Theorem addlid 8311

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 8164	. . 3 ⊢ 0 ∈ ℂ
2		addcom 8309	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐴 + 0) = (0 + 𝐴))
3	1, 2	mpan2 425	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = (0 + 𝐴))
4		addrid 8310	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
5	3, 4	eqtr3d 2264	1 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  (class class class)co 6013  ℂcc 8023  0cc0 8025   + caddc 8028

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211  ax-1cn 8118  ax-icn 8120  ax-addcl 8121  ax-mulcl 8123  ax-addcom 8125  ax-i2m1 8130  ax-0id 8133

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225

This theorem is referenced by:  readdcan  8312  addlidi  8315  addlidd  8322  cnegexlem1  8347  cnegexlem2  8348  addcan  8352  negneg  8422  fz0to4untppr  10352  fzo0addel  10426  fzoaddel2  10428  divfl0  10549  modqid  10604  swrdspsleq  11241  swrds1  11242  sumrbdclem  11931  summodclem2a  11935  fisum0diag2  12001  eftlub  12244  gcdid  12550  cncrng  14576  ptolemy  15541