This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

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 ⊢ + = ( +g ‘ 𝐺 )
grpvlinv.n ⊢ 𝑁 = ( invg ‘ 𝐺 )
grpvlinv.z ⊢ 0 = ( 0g ‘ 𝐺 )
Assertion grpvlinv ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) → ( ( 𝑁 ∘ 𝑋 ) ∘f + 𝑋 ) = ( 𝐼 × { 0 } ) )

Proof

Step Hyp Ref Expression
1 grpvlinv.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 grpvlinv.p ⊢ + = ( +g ‘ 𝐺 )
3 grpvlinv.n ⊢ 𝑁 = ( invg ‘ 𝐺 )
4 grpvlinv.z ⊢ 0 = ( 0g ‘ 𝐺 )
5 elmapex ⊢ ( 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) → ( 𝐵 ∈ V ∧ 𝐼 ∈ V ) )
6 5 simprd ⊢ ( 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) → 𝐼 ∈ V )
7 6 adantl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) → 𝐼 ∈ V )
8 elmapi ⊢ ( 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) → 𝑋 : 𝐼 ⟶ 𝐵 )
9 8 adantl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) → 𝑋 : 𝐼 ⟶ 𝐵 )
10 1 4 grpidcl ⊢ ( 𝐺 ∈ Grp → 0 ∈ 𝐵 )
11 10 adantr ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) → 0 ∈ 𝐵 )
12 1 3 grpinvf ⊢ ( 𝐺 ∈ Grp → 𝑁 : 𝐵 ⟶ 𝐵 )
13 12 adantr ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) → 𝑁 : 𝐵 ⟶ 𝐵 )
14 fcompt ⊢ ( ( 𝑁 : 𝐵 ⟶ 𝐵 ∧ 𝑋 : 𝐼 ⟶ 𝐵 ) → ( 𝑁 ∘ 𝑋 ) = ( 𝑥 ∈ 𝐼 ↦ ( 𝑁 ‘ ( 𝑋 ‘ 𝑥 ) ) ) )
15 12 8 14 syl2an ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) → ( 𝑁 ∘ 𝑋 ) = ( 𝑥 ∈ 𝐼 ↦ ( 𝑁 ‘ ( 𝑋 ‘ 𝑥 ) ) ) )
16 1 2 4 3 grplinv ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑁 ‘ 𝑦 ) + 𝑦 ) = 0 )
17 16 adantlr ⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑁 ‘ 𝑦 ) + 𝑦 ) = 0 )
18 7 9 11 13 15 17 caofinvl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ) → ( ( 𝑁 ∘ 𝑋 ) ∘f + 𝑋 ) = ( 𝐼 × { 0 } ) )