Theorem addcomi 8189

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 8182	. 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 7896   + caddc 7901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-addcom 7998

This theorem is referenced by:  addcomli  8190  add42i  8211  mvlladdi  8263  3m1e2  9129  fztpval  10177  fzo0to42pr  10315  ef01bndlem  11940  modxai  12612  tangtx  15160  lgsdir2lem2  15356  lgsdir2lem3  15357  lgsdir2lem5  15359