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 5033 | . . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ≼ ω ↔ 𝑋 ≼ ω)) | |
4 | disjeq1 5002 | . . . . 5 ⊢ (𝑦 = 𝑋 → (Disj 𝑥 ∈ 𝑦 𝑥 ↔ Disj 𝑥 ∈ 𝑋 𝑥)) | |
5 | 3, 4 | anbi12d 633 | . . . 4 ⊢ (𝑦 = 𝑋 → ((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) ↔ (𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥))) |
6 | unieq 4811 | . . . . . 6 ⊢ (𝑦 = 𝑋 → ∪ 𝑦 = ∪ 𝑋) | |
7 | 6 | fveq2d 6649 | . . . . 5 ⊢ (𝑦 = 𝑋 → (𝑀‘∪ 𝑦) = (𝑀‘∪ 𝑋)) |
8 | reseq2 5813 | . . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑀 ↾ 𝑦) = (𝑀 ↾ 𝑋)) | |
9 | 8 | fveq2d 6649 | . . . . 5 ⊢ (𝑦 = 𝑋 → (Σ^‘(𝑀 ↾ 𝑦)) = (Σ^‘(𝑀 ↾ 𝑋))) |
10 | 7, 9 | eqeq12d 2814 | . . . 4 ⊢ (𝑦 = 𝑋 → ((𝑀‘∪ 𝑦) = (Σ^‘(𝑀 ↾ 𝑦)) ↔ (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋)))) |
11 | 5, 10 | imbi12d 348 | . . 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 43090 | . . . . 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 221 | . . . 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 499 | . . 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 43091 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
18 | meadjuni.x | . . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) | |
19 | 17, 18 | ssexd 5192 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ V) |
20 | 18, 16 | sseqtrdi 3965 | . . . 4 ⊢ (𝜑 → 𝑋 ⊆ dom 𝑀) |
21 | 19, 20 | elpwd 4505 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝒫 dom 𝑀) |
22 | 11, 15, 21 | rspcdva 3573 | . 2 ⊢ (𝜑 → ((𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥) → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋)))) |
23 | 1, 2, 22 | mp2and 698 | 1 ⊢ (𝜑 → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ⊆ wss 3881 ∅c0 4243 𝒫 cpw 4497 ∪ cuni 4800 Disj wdisj 4995 class class class wbr 5030 dom cdm 5519 ↾ cres 5521 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ωcom 7560 ≼ cdom 8490 0cc0 10526 +∞cpnf 10661 [,]cicc 12729 SAlgcsalg 42950 Σ^csumge0 43001 Meascmea 43088 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-disj 4996 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-mea 43089 |
This theorem is referenced by: meadjun 43101 meadjiun 43105 |
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