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Metamath Proof Explorer
Description: Division into a reciprocal. (Contributed by NM, 19-Oct-2007)
|
|
Ref |
Expression |
|
Assertion |
recdiv2 |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 1 / 𝐴 ) / 𝐵 ) = ( 1 / ( 𝐴 · 𝐵 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
| 2 |
|
divdiv1 |
⊢ ( ( 1 ∈ ℂ ∧ ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 1 / 𝐴 ) / 𝐵 ) = ( 1 / ( 𝐴 · 𝐵 ) ) ) |
| 3 |
1 2
|
mp3an1 |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 1 / 𝐴 ) / 𝐵 ) = ( 1 / ( 𝐴 · 𝐵 ) ) ) |