Metamath Proof Explorer


Theorem sge0val

Description: The value of the sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Assertion sge0val ( ( 𝑋𝑉𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) → ( Σ^𝐹 ) = if ( +∞ ∈ ran 𝐹 , +∞ , sup ( ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤𝑦 ( 𝐹𝑤 ) ) , ℝ* , < ) ) )

Proof

Step Hyp Ref Expression
1 df-sumge0 Σ^ = ( 𝑥 ∈ V ↦ if ( +∞ ∈ ran 𝑥 , +∞ , sup ( ran ( 𝑦 ∈ ( 𝒫 dom 𝑥 ∩ Fin ) ↦ Σ 𝑤𝑦 ( 𝑥𝑤 ) ) , ℝ* , < ) ) )
2 1 a1i ( ( 𝑋𝑉𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) → Σ^ = ( 𝑥 ∈ V ↦ if ( +∞ ∈ ran 𝑥 , +∞ , sup ( ran ( 𝑦 ∈ ( 𝒫 dom 𝑥 ∩ Fin ) ↦ Σ 𝑤𝑦 ( 𝑥𝑤 ) ) , ℝ* , < ) ) ) )
3 rneq ( 𝑥 = 𝐹 → ran 𝑥 = ran 𝐹 )
4 3 eleq2d ( 𝑥 = 𝐹 → ( +∞ ∈ ran 𝑥 ↔ +∞ ∈ ran 𝐹 ) )
5 4 adantl ( ( ( 𝑋𝑉𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) ∧ 𝑥 = 𝐹 ) → ( +∞ ∈ ran 𝑥 ↔ +∞ ∈ ran 𝐹 ) )
6 dmeq ( 𝑥 = 𝐹 → dom 𝑥 = dom 𝐹 )
7 6 adantl ( ( 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ∧ 𝑥 = 𝐹 ) → dom 𝑥 = dom 𝐹 )
8 fdm ( 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) → dom 𝐹 = 𝑋 )
9 8 adantr ( ( 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ∧ 𝑥 = 𝐹 ) → dom 𝐹 = 𝑋 )
10 7 9 eqtrd ( ( 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ∧ 𝑥 = 𝐹 ) → dom 𝑥 = 𝑋 )
11 10 pweqd ( ( 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ∧ 𝑥 = 𝐹 ) → 𝒫 dom 𝑥 = 𝒫 𝑋 )
12 11 ineq1d ( ( 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ∧ 𝑥 = 𝐹 ) → ( 𝒫 dom 𝑥 ∩ Fin ) = ( 𝒫 𝑋 ∩ Fin ) )
13 12 mpteq1d ( ( 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ∧ 𝑥 = 𝐹 ) → ( 𝑦 ∈ ( 𝒫 dom 𝑥 ∩ Fin ) ↦ Σ 𝑤𝑦 ( 𝑥𝑤 ) ) = ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤𝑦 ( 𝑥𝑤 ) ) )
14 13 adantll ( ( ( 𝑋𝑉𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) ∧ 𝑥 = 𝐹 ) → ( 𝑦 ∈ ( 𝒫 dom 𝑥 ∩ Fin ) ↦ Σ 𝑤𝑦 ( 𝑥𝑤 ) ) = ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤𝑦 ( 𝑥𝑤 ) ) )
15 fveq1 ( 𝑥 = 𝐹 → ( 𝑥𝑤 ) = ( 𝐹𝑤 ) )
16 15 sumeq2sdv ( 𝑥 = 𝐹 → Σ 𝑤𝑦 ( 𝑥𝑤 ) = Σ 𝑤𝑦 ( 𝐹𝑤 ) )
17 16 mpteq2dv ( 𝑥 = 𝐹 → ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤𝑦 ( 𝑥𝑤 ) ) = ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤𝑦 ( 𝐹𝑤 ) ) )
18 17 adantl ( ( ( 𝑋𝑉𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) ∧ 𝑥 = 𝐹 ) → ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤𝑦 ( 𝑥𝑤 ) ) = ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤𝑦 ( 𝐹𝑤 ) ) )
19 14 18 eqtrd ( ( ( 𝑋𝑉𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) ∧ 𝑥 = 𝐹 ) → ( 𝑦 ∈ ( 𝒫 dom 𝑥 ∩ Fin ) ↦ Σ 𝑤𝑦 ( 𝑥𝑤 ) ) = ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤𝑦 ( 𝐹𝑤 ) ) )
20 19 rneqd ( ( ( 𝑋𝑉𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) ∧ 𝑥 = 𝐹 ) → ran ( 𝑦 ∈ ( 𝒫 dom 𝑥 ∩ Fin ) ↦ Σ 𝑤𝑦 ( 𝑥𝑤 ) ) = ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤𝑦 ( 𝐹𝑤 ) ) )
21 20 supeq1d ( ( ( 𝑋𝑉𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) ∧ 𝑥 = 𝐹 ) → sup ( ran ( 𝑦 ∈ ( 𝒫 dom 𝑥 ∩ Fin ) ↦ Σ 𝑤𝑦 ( 𝑥𝑤 ) ) , ℝ* , < ) = sup ( ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤𝑦 ( 𝐹𝑤 ) ) , ℝ* , < ) )
22 5 21 ifbieq2d ( ( ( 𝑋𝑉𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) ∧ 𝑥 = 𝐹 ) → if ( +∞ ∈ ran 𝑥 , +∞ , sup ( ran ( 𝑦 ∈ ( 𝒫 dom 𝑥 ∩ Fin ) ↦ Σ 𝑤𝑦 ( 𝑥𝑤 ) ) , ℝ* , < ) ) = if ( +∞ ∈ ran 𝐹 , +∞ , sup ( ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤𝑦 ( 𝐹𝑤 ) ) , ℝ* , < ) ) )
23 simpr ( ( 𝑋𝑉𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) → 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) )
24 simpl ( ( 𝑋𝑉𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) → 𝑋𝑉 )
25 23 24 fexd ( ( 𝑋𝑉𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) → 𝐹 ∈ V )
26 pnfxr +∞ ∈ ℝ*
27 26 a1i ( ( 𝑋𝑉𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) → +∞ ∈ ℝ* )
28 xrltso < Or ℝ*
29 28 supex sup ( ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤𝑦 ( 𝐹𝑤 ) ) , ℝ* , < ) ∈ V
30 29 a1i ( ( 𝑋𝑉𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) → sup ( ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤𝑦 ( 𝐹𝑤 ) ) , ℝ* , < ) ∈ V )
31 27 30 ifexd ( ( 𝑋𝑉𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) → if ( +∞ ∈ ran 𝐹 , +∞ , sup ( ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤𝑦 ( 𝐹𝑤 ) ) , ℝ* , < ) ) ∈ V )
32 2 22 25 31 fvmptd ( ( 𝑋𝑉𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) → ( Σ^𝐹 ) = if ( +∞ ∈ ran 𝐹 , +∞ , sup ( ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤𝑦 ( 𝐹𝑤 ) ) , ℝ* , < ) ) )