Theorem addcomi 8464

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 8457	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
4	1, 2, 3	mp2an 430	1 ⊢ (𝐴 + 𝐵) = (𝐵 + 𝐴)

Syntax hints:   = wceq 1402   ∈ wcel 2209  (class class class)co 6079  ℂcc 8171   + caddc 8176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-addcom 8273

This theorem is referenced by:  addcomli  8465  add42i  8486  mvlladdi  8538  3m1e2  9407  fztpval  10473  fzo0to42pr  10621  ef01bndlem  12506  modxai  13178  tangtx  15922  log2ublem2  16067  lgsdir2lem2  16131  lgsdir2lem3  16132  lgsdir2lem5  16134