Metamath Proof Explorer


Theorem dpjcntz

Description: The two subgroups that appear in dpjval commute. (Contributed by Mario Carneiro, 26-Apr-2016)

Ref Expression
Hypotheses dpjfval.1 ( 𝜑𝐺 dom DProd 𝑆 )
dpjfval.2 ( 𝜑 → dom 𝑆 = 𝐼 )
dpjlem.3 ( 𝜑𝑋𝐼 )
dpjcntz.z 𝑍 = ( Cntz ‘ 𝐺 )
Assertion dpjcntz ( 𝜑 → ( 𝑆𝑋 ) ⊆ ( 𝑍 ‘ ( 𝐺 DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ) ) )

Proof

Step Hyp Ref Expression
1 dpjfval.1 ( 𝜑𝐺 dom DProd 𝑆 )
2 dpjfval.2 ( 𝜑 → dom 𝑆 = 𝐼 )
3 dpjlem.3 ( 𝜑𝑋𝐼 )
4 dpjcntz.z 𝑍 = ( Cntz ‘ 𝐺 )
5 1 2 3 dpjlem ( 𝜑 → ( 𝐺 DProd ( 𝑆 ↾ { 𝑋 } ) ) = ( 𝑆𝑋 ) )
6 1 2 dprdf2 ( 𝜑𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) )
7 disjdif ( { 𝑋 } ∩ ( 𝐼 ∖ { 𝑋 } ) ) = ∅
8 7 a1i ( 𝜑 → ( { 𝑋 } ∩ ( 𝐼 ∖ { 𝑋 } ) ) = ∅ )
9 undif2 ( { 𝑋 } ∪ ( 𝐼 ∖ { 𝑋 } ) ) = ( { 𝑋 } ∪ 𝐼 )
10 3 snssd ( 𝜑 → { 𝑋 } ⊆ 𝐼 )
11 ssequn1 ( { 𝑋 } ⊆ 𝐼 ↔ ( { 𝑋 } ∪ 𝐼 ) = 𝐼 )
12 10 11 sylib ( 𝜑 → ( { 𝑋 } ∪ 𝐼 ) = 𝐼 )
13 9 12 eqtr2id ( 𝜑𝐼 = ( { 𝑋 } ∪ ( 𝐼 ∖ { 𝑋 } ) ) )
14 eqid ( 0g𝐺 ) = ( 0g𝐺 )
15 6 8 13 4 14 dmdprdsplit ( 𝜑 → ( 𝐺 dom DProd 𝑆 ↔ ( ( 𝐺 dom DProd ( 𝑆 ↾ { 𝑋 } ) ∧ 𝐺 dom DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ) ∧ ( 𝐺 DProd ( 𝑆 ↾ { 𝑋 } ) ) ⊆ ( 𝑍 ‘ ( 𝐺 DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ) ) ∧ ( ( 𝐺 DProd ( 𝑆 ↾ { 𝑋 } ) ) ∩ ( 𝐺 DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ) ) = { ( 0g𝐺 ) } ) ) )
16 1 15 mpbid ( 𝜑 → ( ( 𝐺 dom DProd ( 𝑆 ↾ { 𝑋 } ) ∧ 𝐺 dom DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ) ∧ ( 𝐺 DProd ( 𝑆 ↾ { 𝑋 } ) ) ⊆ ( 𝑍 ‘ ( 𝐺 DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ) ) ∧ ( ( 𝐺 DProd ( 𝑆 ↾ { 𝑋 } ) ) ∩ ( 𝐺 DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ) ) = { ( 0g𝐺 ) } ) )
17 16 simp2d ( 𝜑 → ( 𝐺 DProd ( 𝑆 ↾ { 𝑋 } ) ) ⊆ ( 𝑍 ‘ ( 𝐺 DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ) ) )
18 5 17 eqsstrrd ( 𝜑 → ( 𝑆𝑋 ) ⊆ ( 𝑍 ‘ ( 𝐺 DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ) ) )