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Metamath Proof Explorer

Theorem flpmodeq

Description: Partition of a division into its integer part and the remainder. (Contributed by Alexander van der Vekens, 14-Apr-2018)

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

Proof

Step Hyp Ref Expression
1 modvalr ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 mod 𝐵 ) = ( 𝐴 − ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) · 𝐵 ) ) )
2 1 eqcomd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 − ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) · 𝐵 ) ) = ( 𝐴 mod 𝐵 ) )
3 recn ⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
4 3 adantr ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → 𝐴 ∈ ℂ )
5 rerpdivcl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 / 𝐵 ) ∈ ℝ )
6 flcl ⊢ ( ( 𝐴 / 𝐵 ) ∈ ℝ → ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ∈ ℤ )
7 6 zcnd ⊢ ( ( 𝐴 / 𝐵 ) ∈ ℝ → ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ∈ ℂ )
8 5 7 syl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ∈ ℂ )
9 rpcn ⊢ ( 𝐵 ∈ ℝ+ → 𝐵 ∈ ℂ )
10 9 adantl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → 𝐵 ∈ ℂ )
11 8 10 mulcld ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) · 𝐵 ) ∈ ℂ )
12 modcl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 mod 𝐵 ) ∈ ℝ )
13 12 recnd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 mod 𝐵 ) ∈ ℂ )
14 4 11 13 subaddd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( 𝐴 − ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) · 𝐵 ) ) = ( 𝐴 mod 𝐵 ) ↔ ( ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) · 𝐵 ) + ( 𝐴 mod 𝐵 ) ) = 𝐴 ) )
15 2 14 mpbid ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) · 𝐵 ) + ( 𝐴 mod 𝐵 ) ) = 𝐴 )