Description: Commutative/associative law that swaps the first and the third factor in a triple product. (Contributed by Glauco Siliprandi, 11-Dec-2019)
Ref | Expression | ||
---|---|---|---|
Hypotheses | mul13d.1 | ⊢ ( 𝜑 → 𝐴 ∈ ℂ ) | |
mul13d.2 | ⊢ ( 𝜑 → 𝐵 ∈ ℂ ) | ||
mul13d.3 | ⊢ ( 𝜑 → 𝐶 ∈ ℂ ) | ||
Assertion | mul13d | ⊢ ( 𝜑 → ( 𝐴 · ( 𝐵 · 𝐶 ) ) = ( 𝐶 · ( 𝐵 · 𝐴 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul13d.1 | ⊢ ( 𝜑 → 𝐴 ∈ ℂ ) | |
2 | mul13d.2 | ⊢ ( 𝜑 → 𝐵 ∈ ℂ ) | |
3 | mul13d.3 | ⊢ ( 𝜑 → 𝐶 ∈ ℂ ) | |
4 | 1 2 3 | mul12d | ⊢ ( 𝜑 → ( 𝐴 · ( 𝐵 · 𝐶 ) ) = ( 𝐵 · ( 𝐴 · 𝐶 ) ) ) |
5 | 2 1 3 | mulassd | ⊢ ( 𝜑 → ( ( 𝐵 · 𝐴 ) · 𝐶 ) = ( 𝐵 · ( 𝐴 · 𝐶 ) ) ) |
6 | 2 1 | mulcld | ⊢ ( 𝜑 → ( 𝐵 · 𝐴 ) ∈ ℂ ) |
7 | 6 3 | mulcomd | ⊢ ( 𝜑 → ( ( 𝐵 · 𝐴 ) · 𝐶 ) = ( 𝐶 · ( 𝐵 · 𝐴 ) ) ) |
8 | 4 5 7 | 3eqtr2d | ⊢ ( 𝜑 → ( 𝐴 · ( 𝐵 · 𝐶 ) ) = ( 𝐶 · ( 𝐵 · 𝐴 ) ) ) |