Description: The norm of the ring of the integers. (Contributed by Thierry Arnoux, 8-Nov-2017) (Revised by AV, 13-Jun-2019)
Ref | Expression | ||
---|---|---|---|
Assertion | zzsnm | ⊢ ( 𝑀 ∈ ℤ → ( abs ‘ 𝑀 ) = ( ( norm ‘ ℤring ) ‘ 𝑀 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zringnm | ⊢ ( norm ‘ ℤring ) = ( abs ↾ ℤ ) | |
2 | 1 | eqcomi | ⊢ ( abs ↾ ℤ ) = ( norm ‘ ℤring ) |
3 | 2 | fveq1i | ⊢ ( ( abs ↾ ℤ ) ‘ 𝑀 ) = ( ( norm ‘ ℤring ) ‘ 𝑀 ) |
4 | fvres | ⊢ ( 𝑀 ∈ ℤ → ( ( abs ↾ ℤ ) ‘ 𝑀 ) = ( abs ‘ 𝑀 ) ) | |
5 | 3 4 | syl5reqr | ⊢ ( 𝑀 ∈ ℤ → ( abs ‘ 𝑀 ) = ( ( norm ‘ ℤring ) ‘ 𝑀 ) ) |