Theorem addlid 8293

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 8146	. . 3 ⊢ 0 ∈ ℂ
2		addcom 8291	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐴 + 0) = (0 + 𝐴))
3	1, 2	mpan2 425	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = (0 + 𝐴))
4		addrid 8292	. 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 6007  ℂcc 8005  0cc0 8007   + caddc 8010

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 8100  ax-icn 8102  ax-addcl 8103  ax-mulcl 8105  ax-addcom 8107  ax-i2m1 8112  ax-0id 8115

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

This theorem is referenced by:  readdcan  8294  addlidi  8297  addlidd  8304  cnegexlem1  8329  cnegexlem2  8330  addcan  8334  negneg  8404  fz0to4untppr  10328  fzo0addel  10402  fzoaddel2  10404  divfl0  10524  modqid  10579  swrdspsleq  11207  swrds1  11208  sumrbdclem  11896  summodclem2a  11900  fisum0diag2  11966  eftlub  12209  gcdid  12515  cncrng  14541  ptolemy  15506