Metamath Proof Explorer
Description: Commutative/associative law that swaps the first two terms in a triple
sum. (Contributed by NM, 21-Jan-1997)
|
|
Ref |
Expression |
|
Hypotheses |
add.1 |
⊢ 𝐴 ∈ ℂ |
|
|
add.2 |
⊢ 𝐵 ∈ ℂ |
|
|
add.3 |
⊢ 𝐶 ∈ ℂ |
|
Assertion |
add12i |
⊢ ( 𝐴 + ( 𝐵 + 𝐶 ) ) = ( 𝐵 + ( 𝐴 + 𝐶 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
add.1 |
⊢ 𝐴 ∈ ℂ |
2 |
|
add.2 |
⊢ 𝐵 ∈ ℂ |
3 |
|
add.3 |
⊢ 𝐶 ∈ ℂ |
4 |
|
add12 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 + ( 𝐵 + 𝐶 ) ) = ( 𝐵 + ( 𝐴 + 𝐶 ) ) ) |
5 |
1 2 3 4
|
mp3an |
⊢ ( 𝐴 + ( 𝐵 + 𝐶 ) ) = ( 𝐵 + ( 𝐴 + 𝐶 ) ) |