Theorem addcomi 8471

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 8464	. 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  8472  add42i  8493  mvlladdi  8545  3m1e2  9426  fztpval  10500  fzo0to42pr  10648  ef01bndlem  12539  modxai  13215  tangtx  15989  log2ublem2  16141  ppiqub  16212  lgsdir2lem2  16267  lgsdir2lem3  16268  lgsdir2lem5  16270