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Metamath Proof Explorer

Theorem modvalp1

Description: The value of the modulo operation (expressed with sum of denominator and nominator). (Contributed by Alexander van der Vekens, 14-Apr-2018)

Ref Expression
Assertion modvalp1 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( 𝐴 + 𝐵 ) − ( ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) + 1 ) · 𝐵 ) ) = ( 𝐴 mod 𝐵 ) )

Proof

Step Hyp Ref Expression
1 recn ⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
2 1 adantr ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → 𝐴 ∈ ℂ )
3 refldivcl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ∈ ℝ )
4 3 recnd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ∈ ℂ )
5 rpcn ⊢ ( 𝐵 ∈ ℝ+ → 𝐵 ∈ ℂ )
6 5 adantl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → 𝐵 ∈ ℂ )
7 4 6 mulcld ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) · 𝐵 ) ∈ ℂ )
8 2 7 6 pnpcan2d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( 𝐴 + 𝐵 ) − ( ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) · 𝐵 ) + 𝐵 ) ) = ( 𝐴 − ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) · 𝐵 ) ) )
9 4 6 adddirp1d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) + 1 ) · 𝐵 ) = ( ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) · 𝐵 ) + 𝐵 ) )
10 9 oveq2d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( 𝐴 + 𝐵 ) − ( ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) + 1 ) · 𝐵 ) ) = ( ( 𝐴 + 𝐵 ) − ( ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) · 𝐵 ) + 𝐵 ) ) )
11 modvalr ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 mod 𝐵 ) = ( 𝐴 − ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) · 𝐵 ) ) )
12 8 10 11 3eqtr4d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( 𝐴 + 𝐵 ) − ( ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) + 1 ) · 𝐵 ) ) = ( 𝐴 mod 𝐵 ) )