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Metamath Proof Explorer

Theorem grpinv11

Description: The group inverse is one-to-one. (Contributed by NM, 22-Mar-2015) (Proof shortened by SN, 8-Jul-2025)

Ref Expression
Hypotheses grpinvinv.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
grpinvinv.n ⊢ 𝑁 = ( invg ‘ 𝐺 )
grpinv11.g ⊢ ( 𝜑 → 𝐺 ∈ Grp )
grpinv11.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
grpinv11.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion grpinv11 ( 𝜑 → ( ( 𝑁 ‘ 𝑋 ) = ( 𝑁 ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) )

Proof

Step Hyp Ref Expression
1 grpinvinv.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 grpinvinv.n ⊢ 𝑁 = ( invg ‘ 𝐺 )
3 grpinv11.g ⊢ ( 𝜑 → 𝐺 ∈ Grp )
4 grpinv11.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5 grpinv11.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6 fveq2 ⊢ ( ( 𝑁 ‘ 𝑋 ) = ( 𝑁 ‘ 𝑌 ) → ( 𝑁 ‘ ( 𝑁 ‘ 𝑋 ) ) = ( 𝑁 ‘ ( 𝑁 ‘ 𝑌 ) ) )
7 1 2 grpinvinv ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝑁 ‘ 𝑋 ) ) = 𝑋 )
8 3 4 7 syl2anc ⊢ ( 𝜑 → ( 𝑁 ‘ ( 𝑁 ‘ 𝑋 ) ) = 𝑋 )
9 1 2 grpinvinv ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝑁 ‘ 𝑌 ) ) = 𝑌 )
10 3 5 9 syl2anc ⊢ ( 𝜑 → ( 𝑁 ‘ ( 𝑁 ‘ 𝑌 ) ) = 𝑌 )
11 8 10 eqeq12d ⊢ ( 𝜑 → ( ( 𝑁 ‘ ( 𝑁 ‘ 𝑋 ) ) = ( 𝑁 ‘ ( 𝑁 ‘ 𝑌 ) ) ↔ 𝑋 = 𝑌 ) )
12 6 11 imbitrid ⊢ ( 𝜑 → ( ( 𝑁 ‘ 𝑋 ) = ( 𝑁 ‘ 𝑌 ) → 𝑋 = 𝑌 ) )
13 fveq2 ⊢ ( 𝑋 = 𝑌 → ( 𝑁 ‘ 𝑋 ) = ( 𝑁 ‘ 𝑌 ) )
14 12 13 impbid1 ⊢ ( 𝜑 → ( ( 𝑁 ‘ 𝑋 ) = ( 𝑁 ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) )