Theorem dpjlsm

Description: The two subgroups that appear in dpjval add to the full direct product. (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 )
	dpjlsm.s	⊢ ⊕ = ( LSSum ‘ 𝐺 )
Assertion	dpjlsm	⊢ ( 𝜑 → ( 𝐺 DProd 𝑆 ) = ( ( 𝑆 ‘ 𝑋 ) ⊕ ( 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		dpjlsm.s	⊢ ⊕ = ( LSSum ‘ 𝐺 )
5	1 2	dprdf2	⊢ ( 𝜑 → 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) )
6		disjdif	⊢ ( { 𝑋 } ∩ ( 𝐼 ∖ { 𝑋 } ) ) = ∅
7	6	a1i	⊢ ( 𝜑 → ( { 𝑋 } ∩ ( 𝐼 ∖ { 𝑋 } ) ) = ∅ )
8		undif2	⊢ ( { 𝑋 } ∪ ( 𝐼 ∖ { 𝑋 } ) ) = ( { 𝑋 } ∪ 𝐼 )
9	3	snssd	⊢ ( 𝜑 → { 𝑋 } ⊆ 𝐼 )
10		ssequn1	⊢ ( { 𝑋 } ⊆ 𝐼 ↔ ( { 𝑋 } ∪ 𝐼 ) = 𝐼 )
11	9 10	sylib	⊢ ( 𝜑 → ( { 𝑋 } ∪ 𝐼 ) = 𝐼 )
12	8 11	syl5req	⊢ ( 𝜑 → 𝐼 = ( { 𝑋 } ∪ ( 𝐼 ∖ { 𝑋 } ) ) )
13	5 7 12 4 1	dprdsplit	⊢ ( 𝜑 → ( 𝐺 DProd 𝑆 ) = ( ( 𝐺 DProd ( 𝑆 ↾ { 𝑋 } ) ) ⊕ ( 𝐺 DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ) ) )
14	1 2 3	dpjlem	⊢ ( 𝜑 → ( 𝐺 DProd ( 𝑆 ↾ { 𝑋 } ) ) = ( 𝑆 ‘ 𝑋 ) )
15	14	oveq1d	⊢ ( 𝜑 → ( ( 𝐺 DProd ( 𝑆 ↾ { 𝑋 } ) ) ⊕ ( 𝐺 DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ) ) = ( ( 𝑆 ‘ 𝑋 ) ⊕ ( 𝐺 DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ) ) )
16	13 15	eqtrd	⊢ ( 𝜑 → ( 𝐺 DProd 𝑆 ) = ( ( 𝑆 ‘ 𝑋 ) ⊕ ( 𝐺 DProd ( 𝑆 ↾ ( 𝐼 ∖ { 𝑋 } ) ) ) ) )