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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 5156 | . . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ≼ ω ↔ 𝑋 ≼ ω)) | |
4 | disjeq1 5125 | . . . . 5 ⊢ (𝑦 = 𝑋 → (Disj 𝑥 ∈ 𝑦 𝑥 ↔ Disj 𝑥 ∈ 𝑋 𝑥)) | |
5 | 3, 4 | anbi12d 630 | . . . 4 ⊢ (𝑦 = 𝑋 → ((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) ↔ (𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥))) |
6 | unieq 4924 | . . . . . 6 ⊢ (𝑦 = 𝑋 → ∪ 𝑦 = ∪ 𝑋) | |
7 | 6 | fveq2d 6905 | . . . . 5 ⊢ (𝑦 = 𝑋 → (𝑀‘∪ 𝑦) = (𝑀‘∪ 𝑋)) |
8 | reseq2 5984 | . . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑀 ↾ 𝑦) = (𝑀 ↾ 𝑋)) | |
9 | 8 | fveq2d 6905 | . . . . 5 ⊢ (𝑦 = 𝑋 → (Σ^‘(𝑀 ↾ 𝑦)) = (Σ^‘(𝑀 ↾ 𝑋))) |
10 | 7, 9 | eqeq12d 2742 | . . . 4 ⊢ (𝑦 = 𝑋 → ((𝑀‘∪ 𝑦) = (Σ^‘(𝑀 ↾ 𝑦)) ↔ (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋)))) |
11 | 5, 10 | imbi12d 343 | . . 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 46072 | . . . . 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 217 | . . . 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 494 | . . 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 46073 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
18 | meadjuni.x | . . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) | |
19 | 17, 18 | ssexd 5329 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ V) |
20 | 18, 16 | sseqtrdi 4030 | . . . 4 ⊢ (𝜑 → 𝑋 ⊆ dom 𝑀) |
21 | 19, 20 | elpwd 4613 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝒫 dom 𝑀) |
22 | 11, 15, 21 | rspcdva 3609 | . 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 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 Vcvv 3462 ⊆ wss 3947 ∅c0 4325 𝒫 cpw 4607 ∪ cuni 4913 Disj wdisj 5118 class class class wbr 5153 dom cdm 5682 ↾ cres 5684 ⟶wf 6550 ‘cfv 6554 (class class class)co 7424 ωcom 7876 ≼ cdom 8972 0cc0 11158 +∞cpnf 11295 [,]cicc 13381 SAlgcsalg 45929 Σ^csumge0 45983 Meascmea 46070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-disj 5119 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-mea 46071 |
This theorem is referenced by: meadjun 46083 meadjiun 46087 |
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