Theorem addcomi 8187

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 8180	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
4	1, 2, 3	mp2an 426	1 ⊢ (𝐴 + 𝐵) = (𝐵 + 𝐴)

Syntax hints:   = wceq 1364   ∈ wcel 2167  (class class class)co 5925  ℂcc 7894   + caddc 7899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-addcom 7996

This theorem is referenced by:  addcomli  8188  add42i  8209  mvlladdi  8261  3m1e2  9127  fztpval  10175  fzo0to42pr  10313  ef01bndlem  11938  modxai  12610  tangtx  15158  lgsdir2lem2  15354  lgsdir2lem3  15355  lgsdir2lem5  15357