Theorem peano2cn 8357

Description: A theorem for complex numbers analogous the second Peano postulate peano2 4699. (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 8168	. 2 ⊢ 1 ∈ ℂ
2		addcl 8200	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) ∈ ℂ)
3	1, 2	mpan2 425	1 ⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)

Syntax hints:   → wi 4   ∈ wcel 2202  (class class class)co 6028  ℂcc 8073  1c1 8076   + caddc 8078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-1cn 8168  ax-addcl 8171

This theorem is referenced by:  xp1d2m1eqxm1d2  9440  nneo  9626  zeo  9628  zeo2  9629  zesq  10964  facndiv  11045  faclbnd  11047  faclbnd6  11050  bcxmas  12111  trireciplem  12122  odd2np1  12495  abssinper  15637  lgseisenlem1  15869  lgsquadlem1  15876