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Metamath Proof Explorer

Theorem modsubmod

Description: The difference of a real number modulo a positive real number and another real number equals the difference of the two real numbers modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018)

Ref Expression
Assertion modsubmod ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 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 1 3adant2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → ( 𝐴 mod 𝑀 ) ∈ ℝ )
3 simp1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → 𝐴 ∈ ℝ )
4 simp2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → 𝐵 ∈ ℝ )
5 simp3 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → 𝑀 ∈ ℝ+ )
6 modabs2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → ( ( 𝐴 mod 𝑀 ) mod 𝑀 ) = ( 𝐴 mod 𝑀 ) )
7 6 3adant2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → ( ( 𝐴 mod 𝑀 ) mod 𝑀 ) = ( 𝐴 mod 𝑀 ) )
8 eqidd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → ( 𝐵 mod 𝑀 ) = ( 𝐵 mod 𝑀 ) )
9 2 3 4 4 5 7 8 modsub12d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → ( ( ( 𝐴 mod 𝑀 ) − 𝐵 ) mod 𝑀 ) = ( ( 𝐴 − 𝐵 ) mod 𝑀 ) )