This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem mod0

Description: A mod B is zero iff A is evenly divisible by B . (Contributed by Jeff Madsen, 2-Sep-2009) (Proof shortened by Fan Zheng, 7-Jun-2016)

Ref Expression
Assertion mod0 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( 𝐴 mod 𝐵 ) = 0 ↔ ( 𝐴 / 𝐵 ) ∈ ℤ ) )

Proof

Step Hyp Ref Expression
1 modval ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 mod 𝐵 ) = ( 𝐴 − ( 𝐵 · ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ) ) )
2 1 eqeq1d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( 𝐴 mod 𝐵 ) = 0 ↔ ( 𝐴 − ( 𝐵 · ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ) ) = 0 ) )
3 recn ⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
4 3 adantr ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → 𝐴 ∈ ℂ )
5 rpre ⊢ ( 𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ )
6 5 adantl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → 𝐵 ∈ ℝ )
7 refldivcl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ∈ ℝ )
8 6 7 remulcld ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐵 · ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ) ∈ ℝ )
9 8 recnd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐵 · ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ) ∈ ℂ )
10 4 9 subeq0ad ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( 𝐴 − ( 𝐵 · ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ) ) = 0 ↔ 𝐴 = ( 𝐵 · ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ) ) )
11 2 10 bitrd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( 𝐴 mod 𝐵 ) = 0 ↔ 𝐴 = ( 𝐵 · ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ) ) )
12 7 recnd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ∈ ℂ )
13 rpcnne0 ⊢ ( 𝐵 ∈ ℝ+ → ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) )
14 13 adantl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) )
15 divmul2 ⊢ ( ( 𝐴 ∈ ℂ ∧ ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 𝐴 / 𝐵 ) = ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ↔ 𝐴 = ( 𝐵 · ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ) ) )
16 4 12 14 15 syl3anc ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( 𝐴 / 𝐵 ) = ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ↔ 𝐴 = ( 𝐵 · ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ) ) )
17 eqcom ⊢ ( ( 𝐴 / 𝐵 ) = ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ↔ ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) = ( 𝐴 / 𝐵 ) )
18 16 17 bitr3di ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 = ( 𝐵 · ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ) ↔ ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) = ( 𝐴 / 𝐵 ) ) )
19 11 18 bitrd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( 𝐴 mod 𝐵 ) = 0 ↔ ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) = ( 𝐴 / 𝐵 ) ) )
20 rerpdivcl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 / 𝐵 ) ∈ ℝ )
21 flidz ⊢ ( ( 𝐴 / 𝐵 ) ∈ ℝ → ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) = ( 𝐴 / 𝐵 ) ↔ ( 𝐴 / 𝐵 ) ∈ ℤ ) )
22 20 21 syl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) = ( 𝐴 / 𝐵 ) ↔ ( 𝐴 / 𝐵 ) ∈ ℤ ) )
23 19 22 bitrd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( 𝐴 mod 𝐵 ) = 0 ↔ ( 𝐴 / 𝐵 ) ∈ ℤ ) )