Metamath Proof Explorer

Theorem sge0rnbnd

Description: The range used in the definition of sum^ is bounded, when the whole sum is a real number. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Hypotheses sge0rnbnd.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
sge0rnbnd.f ⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) )
sge0rnbnd.re ⊢ ( 𝜑 → ( Σ^ ‘ 𝐹 ) ∈ ℝ )
Assertion sge0rnbnd ( 𝜑 → ∃ 𝑧 ∈ ℝ ∀ 𝑤 ∈ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) 𝑤 ≤ 𝑧 )

Proof

Step Hyp Ref Expression
1 sge0rnbnd.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
2 sge0rnbnd.f ⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) )
3 sge0rnbnd.re ⊢ ( 𝜑 → ( Σ^ ‘ 𝐹 ) ∈ ℝ )
4 simpl ⊢ ( ( 𝜑 ∧ 𝑤 ∈ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) ) → 𝜑 )
5 vex ⊢ 𝑤 ∈ V
6 eqid ⊢ ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) = ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) )
7 6 elrnmpt ⊢ ( 𝑤 ∈ V → ( 𝑤 ∈ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) ↔ ∃ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑤 = Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) )
8 5 7 ax-mp ⊢ ( 𝑤 ∈ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) ↔ ∃ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑤 = Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) )
9 8 biimpi ⊢ ( 𝑤 ∈ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) → ∃ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑤 = Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) )
10 9 adantl ⊢ ( ( 𝜑 ∧ 𝑤 ∈ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) ) → ∃ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑤 = Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) )
11 simp3 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ∧ 𝑤 = Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) → 𝑤 = Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) )
12 1 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → 𝑋 ∈ 𝑉 )
13 2 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) )
14 1 2 3 sge0rern ⊢ ( 𝜑 → ¬ +∞ ∈ ran 𝐹 )
15 14 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → ¬ +∞ ∈ ran 𝐹 )
16 13 15 fge0iccico ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → 𝐹 : 𝑋 ⟶ ( 0 [,) +∞ ) )
17 elpwinss ⊢ ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) → 𝑥 ⊆ 𝑋 )
18 17 adantl ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → 𝑥 ⊆ 𝑋 )
19 elinel2 ⊢ ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) → 𝑥 ∈ Fin )
20 19 adantl ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → 𝑥 ∈ Fin )
21 12 16 18 20 fsumlesge0 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ) → Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ≤ ( Σ^ ‘ 𝐹 ) )
22 21 3adant3 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ∧ 𝑤 = Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) → Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ≤ ( Σ^ ‘ 𝐹 ) )
23 11 22 eqbrtrd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ∧ 𝑤 = Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) → 𝑤 ≤ ( Σ^ ‘ 𝐹 ) )
24 23 3exp ⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) → ( 𝑤 = Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) → 𝑤 ≤ ( Σ^ ‘ 𝐹 ) ) ) )
25 24 rexlimdv ⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑤 = Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) → 𝑤 ≤ ( Σ^ ‘ 𝐹 ) ) )
26 4 10 25 sylc ⊢ ( ( 𝜑 ∧ 𝑤 ∈ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) ) → 𝑤 ≤ ( Σ^ ‘ 𝐹 ) )
27 26 ralrimiva ⊢ ( 𝜑 → ∀ 𝑤 ∈ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) 𝑤 ≤ ( Σ^ ‘ 𝐹 ) )
28 brralrspcev ⊢ ( ( ( Σ^ ‘ 𝐹 ) ∈ ℝ ∧ ∀ 𝑤 ∈ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) 𝑤 ≤ ( Σ^ ‘ 𝐹 ) ) → ∃ 𝑧 ∈ ℝ ∀ 𝑤 ∈ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) 𝑤 ≤ 𝑧 )
29 3 27 28 syl2anc ⊢ ( 𝜑 → ∃ 𝑧 ∈ ℝ ∀ 𝑤 ∈ ran ( 𝑥 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ) 𝑤 ≤ 𝑧 )