Theorem logbfval

Description: The general logarithm of a complex number to a fixed base. (Contributed by AV, 11-Jun-2020)

	Ref	Expression
Assertion	logbfval	⊢ ( ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) → ( ( curry logb ‘ 𝐵 ) ‘ 𝑋 ) = ( 𝐵 logb 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		df-logb	⊢ logb = ( 𝑥 ∈ ( ℂ ∖ { 0 , 1 } ) , 𝑦 ∈ ( ℂ ∖ { 0 } ) ↦ ( ( log ‘ 𝑦 ) / ( log ‘ 𝑥 ) ) )
2		ovexd	⊢ ( ( ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) ∧ ( 𝑥 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) ) → ( ( log ‘ 𝑦 ) / ( log ‘ 𝑥 ) ) ∈ V )
3	2	ralrimivva	⊢ ( ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) → ∀ 𝑥 ∈ ( ℂ ∖ { 0 , 1 } ) ∀ 𝑦 ∈ ( ℂ ∖ { 0 } ) ( ( log ‘ 𝑦 ) / ( log ‘ 𝑥 ) ) ∈ V )
4		cnex	⊢ ℂ ∈ V
5		difexg	⊢ ( ℂ ∈ V → ( ℂ ∖ { 0 } ) ∈ V )
6	4 5	mp1i	⊢ ( ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) → ( ℂ ∖ { 0 } ) ∈ V )
7		eldifpr	⊢ ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ↔ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) )
8	7	biimpri	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) → 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) )
9	8	adantr	⊢ ( ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) → 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) )
10		simpr	⊢ ( ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) → 𝑋 ∈ ( ℂ ∖ { 0 } ) )
11	1 3 6 9 10	fvmpocurryd	⊢ ( ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) → ( ( curry logb ‘ 𝐵 ) ‘ 𝑋 ) = ( 𝐵 logb 𝑋 ) )