Theorem dvmptcl

Description: Closure lemma for dvmptcmul and other related theorems. (Contributed by Mario Carneiro, 1-Sep-2014) (Revised by Mario Carneiro, 11-Feb-2015)

	Ref	Expression
Hypotheses	dvmptadd.s	⊢ ( 𝜑 → 𝑆 ∈ { ℝ , ℂ } )
	dvmptadd.a	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → 𝐴 ∈ ℂ )
	dvmptadd.b	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → 𝐵 ∈ 𝑉 )
	dvmptadd.da	⊢ ( 𝜑 → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ 𝐵 ) )
Assertion	dvmptcl	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → 𝐵 ∈ ℂ )

Proof

Step	Hyp	Ref	Expression
1		dvmptadd.s	⊢ ( 𝜑 → 𝑆 ∈ { ℝ , ℂ } )
2		dvmptadd.a	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → 𝐴 ∈ ℂ )
3		dvmptadd.b	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → 𝐵 ∈ 𝑉 )
4		dvmptadd.da	⊢ ( 𝜑 → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ 𝐵 ) )
5		dvfg	⊢ ( 𝑆 ∈ { ℝ , ℂ } → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ) : dom ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ) ⟶ ℂ )
6	1 5	syl	⊢ ( 𝜑 → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ) : dom ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ) ⟶ ℂ )
7	4	dmeqd	⊢ ( 𝜑 → dom ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ) = dom ( 𝑥 ∈ 𝑋 ↦ 𝐵 ) )
8	3	ralrimiva	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑋 𝐵 ∈ 𝑉 )
9		dmmptg	⊢ ( ∀ 𝑥 ∈ 𝑋 𝐵 ∈ 𝑉 → dom ( 𝑥 ∈ 𝑋 ↦ 𝐵 ) = 𝑋 )
10	8 9	syl	⊢ ( 𝜑 → dom ( 𝑥 ∈ 𝑋 ↦ 𝐵 ) = 𝑋 )
11	7 10	eqtrd	⊢ ( 𝜑 → dom ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ) = 𝑋 )
12	11	feq2d	⊢ ( 𝜑 → ( ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ) : dom ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ) ⟶ ℂ ↔ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ) : 𝑋 ⟶ ℂ ) )
13	6 12	mpbid	⊢ ( 𝜑 → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ) : 𝑋 ⟶ ℂ )
14	4	feq1d	⊢ ( 𝜑 → ( ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ) : 𝑋 ⟶ ℂ ↔ ( 𝑥 ∈ 𝑋 ↦ 𝐵 ) : 𝑋 ⟶ ℂ ) )
15	13 14	mpbid	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ 𝐵 ) : 𝑋 ⟶ ℂ )
16	15	fvmptelrn	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → 𝐵 ∈ ℂ )