Theorem addcomi 8470

Description: Addition is commutative. 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 8463	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
4	1, 2, 3	mp2an 430	1 ⊢ (𝐴 + 𝐵) = (𝐵 + 𝐴)

This proof depends on syntax axioms:   = wceq 1402   ∈ wcel 2209  (class class class)co 6085  ℂcc 8177   + caddc 8182

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-addcom 8279

This theorem is used by:  addcomli  8471  add42i  8492  mvlladdi  8544  3m1e2  9425  fztpval  10492  fzo0to42pr  10640  ef01bndlem  12525  modxai  13197  tangtx  15942  log2ublem2  16090  lgsdir2lem2  16160  lgsdir2lem3  16161  lgsdir2lem5  16163