Theorem addcomi 8298

Description: Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)

Hypotheses

Ref	Expression
mul.1	⊢ 𝐴 ∈ ℂ
mul.2	⊢ 𝐵 ∈ ℂ

Assertion

Ref	Expression
addcomi	⊢ (𝐴 + 𝐵) = (𝐵 + 𝐴)

Proof of Theorem addcomi

Step	Hyp	Ref	Expression
1		mul.1	. 2 ⊢ 𝐴 ∈ ℂ
2		mul.2	. 2 ⊢ 𝐵 ∈ ℂ
3		addcom 8291	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
4	1, 2, 3	mp2an 426	1 ⊢ (𝐴 + 𝐵) = (𝐵 + 𝐴)

Syntax hints:   = wceq 1395   ∈ wcel 2200  (class class class)co 6007  ℂcc 8005   + caddc 8010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-addcom 8107

This theorem is referenced by:  addcomli  8299  add42i  8320  mvlladdi  8372  3m1e2  9238  fztpval  10287  fzo0to42pr  10434  ef01bndlem  12275  modxai  12947  tangtx  15520  lgsdir2lem2  15716  lgsdir2lem3  15717  lgsdir2lem5  15719