Theorem grpinvval

Description: The inverse of a group element. (Contributed by NM, 24-Aug-2011) (Revised by Mario Carneiro, 7-Aug-2013)

	Ref	Expression
Hypotheses	grpinvval.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	grpinvval.p	⊢ + = ( +g ‘ 𝐺 )
	grpinvval.o	⊢ 0 = ( 0g ‘ 𝐺 )
	grpinvval.n	⊢ 𝑁 = ( invg ‘ 𝐺 )
Assertion	grpinvval	⊢ ( 𝑋 ∈ 𝐵 → ( 𝑁 ‘ 𝑋 ) = ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑋 ) = 0 ) )

Proof

Step	Hyp	Ref	Expression
1		grpinvval.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		grpinvval.p	⊢ + = ( +g ‘ 𝐺 )
3		grpinvval.o	⊢ 0 = ( 0g ‘ 𝐺 )
4		grpinvval.n	⊢ 𝑁 = ( invg ‘ 𝐺 )
5		oveq2	⊢ ( 𝑥 = 𝑋 → ( 𝑦 + 𝑥 ) = ( 𝑦 + 𝑋 ) )
6	5	eqeq1d	⊢ ( 𝑥 = 𝑋 → ( ( 𝑦 + 𝑥 ) = 0 ↔ ( 𝑦 + 𝑋 ) = 0 ) )
7	6	riotabidv	⊢ ( 𝑥 = 𝑋 → ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) = ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑋 ) = 0 ) )
8	1 2 3 4	grpinvfval	⊢ 𝑁 = ( 𝑥 ∈ 𝐵 ↦ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 0 ) )
9		riotaex	⊢ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑋 ) = 0 ) ∈ V
10	7 8 9	fvmpt	⊢ ( 𝑋 ∈ 𝐵 → ( 𝑁 ‘ 𝑋 ) = ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑋 ) = 0 ) )