Metamath Proof Explorer

Theorem congr

Description: Definition of congruence by integer multiple (see ProofWiki "Congruence (Number Theory)", 11-Jul-2021, https://proofwiki.org/wiki/Definition:Congruence_(Number_Theory) ): An integer A is congruent to an integer B modulo M if their difference is a multiple of M . See also the definition in ApostolNT p. 104: "... a is congruent to b modulo m , and we write a == b (mod m ) if m divides the difference a - b ", or Wikipedia "Modular arithmetic - Congruence", https://en.wikipedia.org/wiki/Modular_arithmetic#Congruence , 11-Jul-2021,: "Given an integer n > 1, called a modulus, two integers are said to be congruent modulo n, if n is a divisor of their difference (i.e., if there is an integer k such that a-b = kn)". (Contributed by AV, 11-Jul-2021)

Ref Expression
Assertion congr ⊢A∈ℤ∧B∈ℤ∧M∈ℕ→AmodM=BmodM↔∃n∈ℤn⋅ M=A−B

Proof

Step Hyp Ref Expression
1 moddvds ⊢M∈ℕ∧A∈ℤ∧B∈ℤ→AmodM=BmodM↔M∥A−B
2 1 3coml ⊢A∈ℤ∧B∈ℤ∧M∈ℕ→AmodM=BmodM↔M∥A−B
3 simp3 ⊢A∈ℤ∧B∈ℤ∧M∈ℕ→M∈ℕ
4 3 nnzd ⊢A∈ℤ∧B∈ℤ∧M∈ℕ→M∈ℤ
5 zsubcl ⊢A∈ℤ∧B∈ℤ→A−B∈ℤ
6 5 3adant3 ⊢A∈ℤ∧B∈ℤ∧M∈ℕ→A−B∈ℤ
7 divides ⊢M∈ℤ∧A−B∈ℤ→M∥A−B↔∃n∈ℤn⋅ M=A−B
8 4 6 7 syl2anc ⊢A∈ℤ∧B∈ℤ∧M∈ℕ→M∥A−B↔∃n∈ℤn⋅ M=A−B
9 2 8 bitrd ⊢A∈ℤ∧B∈ℤ∧M∈ℕ→AmodM=BmodM↔∃n∈ℤn⋅ M=A−B