Theorem addlid 8473

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 8325	. . 3 ⊢ 0 ∈ ℂ
2		addcom 8471	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐴 + 0) = (0 + 𝐴))
3	1, 2	mpan2 429	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = (0 + 𝐴))
4		addrid 8472	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
5	3, 4	eqtr3d 2273	1 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ∈ wcel 2209  (class class class)co 6088  ℂcc 8184  0cc0 8186   + caddc 8189

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220  ax-1cn 8279  ax-icn 8281  ax-addcl 8282  ax-mulcl 8284  ax-addcom 8286  ax-i2m1 8291  ax-0id 8294

This proof depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234

This theorem is used by:  readdcan  8474  addlidi  8477  addlidd  8484  cnegexlem1  8509  cnegexlem2  8510  addcan  8514  negneg  8584  fz0to4untppr  10549  fzo0addel  10624  fzoaddel2  10626  divfl0  10753  modqid  10808  swrdspsleq  11462  swrds1  11463  sumrbdclem  12170  summodclem2a  12174  fisum0diag2  12240  eftlub  12483  gcdid  12789  cncrng  15016  ptolemy  16063