Theorem addid2i 8100

Description: 0 is a left identity for addition. (Contributed by NM, 3-Jan-2013.)

Hypothesis

Ref	Expression
mul.1	⊢ 𝐴 ∈ ℂ

Assertion

Ref	Expression
addid2i	⊢ (0 + 𝐴) = 𝐴

Proof of Theorem addid2i

Step	Hyp	Ref	Expression
1		mul.1	. 2 ⊢ 𝐴 ∈ ℂ
2		addlid 8096	. 2 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ (0 + 𝐴) = 𝐴

Syntax hints:   = wceq 1353   ∈ wcel 2148  (class class class)co 5875  ℂcc 7809  0cc0 7811   + caddc 7814

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159  ax-1cn 7904  ax-icn 7906  ax-addcl 7907  ax-mulcl 7909  ax-addcom 7911  ax-i2m1 7916  ax-0id 7919

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173

This theorem is referenced by:  ine0  8351  inelr  8541  muleqadd  8625  0p1e1  9033  iap0  9142  num0h  9395  nummul1c  9432  decrmac  9441  decmul1  9447  fz0tp  10122  fz0to4untppr  10124  fzo0to3tp  10219  rei  10908  imi  10909  resqrexlemover  11019  ef01bndlem  11764  efhalfpi  14223  sinq34lt0t  14255  ex-fac  14483