Metamath Proof Explorer


Theorem meadjuni

Description: The measure of the disjoint union of a countable set is the extended sum of the measures. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Hypotheses meadjuni.m ( 𝜑𝑀 ∈ Meas )
meadjuni.s 𝑆 = dom 𝑀
meadjuni.x ( 𝜑𝑋𝑆 )
meadjuni.cnb ( 𝜑𝑋 ≼ ω )
meadjuni.dj ( 𝜑Disj 𝑥𝑋 𝑥 )
Assertion meadjuni ( 𝜑 → ( 𝑀 𝑋 ) = ( Σ^ ‘ ( 𝑀𝑋 ) ) )

Proof

Step Hyp Ref Expression
1 meadjuni.m ( 𝜑𝑀 ∈ Meas )
2 meadjuni.s 𝑆 = dom 𝑀
3 meadjuni.x ( 𝜑𝑋𝑆 )
4 meadjuni.cnb ( 𝜑𝑋 ≼ ω )
5 meadjuni.dj ( 𝜑Disj 𝑥𝑋 𝑥 )
6 breq1 ( 𝑦 = 𝑋 → ( 𝑦 ≼ ω ↔ 𝑋 ≼ ω ) )
7 disjeq1 ( 𝑦 = 𝑋 → ( Disj 𝑥𝑦 𝑥Disj 𝑥𝑋 𝑥 ) )
8 6 7 anbi12d ( 𝑦 = 𝑋 → ( ( 𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥 ) ↔ ( 𝑋 ≼ ω ∧ Disj 𝑥𝑋 𝑥 ) ) )
9 unieq ( 𝑦 = 𝑋 𝑦 = 𝑋 )
10 9 fveq2d ( 𝑦 = 𝑋 → ( 𝑀 𝑦 ) = ( 𝑀 𝑋 ) )
11 reseq2 ( 𝑦 = 𝑋 → ( 𝑀𝑦 ) = ( 𝑀𝑋 ) )
12 11 fveq2d ( 𝑦 = 𝑋 → ( Σ^ ‘ ( 𝑀𝑦 ) ) = ( Σ^ ‘ ( 𝑀𝑋 ) ) )
13 10 12 eqeq12d ( 𝑦 = 𝑋 → ( ( 𝑀 𝑦 ) = ( Σ^ ‘ ( 𝑀𝑦 ) ) ↔ ( 𝑀 𝑋 ) = ( Σ^ ‘ ( 𝑀𝑋 ) ) ) )
14 8 13 imbi12d ( 𝑦 = 𝑋 → ( ( ( 𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥 ) → ( 𝑀 𝑦 ) = ( Σ^ ‘ ( 𝑀𝑦 ) ) ) ↔ ( ( 𝑋 ≼ ω ∧ Disj 𝑥𝑋 𝑥 ) → ( 𝑀 𝑋 ) = ( Σ^ ‘ ( 𝑀𝑋 ) ) ) ) )
15 ismea ( 𝑀 ∈ Meas ↔ ( ( ( 𝑀 : dom 𝑀 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑀 ∈ SAlg ) ∧ ( 𝑀 ‘ ∅ ) = 0 ) ∧ ∀ 𝑦 ∈ 𝒫 dom 𝑀 ( ( 𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥 ) → ( 𝑀 𝑦 ) = ( Σ^ ‘ ( 𝑀𝑦 ) ) ) ) )
16 1 15 sylib ( 𝜑 → ( ( ( 𝑀 : dom 𝑀 ⟶ ( 0 [,] +∞ ) ∧ dom 𝑀 ∈ SAlg ) ∧ ( 𝑀 ‘ ∅ ) = 0 ) ∧ ∀ 𝑦 ∈ 𝒫 dom 𝑀 ( ( 𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥 ) → ( 𝑀 𝑦 ) = ( Σ^ ‘ ( 𝑀𝑦 ) ) ) ) )
17 16 simprd ( 𝜑 → ∀ 𝑦 ∈ 𝒫 dom 𝑀 ( ( 𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥 ) → ( 𝑀 𝑦 ) = ( Σ^ ‘ ( 𝑀𝑦 ) ) ) )
18 1 2 dmmeasal ( 𝜑𝑆 ∈ SAlg )
19 18 3 ssexd ( 𝜑𝑋 ∈ V )
20 3 2 sseqtrdi ( 𝜑𝑋 ⊆ dom 𝑀 )
21 19 20 elpwd ( 𝜑𝑋 ∈ 𝒫 dom 𝑀 )
22 14 17 21 rspcdva ( 𝜑 → ( ( 𝑋 ≼ ω ∧ Disj 𝑥𝑋 𝑥 ) → ( 𝑀 𝑋 ) = ( Σ^ ‘ ( 𝑀𝑋 ) ) ) )
23 4 5 22 mp2and ( 𝜑 → ( 𝑀 𝑋 ) = ( Σ^ ‘ ( 𝑀𝑋 ) ) )