Metamath Proof Explorer


Theorem grpvlinv

Description: Tuple-wise left inverse in groups. (Contributed by Stefan O'Rear, 5-Sep-2015)

Ref Expression
Hypotheses grpvlinv.b 𝐵 = ( Base ‘ 𝐺 )
grpvlinv.p + = ( +g𝐺 )
grpvlinv.n 𝑁 = ( invg𝐺 )
grpvlinv.z 0 = ( 0g𝐺 )
Assertion grpvlinv ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵m 𝐼 ) ) → ( ( 𝑁𝑋 ) ∘f + 𝑋 ) = ( 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 elmapex ( 𝑋 ∈ ( 𝐵m 𝐼 ) → ( 𝐵 ∈ V ∧ 𝐼 ∈ V ) )
6 5 simprd ( 𝑋 ∈ ( 𝐵m 𝐼 ) → 𝐼 ∈ V )
7 6 adantl ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵m 𝐼 ) ) → 𝐼 ∈ V )
8 elmapi ( 𝑋 ∈ ( 𝐵m 𝐼 ) → 𝑋 : 𝐼𝐵 )
9 8 adantl ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵m 𝐼 ) ) → 𝑋 : 𝐼𝐵 )
10 1 4 grpidcl ( 𝐺 ∈ Grp → 0𝐵 )
11 10 adantr ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵m 𝐼 ) ) → 0𝐵 )
12 1 3 grpinvf ( 𝐺 ∈ Grp → 𝑁 : 𝐵𝐵 )
13 12 adantr ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵m 𝐼 ) ) → 𝑁 : 𝐵𝐵 )
14 fcompt ( ( 𝑁 : 𝐵𝐵𝑋 : 𝐼𝐵 ) → ( 𝑁𝑋 ) = ( 𝑥𝐼 ↦ ( 𝑁 ‘ ( 𝑋𝑥 ) ) ) )
15 12 8 14 syl2an ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵m 𝐼 ) ) → ( 𝑁𝑋 ) = ( 𝑥𝐼 ↦ ( 𝑁 ‘ ( 𝑋𝑥 ) ) ) )
16 1 2 4 3 grplinv ( ( 𝐺 ∈ Grp ∧ 𝑦𝐵 ) → ( ( 𝑁𝑦 ) + 𝑦 ) = 0 )
17 16 adantlr ( ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵m 𝐼 ) ) ∧ 𝑦𝐵 ) → ( ( 𝑁𝑦 ) + 𝑦 ) = 0 )
18 7 9 11 13 15 17 caofinvl ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵m 𝐼 ) ) → ( ( 𝑁𝑋 ) ∘f + 𝑋 ) = ( 𝐼 × { 0 } ) )