Theorem addassi 8165

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 8140	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
5	1, 2, 3, 4	mp3an 1371	1 ⊢ ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))

Syntax hints:   = wceq 1395   ∈ wcel 2200  (class class class)co 6007  ℂcc 8008   + caddc 8013

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 8112

This theorem depends on definitions:  df-bi 117  df-3an 1004

This theorem is referenced by:  2p2e4  9248  3p2e5  9263  3p3e6  9264  4p2e6  9265  4p3e7  9266  4p4e8  9267  5p2e7  9268  5p3e8  9269  5p4e9  9270  6p2e8  9271  6p3e9  9272  7p2e9  9273  numsuc  9602  nummac  9633  numaddc  9636  6p5lem  9658  5p5e10  9659  6p4e10  9660  7p3e10  9663  8p2e10  9668  binom2i  10882  resqrexlemover  11536  3dvdsdec  12391  3dvds2dec  12392  decsplit  12967  lgsdir2lem2  15723  2lgsoddprmlem3d  15804