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Metamath Proof Explorer

Theorem grpinvcnv

Description: The group inverse is its own inverse function. (Contributed by Mario Carneiro, 14-Aug-2015)

Ref Expression
Hypotheses grpinvinv.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
grpinvinv.n ⊢ 𝑁 = ( invg ‘ 𝐺 )
Assertion grpinvcnv ( 𝐺 ∈ Grp → ◡ 𝑁 = 𝑁 )

Proof

Step Hyp Ref Expression
1 grpinvinv.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 grpinvinv.n ⊢ 𝑁 = ( invg ‘ 𝐺 )
3 eqid ⊢ ( 𝑥 ∈ 𝐵 ↦ ( 𝑁 ‘ 𝑥 ) ) = ( 𝑥 ∈ 𝐵 ↦ ( 𝑁 ‘ 𝑥 ) )
4 1 2 grpinvcl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑥 ) ∈ 𝐵 )
5 1 2 grpinvcl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑦 ) ∈ 𝐵 )
6 eqid ⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
7 eqid ⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
8 1 6 7 2 grpinvid1 ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( ( 𝑁 ‘ 𝑦 ) = 𝑥 ↔ ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) ) )
9 8 3com23 ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑁 ‘ 𝑦 ) = 𝑥 ↔ ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) ) )
10 1 6 7 2 grpinvid2 ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑁 ‘ 𝑥 ) = 𝑦 ↔ ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) ) )
11 9 10 bitr4d ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑁 ‘ 𝑦 ) = 𝑥 ↔ ( 𝑁 ‘ 𝑥 ) = 𝑦 ) )
12 11 3expb ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝑁 ‘ 𝑦 ) = 𝑥 ↔ ( 𝑁 ‘ 𝑥 ) = 𝑦 ) )
13 eqcom ⊢ ( 𝑥 = ( 𝑁 ‘ 𝑦 ) ↔ ( 𝑁 ‘ 𝑦 ) = 𝑥 )
14 eqcom ⊢ ( 𝑦 = ( 𝑁 ‘ 𝑥 ) ↔ ( 𝑁 ‘ 𝑥 ) = 𝑦 )
15 12 13 14 3bitr4g ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 = ( 𝑁 ‘ 𝑦 ) ↔ 𝑦 = ( 𝑁 ‘ 𝑥 ) ) )
16 3 4 5 15 f1ocnv2d ⊢ ( 𝐺 ∈ Grp → ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑁 ‘ 𝑥 ) ) : 𝐵 –1-1-onto→ 𝐵 ∧ ◡ ( 𝑥 ∈ 𝐵 ↦ ( 𝑁 ‘ 𝑥 ) ) = ( 𝑦 ∈ 𝐵 ↦ ( 𝑁 ‘ 𝑦 ) ) ) )
17 16 simprd ⊢ ( 𝐺 ∈ Grp → ◡ ( 𝑥 ∈ 𝐵 ↦ ( 𝑁 ‘ 𝑥 ) ) = ( 𝑦 ∈ 𝐵 ↦ ( 𝑁 ‘ 𝑦 ) ) )
18 1 2 grpinvf ⊢ ( 𝐺 ∈ Grp → 𝑁 : 𝐵 ⟶ 𝐵 )
19 18 feqmptd ⊢ ( 𝐺 ∈ Grp → 𝑁 = ( 𝑥 ∈ 𝐵 ↦ ( 𝑁 ‘ 𝑥 ) ) )
20 19 cnveqd ⊢ ( 𝐺 ∈ Grp → ◡ 𝑁 = ◡ ( 𝑥 ∈ 𝐵 ↦ ( 𝑁 ‘ 𝑥 ) ) )
21 18 feqmptd ⊢ ( 𝐺 ∈ Grp → 𝑁 = ( 𝑦 ∈ 𝐵 ↦ ( 𝑁 ‘ 𝑦 ) ) )
22 17 20 21 3eqtr4d ⊢ ( 𝐺 ∈ Grp → ◡ 𝑁 = 𝑁 )