Theorem peano2cn 8469

Description: A theorem for complex numbers analogous the second Peano postulate peano2 4744. (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 8279	. 2 ⊢ 1 ∈ ℂ
2		addcl 8311	. 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 6088  ℂcc 8184  1c1 8187   + caddc 8189

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-1cn 8279  ax-addcl 8282

This theorem is used by:  xp1d2m1eqxm1d2  9570  nneo  9761  zeo  9763  zeo2  9764  zesq  11118  facndiv  11200  faclbnd  11202  faclbnd6  11205  bcxmas  12282  trireciplem  12293  odd2np1  12666  abssinper  16085  lgseisenlem1  16406  lgsquadlem1  16413