39 4
|
rspcdva |
⊢ ( 𝜑 → 𝐶 ∈ ℂ ) |
41 |
1
|
sumsn |
⊢ ( ( 1 ∈ 𝐴 ∧ 𝐶 ∈ ℂ ) → Σ 𝑚 ∈ { 1 } 𝐵 = 𝐶 ) |
42 |
4 40 41
|
syl2anc |
⊢ ( 𝜑 → Σ 𝑚 ∈ { 1 } 𝐵 = 𝐶 ) |
43 |
30 37 42
|
3eqtr2d |
⊢ ( 𝜑 → Σ 𝑚 ∈ 𝐴 Σ 𝑘 ∈ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚 } ( ( μ ‘ 𝑘 ) · 𝐵 ) = 𝐶 ) |