Theorem peano2cn 8461

Description: A theorem for complex numbers analogous the second Peano postulate peano2 4742. (Contributed by NM, 17-Aug-2005.)

Assertion

Ref	Expression
peano2cn	⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)

Proof of Theorem peano2cn

Step	Hyp	Ref	Expression
1		ax-1cn 8272	. 2 ⊢ 1 ∈ ℂ
2		addcl 8304	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) ∈ ℂ)
3	1, 2	mpan2 429	1 ⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2209  (class class class)co 6085  ℂcc 8177  1c1 8180   + caddc 8182

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-1cn 8272  ax-addcl 8275

This theorem is used by:  xp1d2m1eqxm1d2  9560  nneo  9751  zeo  9753  zeo2  9754  zesq  11098  facndiv  11179  faclbnd  11181  faclbnd6  11184  bcxmas  12258  trireciplem  12269  odd2np1  12642  abssinper  15950  lgseisenlem1  16201  lgsquadlem1  16208