Theorem addlid 8165

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 8018	. . 3 ⊢ 0 ∈ ℂ
2		addcom 8163	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐴 + 0) = (0 + 𝐴))
3	1, 2	mpan2 425	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = (0 + 𝐴))
4		addrid 8164	. 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 5922  ℂcc 7877  0cc0 7879   + caddc 7882

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 7972  ax-icn 7974  ax-addcl 7975  ax-mulcl 7977  ax-addcom 7979  ax-i2m1 7984  ax-0id 7987

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

This theorem is referenced by:  readdcan  8166  addlidi  8169  addlidd  8176  cnegexlem1  8201  cnegexlem2  8202  addcan  8206  negneg  8276  fz0to4untppr  10199  fzoaddel2  10269  divfl0  10386  modqid  10441  sumrbdclem  11542  summodclem2a  11546  fisum0diag2  11612  eftlub  11855  gcdid  12153  cncrng  14125  ptolemy  15060