Metamath Proof Explorer


Theorem dpjghm2

Description: The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 26-Apr-2016)

Ref Expression
Hypotheses dpjfval.1 ( 𝜑𝐺 dom DProd 𝑆 )
dpjfval.2 ( 𝜑 → dom 𝑆 = 𝐼 )
dpjfval.p 𝑃 = ( 𝐺 dProj 𝑆 )
dpjlid.3 ( 𝜑𝑋𝐼 )
Assertion dpjghm2 ( 𝜑 → ( 𝑃𝑋 ) ∈ ( ( 𝐺s ( 𝐺 DProd 𝑆 ) ) GrpHom ( 𝐺s ( 𝑆𝑋 ) ) ) )

Proof

Step Hyp Ref Expression
1 dpjfval.1 ( 𝜑𝐺 dom DProd 𝑆 )
2 dpjfval.2 ( 𝜑 → dom 𝑆 = 𝐼 )
3 dpjfval.p 𝑃 = ( 𝐺 dProj 𝑆 )
4 dpjlid.3 ( 𝜑𝑋𝐼 )
5 1 2 3 4 dpjghm ( 𝜑 → ( 𝑃𝑋 ) ∈ ( ( 𝐺s ( 𝐺 DProd 𝑆 ) ) GrpHom 𝐺 ) )
6 1 2 dprdf2 ( 𝜑𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) )
7 6 4 ffvelrnd ( 𝜑 → ( 𝑆𝑋 ) ∈ ( SubGrp ‘ 𝐺 ) )
8 1 2 3 4 dpjf ( 𝜑 → ( 𝑃𝑋 ) : ( 𝐺 DProd 𝑆 ) ⟶ ( 𝑆𝑋 ) )
9 8 frnd ( 𝜑 → ran ( 𝑃𝑋 ) ⊆ ( 𝑆𝑋 ) )
10 eqid ( 𝐺s ( 𝑆𝑋 ) ) = ( 𝐺s ( 𝑆𝑋 ) )
11 10 resghm2b ( ( ( 𝑆𝑋 ) ∈ ( SubGrp ‘ 𝐺 ) ∧ ran ( 𝑃𝑋 ) ⊆ ( 𝑆𝑋 ) ) → ( ( 𝑃𝑋 ) ∈ ( ( 𝐺s ( 𝐺 DProd 𝑆 ) ) GrpHom 𝐺 ) ↔ ( 𝑃𝑋 ) ∈ ( ( 𝐺s ( 𝐺 DProd 𝑆 ) ) GrpHom ( 𝐺s ( 𝑆𝑋 ) ) ) ) )
12 7 9 11 syl2anc ( 𝜑 → ( ( 𝑃𝑋 ) ∈ ( ( 𝐺s ( 𝐺 DProd 𝑆 ) ) GrpHom 𝐺 ) ↔ ( 𝑃𝑋 ) ∈ ( ( 𝐺s ( 𝐺 DProd 𝑆 ) ) GrpHom ( 𝐺s ( 𝑆𝑋 ) ) ) ) )
13 5 12 mpbid ( 𝜑 → ( 𝑃𝑋 ) ∈ ( ( 𝐺s ( 𝐺 DProd 𝑆 ) ) GrpHom ( 𝐺s ( 𝑆𝑋 ) ) ) )