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General helpful statements
relogbzexpd
Metamath Proof Explorer
Description: Power law for the general logarithm for integer powers: The logarithm
of a positive real number to the power of an integer is equal to the
product of the exponent and the logarithm of the base of the power, a
deduction version. (Contributed by metakunt , 22-May-2024)
Ref
Expression
Hypotheses
relogbzexpd.1
⊢ φ → B ∈ ℝ +
relogbzexpd.2
⊢ φ → B ≠ 1
relogbzexpd.3
⊢ φ → C ∈ ℝ +
relogbzexpd.4
⊢ φ → N ∈ ℤ
Assertion
relogbzexpd
⊢ φ → log B C N = N ⁢ log B C
Proof
Step
Hyp
Ref
Expression
1
relogbzexpd.1
⊢ φ → B ∈ ℝ +
2
relogbzexpd.2
⊢ φ → B ≠ 1
3
relogbzexpd.3
⊢ φ → C ∈ ℝ +
4
relogbzexpd.4
⊢ φ → N ∈ ℤ
5
1
rpcnd
⊢ φ → B ∈ ℂ
6
1
rpne0d
⊢ φ → B ≠ 0
7
6 2
nelprd
⊢ φ → ¬ B ∈ 0 1
8
5 7
eldifd
⊢ φ → B ∈ ℂ ∖ 0 1
9
8 3 4
3jca
⊢ φ → B ∈ ℂ ∖ 0 1 ∧ C ∈ ℝ + ∧ N ∈ ℤ
10
relogbzexp
⊢ B ∈ ℂ ∖ 0 1 ∧ C ∈ ℝ + ∧ N ∈ ℤ → log B C N = N ⁢ log B C
11
9 10
syl
⊢ φ → log B C N = N ⁢ log B C