Theorem addassi 8029

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 8004	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
5	1, 2, 3, 4	mp3an 1348	1 ⊢ ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))

Syntax hints:   = wceq 1364   ∈ wcel 2164  (class class class)co 5919  ℂcc 7872   + caddc 7877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-addass 7976

This theorem depends on definitions:  df-bi 117  df-3an 982

This theorem is referenced by:  2p2e4  9111  3p2e5  9126  3p3e6  9127  4p2e6  9128  4p3e7  9129  4p4e8  9130  5p2e7  9131  5p3e8  9132  5p4e9  9133  6p2e8  9134  6p3e9  9135  7p2e9  9136  numsuc  9464  nummac  9495  numaddc  9498  6p5lem  9520  5p5e10  9521  6p4e10  9522  7p3e10  9525  8p2e10  9530  binom2i  10722  resqrexlemover  11157  3dvdsdec  12009  3dvds2dec  12010  lgsdir2lem2  15177  2lgsoddprmlem3d  15258