Theorem addlid 8323

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 8176	. . 3 ⊢ 0 ∈ ℂ
2		addcom 8321	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐴 + 0) = (0 + 𝐴))
3	1, 2	mpan2 425	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = (0 + 𝐴))
4		addrid 8322	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
5	3, 4	eqtr3d 2265	1 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1397   ∈ wcel 2201  (class class class)co 6023  ℂcc 8035  0cc0 8037   + caddc 8040

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-ext 2212  ax-1cn 8130  ax-icn 8132  ax-addcl 8133  ax-mulcl 8135  ax-addcom 8137  ax-i2m1 8142  ax-0id 8145

This theorem depends on definitions:  df-bi 117  df-cleq 2223  df-clel 2226

This theorem is referenced by:  readdcan  8324  addlidi  8327  addlidd  8334  cnegexlem1  8359  cnegexlem2  8360  addcan  8364  negneg  8434  fz0to4untppr  10364  fzo0addel  10439  fzoaddel2  10441  divfl0  10562  modqid  10617  swrdspsleq  11257  swrds1  11258  sumrbdclem  11961  summodclem2a  11965  fisum0diag2  12031  eftlub  12274  gcdid  12580  cncrng  14607  ptolemy  15577