Theorem dprdub

Description: Each factor is a subset of the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016)

	Ref	Expression
Hypotheses	dprdub.1	⊢ ( 𝜑 → 𝐺 dom DProd 𝑆 )
	dprdub.2	⊢ ( 𝜑 → dom 𝑆 = 𝐼 )
	dprdub.3	⊢ ( 𝜑 → 𝑋 ∈ 𝐼 )
Assertion	dprdub	⊢ ( 𝜑 → ( 𝑆 ‘ 𝑋 ) ⊆ ( 𝐺 DProd 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		dprdub.1	⊢ ( 𝜑 → 𝐺 dom DProd 𝑆 )
2		dprdub.2	⊢ ( 𝜑 → dom 𝑆 = 𝐼 )
3		dprdub.3	⊢ ( 𝜑 → 𝑋 ∈ 𝐼 )
4		eqid	⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
5		eqid	⊢ { ℎ ∈ X 𝑖 ∈ 𝐼 ( 𝑆 ‘ 𝑖 ) ∣ ℎ finSupp ( 0g ‘ 𝐺 ) } = { ℎ ∈ X 𝑖 ∈ 𝐼 ( 𝑆 ‘ 𝑖 ) ∣ ℎ finSupp ( 0g ‘ 𝐺 ) }
6	1	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑋 ) ) → 𝐺 dom DProd 𝑆 )
7	2	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑋 ) ) → dom 𝑆 = 𝐼 )
8	3	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑋 ) ) → 𝑋 ∈ 𝐼 )
9		simpr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑋 ) ) → 𝑥 ∈ ( 𝑆 ‘ 𝑋 ) )
10		eqid	⊢ ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 = 𝑋 , 𝑥 , ( 0g ‘ 𝐺 ) ) ) = ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 = 𝑋 , 𝑥 , ( 0g ‘ 𝐺 ) ) )
11	4 5 6 7 8 9 10	dprdfid	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑋 ) ) → ( ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 = 𝑋 , 𝑥 , ( 0g ‘ 𝐺 ) ) ) ∈ { ℎ ∈ X 𝑖 ∈ 𝐼 ( 𝑆 ‘ 𝑖 ) ∣ ℎ finSupp ( 0g ‘ 𝐺 ) } ∧ ( 𝐺 Σg ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 = 𝑋 , 𝑥 , ( 0g ‘ 𝐺 ) ) ) ) = 𝑥 ) )
12	11	simprd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑋 ) ) → ( 𝐺 Σg ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 = 𝑋 , 𝑥 , ( 0g ‘ 𝐺 ) ) ) ) = 𝑥 )
13	11	simpld	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑋 ) ) → ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 = 𝑋 , 𝑥 , ( 0g ‘ 𝐺 ) ) ) ∈ { ℎ ∈ X 𝑖 ∈ 𝐼 ( 𝑆 ‘ 𝑖 ) ∣ ℎ finSupp ( 0g ‘ 𝐺 ) } )
14	4 5 6 7 13	eldprdi	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑋 ) ) → ( 𝐺 Σg ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 = 𝑋 , 𝑥 , ( 0g ‘ 𝐺 ) ) ) ) ∈ ( 𝐺 DProd 𝑆 ) )
15	12 14	eqeltrrd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑋 ) ) → 𝑥 ∈ ( 𝐺 DProd 𝑆 ) )
16	15	ex	⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝑆 ‘ 𝑋 ) → 𝑥 ∈ ( 𝐺 DProd 𝑆 ) ) )
17	16	ssrdv	⊢ ( 𝜑 → ( 𝑆 ‘ 𝑋 ) ⊆ ( 𝐺 DProd 𝑆 ) )