Metamath Proof Explorer
Description: Multiplication of a number and its reciprocal. (Contributed by Mario
Carneiro, 27-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
div1d.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
|
reccld.2 |
⊢ ( 𝜑 → 𝐴 ≠ 0 ) |
|
Assertion |
recid2d |
⊢ ( 𝜑 → ( ( 1 / 𝐴 ) · 𝐴 ) = 1 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
div1d.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
reccld.2 |
⊢ ( 𝜑 → 𝐴 ≠ 0 ) |
3 |
|
recid2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( 1 / 𝐴 ) · 𝐴 ) = 1 ) |
4 |
1 2 3
|
syl2anc |
⊢ ( 𝜑 → ( ( 1 / 𝐴 ) · 𝐴 ) = 1 ) |