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Metamath Proof Explorer

Theorem logbid1

Description: General logarithm is 1 when base and arg match. Property 1(a) of Cohen4 p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014) (Revised by David A. Wheeler, 22-Jul-2017)

Ref Expression
Assertion logbid1 ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1 ) → ( 𝐴 logb 𝐴 ) = 1 )

Proof

Step Hyp Ref Expression
1 eldifpr ⊢ ( 𝐴 ∈ ( ℂ ∖ { 0 , 1 } ) ↔ ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1 ) )
2 1 biimpri ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1 ) → 𝐴 ∈ ( ℂ ∖ { 0 , 1 } ) )
3 eldifsn ⊢ ( 𝐴 ∈ ( ℂ ∖ { 0 } ) ↔ ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) )
4 3 biimpri ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → 𝐴 ∈ ( ℂ ∖ { 0 } ) )
5 4 3adant3 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1 ) → 𝐴 ∈ ( ℂ ∖ { 0 } ) )
6 logbval ⊢ ( ( 𝐴 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝐴 ∈ ( ℂ ∖ { 0 } ) ) → ( 𝐴 logb 𝐴 ) = ( ( log ‘ 𝐴 ) / ( log ‘ 𝐴 ) ) )
7 2 5 6 syl2anc ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1 ) → ( 𝐴 logb 𝐴 ) = ( ( log ‘ 𝐴 ) / ( log ‘ 𝐴 ) ) )
8 logcl ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( log ‘ 𝐴 ) ∈ ℂ )
9 8 3adant3 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1 ) → ( log ‘ 𝐴 ) ∈ ℂ )
10 logccne0 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1 ) → ( log ‘ 𝐴 ) ≠ 0 )
11 9 10 dividd ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1 ) → ( ( log ‘ 𝐴 ) / ( log ‘ 𝐴 ) ) = 1 )
12 7 11 eqtrd ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1 ) → ( 𝐴 logb 𝐴 ) = 1 )