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Metamath Proof Explorer

Theorem modvalr

Description: The value of the modulo operation (multiplication in reversed order). (Contributed by Alexander van der Vekens, 14-Apr-2018)

Ref Expression
Assertion modvalr ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 mod 𝐵 ) = ( 𝐴 − ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) · 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 modval ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 mod 𝐵 ) = ( 𝐴 − ( 𝐵 · ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ) ) )
2 rpcn ⊢ ( 𝐵 ∈ ℝ+ → 𝐵 ∈ ℂ )
3 2 adantl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → 𝐵 ∈ ℂ )
4 rerpdivcl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 / 𝐵 ) ∈ ℝ )
5 reflcl ⊢ ( ( 𝐴 / 𝐵 ) ∈ ℝ → ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ∈ ℝ )
6 5 recnd ⊢ ( ( 𝐴 / 𝐵 ) ∈ ℝ → ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ∈ ℂ )
7 4 6 syl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ∈ ℂ )
8 3 7 mulcomd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐵 · ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ) = ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) · 𝐵 ) )
9 8 oveq2d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 − ( 𝐵 · ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ) ) = ( 𝐴 − ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) · 𝐵 ) ) )
10 1 9 eqtrd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 mod 𝐵 ) = ( 𝐴 − ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) · 𝐵 ) ) )