Theorem cnaddinv

Description: Value of the group inverse of complex number addition. See also cnfldneg . (Contributed by Steve Rodriguez, 3-Dec-2006) (Revised by AV, 26-Aug-2021) (New usage is discouraged.)

	Ref	Expression
Hypothesis	cnaddabl.g	⊢ 𝐺 = { ⟨ ( Base ‘ ndx ) , ℂ ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ }
Assertion	cnaddinv	⊢ ( 𝐴 ∈ ℂ → ( ( invg ‘ 𝐺 ) ‘ 𝐴 ) = - 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		cnaddabl.g	⊢ 𝐺 = { ⟨ ( Base ‘ ndx ) , ℂ ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ }
2		negid	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 + - 𝐴 ) = 0 )
3	1	cnaddabl	⊢ 𝐺 ∈ Abel
4		ablgrp	⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ Grp )
5	3 4	ax-mp	⊢ 𝐺 ∈ Grp
6		id	⊢ ( 𝐴 ∈ ℂ → 𝐴 ∈ ℂ )
7		negcl	⊢ ( 𝐴 ∈ ℂ → - 𝐴 ∈ ℂ )
8		cnex	⊢ ℂ ∈ V
9	1	grpbase	⊢ ( ℂ ∈ V → ℂ = ( Base ‘ 𝐺 ) )
10	8 9	ax-mp	⊢ ℂ = ( Base ‘ 𝐺 )
11		addex	⊢ + ∈ V
12	1	grpplusg	⊢ ( + ∈ V → + = ( +g ‘ 𝐺 ) )
13	11 12	ax-mp	⊢ + = ( +g ‘ 𝐺 )
14	1	cnaddid	⊢ ( 0g ‘ 𝐺 ) = 0
15	14	eqcomi	⊢ 0 = ( 0g ‘ 𝐺 )
16		eqid	⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 )
17	10 13 15 16	grpinvid1	⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ ℂ ∧ - 𝐴 ∈ ℂ ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝐴 ) = - 𝐴 ↔ ( 𝐴 + - 𝐴 ) = 0 ) )
18	5 6 7 17	mp3an2i	⊢ ( 𝐴 ∈ ℂ → ( ( ( invg ‘ 𝐺 ) ‘ 𝐴 ) = - 𝐴 ↔ ( 𝐴 + - 𝐴 ) = 0 ) )
19	2 18	mpbird	⊢ ( 𝐴 ∈ ℂ → ( ( invg ‘ 𝐺 ) ‘ 𝐴 ) = - 𝐴 )