Theorem grplrinv

Description: In a group, every member has a left and right inverse. (Contributed by AV, 1-Sep-2021)

	Ref	Expression
Hypotheses	grplrinv.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	grplrinv.p	⊢ + = ( +g ‘ 𝐺 )
	grplrinv.i	⊢ 0 = ( 0g ‘ 𝐺 )
Assertion	grplrinv	⊢ ( 𝐺 ∈ Grp → ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 0 ∧ ( 𝑥 + 𝑦 ) = 0 ) )

Proof

Step	Hyp	Ref	Expression
1		grplrinv.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		grplrinv.p	⊢ + = ( +g ‘ 𝐺 )
3		grplrinv.i	⊢ 0 = ( 0g ‘ 𝐺 )
4		eqid	⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 )
5	1 4	grpinvcl	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝐵 )
6		oveq1	⊢ ( 𝑦 = ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) → ( 𝑦 + 𝑥 ) = ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) + 𝑥 ) )
7	6	eqeq1d	⊢ ( 𝑦 = ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) → ( ( 𝑦 + 𝑥 ) = 0 ↔ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) + 𝑥 ) = 0 ) )
8		oveq2	⊢ ( 𝑦 = ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) → ( 𝑥 + 𝑦 ) = ( 𝑥 + ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) )
9	8	eqeq1d	⊢ ( 𝑦 = ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) → ( ( 𝑥 + 𝑦 ) = 0 ↔ ( 𝑥 + ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) = 0 ) )
10	7 9	anbi12d	⊢ ( 𝑦 = ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) → ( ( ( 𝑦 + 𝑥 ) = 0 ∧ ( 𝑥 + 𝑦 ) = 0 ) ↔ ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) + 𝑥 ) = 0 ∧ ( 𝑥 + ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) = 0 ) ) )
11	10	adantl	⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 = ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) → ( ( ( 𝑦 + 𝑥 ) = 0 ∧ ( 𝑥 + 𝑦 ) = 0 ) ↔ ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) + 𝑥 ) = 0 ∧ ( 𝑥 + ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) = 0 ) ) )
12	1 2 3 4	grplinv	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) + 𝑥 ) = 0 )
13	1 2 3 4	grprinv	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 + ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) = 0 )
14	12 13	jca	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ) → ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) + 𝑥 ) = 0 ∧ ( 𝑥 + ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) = 0 ) )
15	5 11 14	rspcedvd	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ) → ∃ 𝑦 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 0 ∧ ( 𝑥 + 𝑦 ) = 0 ) )
16	15	ralrimiva	⊢ ( 𝐺 ∈ Grp → ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 0 ∧ ( 𝑥 + 𝑦 ) = 0 ) )