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Metamath Proof Explorer

Theorem grpinv11OLD

Description: Obsolete version of grpinv11 as of 8-Jul-2025. (Contributed by NM, 22-Mar-2015) (Proof modification is discouraged.) (New usage is discouraged.)

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 grpinv11OLD ( 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 6 adantl ⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ 𝑋 ) = ( 𝑁 ‘ 𝑌 ) ) → ( 𝑁 ‘ ( 𝑁 ‘ 𝑋 ) ) = ( 𝑁 ‘ ( 𝑁 ‘ 𝑌 ) ) )
8 1 2 grpinvinv ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝑁 ‘ 𝑋 ) ) = 𝑋 )
9 3 4 8 syl2anc ⊢ ( 𝜑 → ( 𝑁 ‘ ( 𝑁 ‘ 𝑋 ) ) = 𝑋 )
10 9 adantr ⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ 𝑋 ) = ( 𝑁 ‘ 𝑌 ) ) → ( 𝑁 ‘ ( 𝑁 ‘ 𝑋 ) ) = 𝑋 )
11 1 2 grpinvinv ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝑁 ‘ 𝑌 ) ) = 𝑌 )
12 3 5 11 syl2anc ⊢ ( 𝜑 → ( 𝑁 ‘ ( 𝑁 ‘ 𝑌 ) ) = 𝑌 )
13 12 adantr ⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ 𝑋 ) = ( 𝑁 ‘ 𝑌 ) ) → ( 𝑁 ‘ ( 𝑁 ‘ 𝑌 ) ) = 𝑌 )
14 7 10 13 3eqtr3d ⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ 𝑋 ) = ( 𝑁 ‘ 𝑌 ) ) → 𝑋 = 𝑌 )
15 14 ex ⊢ ( 𝜑 → ( ( 𝑁 ‘ 𝑋 ) = ( 𝑁 ‘ 𝑌 ) → 𝑋 = 𝑌 ) )
16 fveq2 ⊢ ( 𝑋 = 𝑌 → ( 𝑁 ‘ 𝑋 ) = ( 𝑁 ‘ 𝑌 ) )
17 15 16 impbid1 ⊢ ( 𝜑 → ( ( 𝑁 ‘ 𝑋 ) = ( 𝑁 ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) )