Step |
Hyp |
Ref |
Expression |
1 |
|
breq2 |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑝 ↑ 2 ) ∥ 𝑥 ↔ ( 𝑝 ↑ 2 ) ∥ 𝐴 ) ) |
2 |
1
|
rexbidv |
⊢ ( 𝑥 = 𝐴 → ( ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ 𝑥 ↔ ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ 𝐴 ) ) |
3 |
|
breq2 |
⊢ ( 𝑥 = 𝐴 → ( 𝑝 ∥ 𝑥 ↔ 𝑝 ∥ 𝐴 ) ) |
4 |
3
|
rabbidv |
⊢ ( 𝑥 = 𝐴 → { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥 } = { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴 } ) |
5 |
4
|
fveq2d |
⊢ ( 𝑥 = 𝐴 → ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥 } ) = ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴 } ) ) |
6 |
5
|
oveq2d |
⊢ ( 𝑥 = 𝐴 → ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥 } ) ) = ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴 } ) ) ) |
7 |
2 6
|
ifbieq2d |
⊢ ( 𝑥 = 𝐴 → if ( ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ 𝑥 , 0 , ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥 } ) ) ) = if ( ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ 𝐴 , 0 , ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴 } ) ) ) ) |
8 |
|
df-mu |
⊢ μ = ( 𝑥 ∈ ℕ ↦ if ( ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ 𝑥 , 0 , ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥 } ) ) ) ) |
9 |
|
c0ex |
⊢ 0 ∈ V |
10 |
|
ovex |
⊢ ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴 } ) ) ∈ V |
11 |
9 10
|
ifex |
⊢ if ( ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ 𝐴 , 0 , ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴 } ) ) ) ∈ V |
12 |
7 8 11
|
fvmpt |
⊢ ( 𝐴 ∈ ℕ → ( μ ‘ 𝐴 ) = if ( ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ 𝐴 , 0 , ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴 } ) ) ) ) |