Metamath Proof Explorer


Theorem fprodsubrecnncnv

Description: The sequence S of finite products, where every factor is subtracted an "always smaller" amount, converges to the finite product of the factors. (Contributed by Glauco Siliprandi, 8-Apr-2021)

Ref Expression
Hypotheses fprodsubrecnncnv.1 𝑘 𝜑
fprodsubrecnncnv.2 ( 𝜑𝑋 ∈ Fin )
fprodsubrecnncnv.3 ( ( 𝜑𝑘𝑋 ) → 𝐴 ∈ ℂ )
fprodsubrecnncnv.4 𝑆 = ( 𝑛 ∈ ℕ ↦ ∏ 𝑘𝑋 ( 𝐴 − ( 1 / 𝑛 ) ) )
Assertion fprodsubrecnncnv ( 𝜑𝑆 ⇝ ∏ 𝑘𝑋 𝐴 )

Proof

Step Hyp Ref Expression
1 fprodsubrecnncnv.1 𝑘 𝜑
2 fprodsubrecnncnv.2 ( 𝜑𝑋 ∈ Fin )
3 fprodsubrecnncnv.3 ( ( 𝜑𝑘𝑋 ) → 𝐴 ∈ ℂ )
4 fprodsubrecnncnv.4 𝑆 = ( 𝑛 ∈ ℕ ↦ ∏ 𝑘𝑋 ( 𝐴 − ( 1 / 𝑛 ) ) )
5 eqid ( 𝑥 ∈ ℂ ↦ ∏ 𝑘𝑋 ( 𝐴𝑥 ) ) = ( 𝑥 ∈ ℂ ↦ ∏ 𝑘𝑋 ( 𝐴𝑥 ) )
6 oveq2 ( 𝑚 = 𝑛 → ( 1 / 𝑚 ) = ( 1 / 𝑛 ) )
7 6 cbvmptv ( 𝑚 ∈ ℕ ↦ ( 1 / 𝑚 ) ) = ( 𝑛 ∈ ℕ ↦ ( 1 / 𝑛 ) )
8 1 2 3 4 5 7 fprodsubrecnncnvlem ( 𝜑𝑆 ⇝ ∏ 𝑘𝑋 𝐴 )