Theorem grpsubadd0sub

Description: Subtraction expressed as addition of the difference of the identity element and the subtrahend. (Contributed by AV, 9-Nov-2019)

	Ref	Expression
Hypotheses	grpsubid.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	grpsubid.o	⊢ 0 = ( 0g ‘ 𝐺 )
	grpsubid.m	⊢ − = ( -g ‘ 𝐺 )
	grpsubadd0sub.p	⊢ + = ( +g ‘ 𝐺 )
Assertion	grpsubadd0sub	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 − 𝑌 ) = ( 𝑋 + ( 0 − 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		grpsubid.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		grpsubid.o	⊢ 0 = ( 0g ‘ 𝐺 )
3		grpsubid.m	⊢ − = ( -g ‘ 𝐺 )
4		grpsubadd0sub.p	⊢ + = ( +g ‘ 𝐺 )
5		eqid	⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 )
6	1 4 5 3	grpsubval	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 − 𝑌 ) = ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) )
7	6	3adant1	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 − 𝑌 ) = ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) )
8	1 3 5 2	grpinvval2	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) = ( 0 − 𝑌 ) )
9	8	3adant2	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) = ( 0 − 𝑌 ) )
10	9	oveq2d	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + ( ( invg ‘ 𝐺 ) ‘ 𝑌 ) ) = ( 𝑋 + ( 0 − 𝑌 ) ) )
11	7 10	eqtrd	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 − 𝑌 ) = ( 𝑋 + ( 0 − 𝑌 ) ) )