Theorem peano2cn 8289

Description: A theorem for complex numbers analogous the second Peano postulate peano2 4687. (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 8100	. 2 ⊢ 1 ∈ ℂ
2		addcl 8132	. 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 6007  ℂcc 8005  1c1 8008   + caddc 8010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-1cn 8100  ax-addcl 8103

This theorem is referenced by:  xp1d2m1eqxm1d2  9372  nneo  9558  zeo  9560  zeo2  9561  zesq  10888  facndiv  10969  faclbnd  10971  faclbnd6  10974  bcxmas  12008  trireciplem  12019  odd2np1  12392  abssinper  15528  lgseisenlem1  15757  lgsquadlem1  15764