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Metamath Proof Explorer

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 ) ) )