Metamath Proof Explorer


Theorem grpvrinv

Description: Tuple-wise right inverse in groups. (Contributed by Mario Carneiro, 22-Sep-2015)

Ref Expression
Hypotheses grpvlinv.b 𝐵 = ( Base ‘ 𝐺 )
grpvlinv.p + = ( +g𝐺 )
grpvlinv.n 𝑁 = ( invg𝐺 )
grpvlinv.z 0 = ( 0g𝐺 )
Assertion grpvrinv ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵m 𝐼 ) ) → ( 𝑋f + ( 𝑁𝑋 ) ) = ( 𝐼 × { 0 } ) )

Proof

Step Hyp Ref Expression
1 grpvlinv.b 𝐵 = ( Base ‘ 𝐺 )
2 grpvlinv.p + = ( +g𝐺 )
3 grpvlinv.n 𝑁 = ( invg𝐺 )
4 grpvlinv.z 0 = ( 0g𝐺 )
5 simpll ( ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵m 𝐼 ) ) ∧ 𝑥𝐼 ) → 𝐺 ∈ Grp )
6 elmapi ( 𝑋 ∈ ( 𝐵m 𝐼 ) → 𝑋 : 𝐼𝐵 )
7 6 adantl ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵m 𝐼 ) ) → 𝑋 : 𝐼𝐵 )
8 7 ffvelrnda ( ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵m 𝐼 ) ) ∧ 𝑥𝐼 ) → ( 𝑋𝑥 ) ∈ 𝐵 )
9 1 2 4 3 grprinv ( ( 𝐺 ∈ Grp ∧ ( 𝑋𝑥 ) ∈ 𝐵 ) → ( ( 𝑋𝑥 ) + ( 𝑁 ‘ ( 𝑋𝑥 ) ) ) = 0 )
10 5 8 9 syl2anc ( ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵m 𝐼 ) ) ∧ 𝑥𝐼 ) → ( ( 𝑋𝑥 ) + ( 𝑁 ‘ ( 𝑋𝑥 ) ) ) = 0 )
11 10 mpteq2dva ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵m 𝐼 ) ) → ( 𝑥𝐼 ↦ ( ( 𝑋𝑥 ) + ( 𝑁 ‘ ( 𝑋𝑥 ) ) ) ) = ( 𝑥𝐼0 ) )
12 elmapex ( 𝑋 ∈ ( 𝐵m 𝐼 ) → ( 𝐵 ∈ V ∧ 𝐼 ∈ V ) )
13 12 simprd ( 𝑋 ∈ ( 𝐵m 𝐼 ) → 𝐼 ∈ V )
14 13 adantl ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵m 𝐼 ) ) → 𝐼 ∈ V )
15 fvexd ( ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵m 𝐼 ) ) ∧ 𝑥𝐼 ) → ( 𝑁 ‘ ( 𝑋𝑥 ) ) ∈ V )
16 7 feqmptd ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵m 𝐼 ) ) → 𝑋 = ( 𝑥𝐼 ↦ ( 𝑋𝑥 ) ) )
17 1 3 grpinvf ( 𝐺 ∈ Grp → 𝑁 : 𝐵𝐵 )
18 fcompt ( ( 𝑁 : 𝐵𝐵𝑋 : 𝐼𝐵 ) → ( 𝑁𝑋 ) = ( 𝑥𝐼 ↦ ( 𝑁 ‘ ( 𝑋𝑥 ) ) ) )
19 17 6 18 syl2an ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵m 𝐼 ) ) → ( 𝑁𝑋 ) = ( 𝑥𝐼 ↦ ( 𝑁 ‘ ( 𝑋𝑥 ) ) ) )
20 14 8 15 16 19 offval2 ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵m 𝐼 ) ) → ( 𝑋f + ( 𝑁𝑋 ) ) = ( 𝑥𝐼 ↦ ( ( 𝑋𝑥 ) + ( 𝑁 ‘ ( 𝑋𝑥 ) ) ) ) )
21 fconstmpt ( 𝐼 × { 0 } ) = ( 𝑥𝐼0 )
22 21 a1i ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵m 𝐼 ) ) → ( 𝐼 × { 0 } ) = ( 𝑥𝐼0 ) )
23 11 20 22 3eqtr4d ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵m 𝐼 ) ) → ( 𝑋f + ( 𝑁𝑋 ) ) = ( 𝐼 × { 0 } ) )