Theorem dpjcntz

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

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

Proof

Step	Hyp	Ref	Expression
1		dpjfval.1	⊢ ( 𝜑 → 𝐺 dom DProd 𝑆 )
2		dpjfval.2	⊢ ( 𝜑 → dom 𝑆 = 𝐼 )
3		dpjlem.3	⊢ ( 𝜑 → 𝑋 ∈ 𝐼 )
4		dpjcntz.z	⊢ 𝑍 = ( Cntz ‘ 𝐺 )
5	1 2 3	dpjlem	⊢ ( 𝜑 → ( 𝐺 DProd ( 𝑆 ↾ { 𝑋 } ) ) = ( 𝑆 ‘ 𝑋 ) )
6	1 2	dprdf2	⊢ ( 𝜑 → 𝑆 : 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	syl5req	⊢ ( 𝜑 → 𝐼 = ( { 𝑋 } ∪ ( 𝐼 ∖ { 𝑋 } ) ) )
14		eqid	⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
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 } ) ) ) ) = { ( 0g ‘ 𝐺 ) } ) ) )
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 } ) ) ) ) = { ( 0g ‘ 𝐺 ) } ) )
17	16	simp2d	⊢ ( 𝜑 → ( 𝐺 DProd ( 𝑆 ↾ { 𝑋 } ) ) ⊆ ( 𝑍 ‘ ( 𝐺 DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ) ) )
18	5 17	eqsstrrd	⊢ ( 𝜑 → ( 𝑆 ‘ 𝑋 ) ⊆ ( 𝑍 ‘ ( 𝐺 DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ) ) )