Theorem addassi 8334

Description: Associative law for addition. (Contributed by NM, 23-Nov-1994.)

Hypotheses

Ref	Expression
axi.1	⊢ 𝐴 ∈ ℂ
axi.2	⊢ 𝐵 ∈ ℂ
axi.3	⊢ 𝐶 ∈ ℂ

Assertion

Ref	Expression
addassi	⊢ ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))

Proof of Theorem addassi

Step	Hyp	Ref	Expression
1		axi.1	. 2 ⊢ 𝐴 ∈ ℂ
2		axi.2	. 2 ⊢ 𝐵 ∈ ℂ
3		axi.3	. 2 ⊢ 𝐶 ∈ ℂ
4		addass 8309	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
5	1, 2, 3, 4	mp3an 1378	1 ⊢ ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))

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-ia1 106  ax-ia2 107  ax-ia3 108  ax-addass 8281

This proof depends on definitions:  df-bi 117  df-3an 1011

This theorem is used by:  2p2e4  9433  3p2e5  9448  3p3e6  9449  4p2e6  9450  4p3e7  9451  4p4e8  9452  5p2e7  9453  5p3e8  9454  5p4e9  9455  6p2e8  9456  6p3e9  9457  7p2e9  9458  numsuc  9794  nummac  9830  numaddc  9833  6p5lem  9855  5p5e10  9856  6p4e10  9857  7p3e10  9860  8p2e10  9865  binom2i  11098  resqrexlemover  11790  3dvdsdec  12648  3dvds2dec  12649  mod2xnegi  13218  decsplit  13229  lgsdir2lem2  16246  2lgsoddprmlem3d  16327