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 = ( invg𝐺 )
Assertion grpinvcnv ( 𝐺 ∈ Grp → 𝑁 = 𝑁 )

Proof

Step Hyp Ref Expression
1 grpinvinv.b 𝐵 = ( Base ‘ 𝐺 )
2 grpinvinv.n 𝑁 = ( invg𝐺 )
3 eqid ( 𝑥𝐵 ↦ ( 𝑁𝑥 ) ) = ( 𝑥𝐵 ↦ ( 𝑁𝑥 ) )
4 1 2 grpinvcl ( ( 𝐺 ∈ Grp ∧ 𝑥𝐵 ) → ( 𝑁𝑥 ) ∈ 𝐵 )
5 1 2 grpinvcl ( ( 𝐺 ∈ Grp ∧ 𝑦𝐵 ) → ( 𝑁𝑦 ) ∈ 𝐵 )
6 eqid ( +g𝐺 ) = ( +g𝐺 )
7 eqid ( 0g𝐺 ) = ( 0g𝐺 )
8 1 6 7 2 grpinvid1 ( ( 𝐺 ∈ Grp ∧ 𝑦𝐵𝑥𝐵 ) → ( ( 𝑁𝑦 ) = 𝑥 ↔ ( 𝑦 ( +g𝐺 ) 𝑥 ) = ( 0g𝐺 ) ) )
9 8 3com23 ( ( 𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵 ) → ( ( 𝑁𝑦 ) = 𝑥 ↔ ( 𝑦 ( +g𝐺 ) 𝑥 ) = ( 0g𝐺 ) ) )
10 1 6 7 2 grpinvid2 ( ( 𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵 ) → ( ( 𝑁𝑥 ) = 𝑦 ↔ ( 𝑦 ( +g𝐺 ) 𝑥 ) = ( 0g𝐺 ) ) )
11 9 10 bitr4d ( ( 𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵 ) → ( ( 𝑁𝑦 ) = 𝑥 ↔ ( 𝑁𝑥 ) = 𝑦 ) )
12 11 3expb ( ( 𝐺 ∈ Grp ∧ ( 𝑥𝐵𝑦𝐵 ) ) → ( ( 𝑁𝑦 ) = 𝑥 ↔ ( 𝑁𝑥 ) = 𝑦 ) )
13 eqcom ( 𝑥 = ( 𝑁𝑦 ) ↔ ( 𝑁𝑦 ) = 𝑥 )
14 eqcom ( 𝑦 = ( 𝑁𝑥 ) ↔ ( 𝑁𝑥 ) = 𝑦 )
15 12 13 14 3bitr4g ( ( 𝐺 ∈ Grp ∧ ( 𝑥𝐵𝑦𝐵 ) ) → ( 𝑥 = ( 𝑁𝑦 ) ↔ 𝑦 = ( 𝑁𝑥 ) ) )
16 3 4 5 15 f1ocnv2d ( 𝐺 ∈ Grp → ( ( 𝑥𝐵 ↦ ( 𝑁𝑥 ) ) : 𝐵1-1-onto𝐵 ( 𝑥𝐵 ↦ ( 𝑁𝑥 ) ) = ( 𝑦𝐵 ↦ ( 𝑁𝑦 ) ) ) )
17 16 simprd ( 𝐺 ∈ Grp → ( 𝑥𝐵 ↦ ( 𝑁𝑥 ) ) = ( 𝑦𝐵 ↦ ( 𝑁𝑦 ) ) )
18 1 2 grpinvf ( 𝐺 ∈ Grp → 𝑁 : 𝐵𝐵 )
19 18 feqmptd ( 𝐺 ∈ Grp → 𝑁 = ( 𝑥𝐵 ↦ ( 𝑁𝑥 ) ) )
20 19 cnveqd ( 𝐺 ∈ Grp → 𝑁 = ( 𝑥𝐵 ↦ ( 𝑁𝑥 ) ) )
21 18 feqmptd ( 𝐺 ∈ Grp → 𝑁 = ( 𝑦𝐵 ↦ ( 𝑁𝑦 ) ) )
22 17 20 21 3eqtr4d ( 𝐺 ∈ Grp → 𝑁 = 𝑁 )