Theorem addlid 8184

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 8037	. . 3 ⊢ 0 ∈ ℂ
2		addcom 8182	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐴 + 0) = (0 + 𝐴))
3	1, 2	mpan2 425	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = (0 + 𝐴))
4		addrid 8183	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
5	3, 4	eqtr3d 2231	1 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2167  (class class class)co 5925  ℂcc 7896  0cc0 7898   + caddc 7901

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178  ax-1cn 7991  ax-icn 7993  ax-addcl 7994  ax-mulcl 7996  ax-addcom 7998  ax-i2m1 8003  ax-0id 8006

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192

This theorem is referenced by:  readdcan  8185  addlidi  8188  addlidd  8195  cnegexlem1  8220  cnegexlem2  8221  addcan  8225  negneg  8295  fz0to4untppr  10218  fzoaddel2  10288  divfl0  10405  modqid  10460  sumrbdclem  11561  summodclem2a  11565  fisum0diag2  11631  eftlub  11874  gcdid  12180  cncrng  14203  ptolemy  15146