Theorem addassi 8340

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 8315	. 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 6088  ℂcc 8183   + caddc 8188

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 8287

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

This theorem is used by:  2p2e4  9440  3p2e5  9455  3p3e6  9456  4p2e6  9457  4p3e7  9458  4p4e8  9459  5p2e7  9460  5p3e8  9461  5p4e9  9462  6p2e8  9463  6p3e9  9464  7p2e9  9465  numsuc  9800  nummac  9836  numaddc  9839  6p5lem  9861  5p5e10  9862  6p4e10  9863  7p3e10  9866  8p2e10  9871  binom2i  11105  resqrexlemover  11797  3dvdsdec  12656  3dvds2dec  12657  mod2xnegi  13226  decsplit  13237  lgsdir2lem2  16370  2lgsoddprmlem3d  16451