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Metamath Proof Explorer

Theorem modmulmod

Description: The product of a real number modulo a positive real number and an integer equals the product of the real number and the integer modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018)

Ref Expression
Assertion modmulmod ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ∧ 𝑀 ∈ ℝ+ ) → ( ( ( 𝐴 mod 𝑀 ) · 𝐵 ) mod 𝑀 ) = ( ( 𝐴 · 𝐵 ) mod 𝑀 ) )

Proof

Step Hyp Ref Expression
1 modcl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → ( 𝐴 mod 𝑀 ) ∈ ℝ )
2 simpl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → 𝐴 ∈ ℝ )
3 1 2 jca ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → ( ( 𝐴 mod 𝑀 ) ∈ ℝ ∧ 𝐴 ∈ ℝ ) )
4 3 3adant2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ∧ 𝑀 ∈ ℝ+ ) → ( ( 𝐴 mod 𝑀 ) ∈ ℝ ∧ 𝐴 ∈ ℝ ) )
5 3simpc ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ∧ 𝑀 ∈ ℝ+ ) → ( 𝐵 ∈ ℤ ∧ 𝑀 ∈ ℝ+ ) )
6 modabs2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → ( ( 𝐴 mod 𝑀 ) mod 𝑀 ) = ( 𝐴 mod 𝑀 ) )
7 6 3adant2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ∧ 𝑀 ∈ ℝ+ ) → ( ( 𝐴 mod 𝑀 ) mod 𝑀 ) = ( 𝐴 mod 𝑀 ) )
8 modmul1 ⊢ ( ( ( ( 𝐴 mod 𝑀 ) ∈ ℝ ∧ 𝐴 ∈ ℝ ) ∧ ( 𝐵 ∈ ℤ ∧ 𝑀 ∈ ℝ+ ) ∧ ( ( 𝐴 mod 𝑀 ) mod 𝑀 ) = ( 𝐴 mod 𝑀 ) ) → ( ( ( 𝐴 mod 𝑀 ) · 𝐵 ) mod 𝑀 ) = ( ( 𝐴 · 𝐵 ) mod 𝑀 ) )
9 4 5 7 8 syl3anc ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ∧ 𝑀 ∈ ℝ+ ) → ( ( ( 𝐴 mod 𝑀 ) · 𝐵 ) mod 𝑀 ) = ( ( 𝐴 · 𝐵 ) mod 𝑀 ) )