Theorem peano2cn 8269

Description: A theorem for complex numbers analogous the second Peano postulate peano2 4684. (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 8080	. 2 ⊢ 1 ∈ ℂ
2		addcl 8112	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) ∈ ℂ)
3	1, 2	mpan2 425	1 ⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)

Syntax hints:   → wi 4   ∈ wcel 2200  (class class class)co 5994  ℂcc 7985  1c1 7988   + caddc 7990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-1cn 8080  ax-addcl 8083

This theorem is referenced by:  xp1d2m1eqxm1d2  9352  nneo  9538  zeo  9540  zeo2  9541  zesq  10867  facndiv  10948  faclbnd  10950  faclbnd6  10953  bcxmas  11986  trireciplem  11997  odd2np1  12370  abssinper  15505  lgseisenlem1  15734  lgsquadlem1  15741