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Metamath Proof Explorer
Description: Two nonnegative integers less than the modulus are equal iff they are
equal modulo the modulus. (Contributed by AV, 14-Mar-2021)
|
|
Ref |
Expression |
|
Assertion |
modlteq |
⊢ ( ( 𝐼 ∈ ( 0 ..^ 𝑁 ) ∧ 𝐽 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝐼 mod 𝑁 ) = ( 𝐽 mod 𝑁 ) ↔ 𝐼 = 𝐽 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
zmodidfzoimp |
⊢ ( 𝐼 ∈ ( 0 ..^ 𝑁 ) → ( 𝐼 mod 𝑁 ) = 𝐼 ) |
| 2 |
|
zmodidfzoimp |
⊢ ( 𝐽 ∈ ( 0 ..^ 𝑁 ) → ( 𝐽 mod 𝑁 ) = 𝐽 ) |
| 3 |
1 2
|
eqeqan12d |
⊢ ( ( 𝐼 ∈ ( 0 ..^ 𝑁 ) ∧ 𝐽 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝐼 mod 𝑁 ) = ( 𝐽 mod 𝑁 ) ↔ 𝐼 = 𝐽 ) ) |