Metamath Proof Explorer

Theorem dpjlid

Description: The X -th index projection acts as the identity on elements of the X -th factor. (Contributed by Mario Carneiro, 26-Apr-2016)

Ref Expression
Hypotheses dpjfval.1 ⊢ ( 𝜑 → 𝐺 dom DProd 𝑆 )
dpjfval.2 ⊢ ( 𝜑 → dom 𝑆 = 𝐼 )
dpjfval.p ⊢ 𝑃 = ( 𝐺 dProj 𝑆 )
dpjlid.3 ⊢ ( 𝜑 → 𝑋 ∈ 𝐼 )
dpjlid.4 ⊢ ( 𝜑 → 𝐴 ∈ ( 𝑆 ‘ 𝑋 ) )
Assertion dpjlid ( 𝜑 → ( ( 𝑃 ‘ 𝑋 ) ‘ 𝐴 ) = 𝐴 )

Proof

Step Hyp Ref Expression
1 dpjfval.1 ⊢ ( 𝜑 → 𝐺 dom DProd 𝑆 )
2 dpjfval.2 ⊢ ( 𝜑 → dom 𝑆 = 𝐼 )
3 dpjfval.p ⊢ 𝑃 = ( 𝐺 dProj 𝑆 )
4 dpjlid.3 ⊢ ( 𝜑 → 𝑋 ∈ 𝐼 )
5 dpjlid.4 ⊢ ( 𝜑 → 𝐴 ∈ ( 𝑆 ‘ 𝑋 ) )
6 eqid ⊢ ( proj1 ‘ 𝐺 ) = ( proj1 ‘ 𝐺 )
7 1 2 3 6 4 dpjval ⊢ ( 𝜑 → ( 𝑃 ‘ 𝑋 ) = ( ( 𝑆 ‘ 𝑋 ) ( proj1 ‘ 𝐺 ) ( 𝐺 DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ) ) )
8 7 fveq1d ⊢ ( 𝜑 → ( ( 𝑃 ‘ 𝑋 ) ‘ 𝐴 ) = ( ( ( 𝑆 ‘ 𝑋 ) ( proj1 ‘ 𝐺 ) ( 𝐺 DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ) ) ‘ 𝐴 ) )
9 eqid ⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
10 eqid ⊢ ( LSSum ‘ 𝐺 ) = ( LSSum ‘ 𝐺 )
11 eqid ⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
12 eqid ⊢ ( Cntz ‘ 𝐺 ) = ( Cntz ‘ 𝐺 )
13 1 2 dprdf2 ⊢ ( 𝜑 → 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) )
14 13 4 ffvelrnd ⊢ ( 𝜑 → ( 𝑆 ‘ 𝑋 ) ∈ ( SubGrp ‘ 𝐺 ) )
15 difssd ⊢ ( 𝜑 → ( 𝐼 ∖ { 𝑋 } ) ⊆ 𝐼 )
16 1 2 15 dprdres ⊢ ( 𝜑 → ( 𝐺 dom DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ∧ ( 𝐺 DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ) ⊆ ( 𝐺 DProd 𝑆 ) ) )
17 16 simpld ⊢ ( 𝜑 → 𝐺 dom DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) )
18 dprdsubg ⊢ ( 𝐺 dom DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) → ( 𝐺 DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ) ∈ ( SubGrp ‘ 𝐺 ) )
19 17 18 syl ⊢ ( 𝜑 → ( 𝐺 DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ) ∈ ( SubGrp ‘ 𝐺 ) )
20 1 2 4 11 dpjdisj ⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑋 ) ∩ ( 𝐺 DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ) ) = { ( 0g ‘ 𝐺 ) } )
21 1 2 4 12 dpjcntz ⊢ ( 𝜑 → ( 𝑆 ‘ 𝑋 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝐺 DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ) ) )
22 9 10 11 12 14 19 20 21 6 pj1lid ⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑆 ‘ 𝑋 ) ) → ( ( ( 𝑆 ‘ 𝑋 ) ( proj1 ‘ 𝐺 ) ( 𝐺 DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ) ) ‘ 𝐴 ) = 𝐴 )
23 5 22 mpdan ⊢ ( 𝜑 → ( ( ( 𝑆 ‘ 𝑋 ) ( proj1 ‘ 𝐺 ) ( 𝐺 DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ) ) ‘ 𝐴 ) = 𝐴 )
24 8 23 eqtrd ⊢ ( 𝜑 → ( ( 𝑃 ‘ 𝑋 ) ‘ 𝐴 ) = 𝐴 )