Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > meadjuni | Structured version Visualization version GIF version |
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 |
---|---|
meadjuni.m | ⊢ (𝜑 → 𝑀 ∈ Meas) |
meadjuni.s | ⊢ 𝑆 = dom 𝑀 |
meadjuni.x | ⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
meadjuni.cnb | ⊢ (𝜑 → 𝑋 ≼ ω) |
meadjuni.dj | ⊢ (𝜑 → Disj 𝑥 ∈ 𝑋 𝑥) |
Ref | Expression |
---|---|
meadjuni | ⊢ (𝜑 → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | meadjuni.cnb | . 2 ⊢ (𝜑 → 𝑋 ≼ ω) | |
2 | meadjuni.dj | . 2 ⊢ (𝜑 → Disj 𝑥 ∈ 𝑋 𝑥) | |
3 | breq1 5061 | . . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ≼ ω ↔ 𝑋 ≼ ω)) | |
4 | disjeq1 5030 | . . . . 5 ⊢ (𝑦 = 𝑋 → (Disj 𝑥 ∈ 𝑦 𝑥 ↔ Disj 𝑥 ∈ 𝑋 𝑥)) | |
5 | 3, 4 | anbi12d 632 | . . . 4 ⊢ (𝑦 = 𝑋 → ((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) ↔ (𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥))) |
6 | unieq 4839 | . . . . . 6 ⊢ (𝑦 = 𝑋 → ∪ 𝑦 = ∪ 𝑋) | |
7 | 6 | fveq2d 6668 | . . . . 5 ⊢ (𝑦 = 𝑋 → (𝑀‘∪ 𝑦) = (𝑀‘∪ 𝑋)) |
8 | reseq2 5842 | . . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑀 ↾ 𝑦) = (𝑀 ↾ 𝑋)) | |
9 | 8 | fveq2d 6668 | . . . . 5 ⊢ (𝑦 = 𝑋 → (Σ^‘(𝑀 ↾ 𝑦)) = (Σ^‘(𝑀 ↾ 𝑋))) |
10 | 7, 9 | eqeq12d 2837 | . . . 4 ⊢ (𝑦 = 𝑋 → ((𝑀‘∪ 𝑦) = (Σ^‘(𝑀 ↾ 𝑦)) ↔ (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋)))) |
11 | 5, 10 | imbi12d 347 | . . 3 ⊢ (𝑦 = 𝑋 → (((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = (Σ^‘(𝑀 ↾ 𝑦))) ↔ ((𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥) → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋))))) |
12 | meadjuni.m | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ Meas) | |
13 | ismea 42727 | . . . . 5 ⊢ (𝑀 ∈ Meas ↔ (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑀((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = (Σ^‘(𝑀 ↾ 𝑦))))) | |
14 | 12, 13 | sylib 220 | . . . 4 ⊢ (𝜑 → (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑀((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = (Σ^‘(𝑀 ↾ 𝑦))))) |
15 | 14 | simprd 498 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 dom 𝑀((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = (Σ^‘(𝑀 ↾ 𝑦)))) |
16 | meadjuni.s | . . . . . 6 ⊢ 𝑆 = dom 𝑀 | |
17 | 12, 16 | dmmeasal 42728 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
18 | meadjuni.x | . . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) | |
19 | 17, 18 | ssexd 5220 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ V) |
20 | 18, 16 | sseqtrdi 4016 | . . . 4 ⊢ (𝜑 → 𝑋 ⊆ dom 𝑀) |
21 | 19, 20 | elpwd 4549 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝒫 dom 𝑀) |
22 | 11, 15, 21 | rspcdva 3624 | . 2 ⊢ (𝜑 → ((𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥) → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋)))) |
23 | 1, 2, 22 | mp2and 697 | 1 ⊢ (𝜑 → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ⊆ wss 3935 ∅c0 4290 𝒫 cpw 4538 ∪ cuni 4831 Disj wdisj 5023 class class class wbr 5058 dom cdm 5549 ↾ cres 5551 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ωcom 7574 ≼ cdom 8501 0cc0 10531 +∞cpnf 10666 [,]cicc 12735 SAlgcsalg 42587 Σ^csumge0 42638 Meascmea 42725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-disj 5024 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-mea 42726 |
This theorem is referenced by: meadjun 42738 meadjiun 42742 |
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