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Metamath Proof Explorer

Theorem rerecid2d

Description: Multiplication of a number and its reciprocal. (Contributed by SN, 25-Nov-2025)

Ref Expression
Hypotheses sn-rereccld.a ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
sn-rereccld.z ⊢ ( 𝜑 → 𝐴 ≠ 0 )
Assertion rerecid2d ( 𝜑 → ( ( 1 /ℝ 𝐴 ) · 𝐴 ) = 1 )

Proof

Step Hyp Ref Expression
1 sn-rereccld.a ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 sn-rereccld.z ⊢ ( 𝜑 → 𝐴 ≠ 0 )
3 1 2 sn-rereccld ⊢ ( 𝜑 → ( 1 /ℝ 𝐴 ) ∈ ℝ )
4 1 2 rerecidd ⊢ ( 𝜑 → ( 𝐴 · ( 1 /ℝ 𝐴 ) ) = 1 )
5 1 3 4 remulinvcom ⊢ ( 𝜑 → ( ( 1 /ℝ 𝐴 ) · 𝐴 ) = 1 )