Theorem lcmineqlem5

Description: Technical lemma for reciprocal multiplication in deduction form. (Contributed by metakunt, 10-May-2024)

	Ref	Expression
Hypotheses	lcmineqlem5.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	lcmineqlem5.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	lcmineqlem5.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	lcmineqlem5.4	⊢ ( 𝜑 → 𝐶 ≠ 0 )
Assertion	lcmineqlem5	⊢ ( 𝜑 → ( 𝐴 · ( 𝐵 · ( 1 / 𝐶 ) ) ) = ( 𝐵 · ( 𝐴 / 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		lcmineqlem5.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		lcmineqlem5.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		lcmineqlem5.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		lcmineqlem5.4	⊢ ( 𝜑 → 𝐶 ≠ 0 )
5	3 4	reccld	⊢ ( 𝜑 → ( 1 / 𝐶 ) ∈ ℂ )
6	1 2 5	mulassd	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) · ( 1 / 𝐶 ) ) = ( 𝐴 · ( 𝐵 · ( 1 / 𝐶 ) ) ) )
7	1 2	mulcomd	⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )
8	7	oveq1d	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) · ( 1 / 𝐶 ) ) = ( ( 𝐵 · 𝐴 ) · ( 1 / 𝐶 ) ) )
9	6 8	eqtr3d	⊢ ( 𝜑 → ( 𝐴 · ( 𝐵 · ( 1 / 𝐶 ) ) ) = ( ( 𝐵 · 𝐴 ) · ( 1 / 𝐶 ) ) )
10	2 1 5	mulassd	⊢ ( 𝜑 → ( ( 𝐵 · 𝐴 ) · ( 1 / 𝐶 ) ) = ( 𝐵 · ( 𝐴 · ( 1 / 𝐶 ) ) ) )
11	9 10	eqtrd	⊢ ( 𝜑 → ( 𝐴 · ( 𝐵 · ( 1 / 𝐶 ) ) ) = ( 𝐵 · ( 𝐴 · ( 1 / 𝐶 ) ) ) )
12	1 3 4	divrecd	⊢ ( 𝜑 → ( 𝐴 / 𝐶 ) = ( 𝐴 · ( 1 / 𝐶 ) ) )
13	12	oveq2d	⊢ ( 𝜑 → ( 𝐵 · ( 𝐴 / 𝐶 ) ) = ( 𝐵 · ( 𝐴 · ( 1 / 𝐶 ) ) ) )
14	11 13	eqtr4d	⊢ ( 𝜑 → ( 𝐴 · ( 𝐵 · ( 1 / 𝐶 ) ) ) = ( 𝐵 · ( 𝐴 / 𝐶 ) ) )