Theorem mul4r

Description: Rearrangement of 4 factors: swap the right factors in the factors of a product of two products. (Contributed by AV, 4-Mar-2023)

	Ref	Expression
Assertion	mul4r	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 · 𝐵 ) · ( 𝐶 · 𝐷 ) ) = ( ( 𝐴 · 𝐷 ) · ( 𝐶 · 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		mulcom	⊢ ( ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) → ( 𝐶 · 𝐷 ) = ( 𝐷 · 𝐶 ) )
2	1	adantl	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( 𝐶 · 𝐷 ) = ( 𝐷 · 𝐶 ) )
3	2	oveq2d	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 · 𝐵 ) · ( 𝐶 · 𝐷 ) ) = ( ( 𝐴 · 𝐵 ) · ( 𝐷 · 𝐶 ) ) )
4		mul4	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐷 ∈ ℂ ∧ 𝐶 ∈ ℂ ) ) → ( ( 𝐴 · 𝐵 ) · ( 𝐷 · 𝐶 ) ) = ( ( 𝐴 · 𝐷 ) · ( 𝐵 · 𝐶 ) ) )
5	4	ancom2s	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 · 𝐵 ) · ( 𝐷 · 𝐶 ) ) = ( ( 𝐴 · 𝐷 ) · ( 𝐵 · 𝐶 ) ) )
6		simplr	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → 𝐵 ∈ ℂ )
7		simprl	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → 𝐶 ∈ ℂ )
8	6 7	mulcomd	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
9	8	oveq2d	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 · 𝐷 ) · ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 · 𝐷 ) · ( 𝐶 · 𝐵 ) ) )
10	3 5 9	3eqtrd	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 · 𝐵 ) · ( 𝐶 · 𝐷 ) ) = ( ( 𝐴 · 𝐷 ) · ( 𝐶 · 𝐵 ) ) )