Theorem addlid 8160

Description: 0 is a left identity for addition. (Contributed by Scott Fenton, 3-Jan-2013.)

Assertion

Ref	Expression
addlid	⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)

Proof of Theorem addlid

Step	Hyp	Ref	Expression
1		0cn 8013	. . 3 ⊢ 0 ∈ ℂ
2		addcom 8158	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐴 + 0) = (0 + 𝐴))
3	1, 2	mpan2 425	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = (0 + 𝐴))
4		addrid 8159	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
5	3, 4	eqtr3d 2228	1 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2164  (class class class)co 5919  ℂcc 7872  0cc0 7874   + 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-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2175  ax-1cn 7967  ax-icn 7969  ax-addcl 7970  ax-mulcl 7972  ax-addcom 7974  ax-i2m1 7979  ax-0id 7982

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-clel 2189

This theorem is referenced by:  readdcan  8161  addid2i  8164  addlidd  8171  cnegexlem1  8196  cnegexlem2  8197  addcan  8201  negneg  8271  fz0to4untppr  10193  fzoaddel2  10263  divfl0  10368  modqid  10423  sumrbdclem  11523  summodclem2a  11527  fisum0diag2  11593  eftlub  11836  gcdid  12126  cncrng  14068  ptolemy  15000