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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 5155 | . . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ≼ ω ↔ 𝑋 ≼ ω)) | |
4 | disjeq1 5124 | . . . . 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 4923 | . . . . . 6 ⊢ (𝑦 = 𝑋 → ∪ 𝑦 = ∪ 𝑋) | |
7 | 6 | fveq2d 6906 | . . . . 5 ⊢ (𝑦 = 𝑋 → (𝑀‘∪ 𝑦) = (𝑀‘∪ 𝑋)) |
8 | reseq2 5984 | . . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑀 ↾ 𝑦) = (𝑀 ↾ 𝑋)) | |
9 | 8 | fveq2d 6906 | . . . . 5 ⊢ (𝑦 = 𝑋 → (Σ^‘(𝑀 ↾ 𝑦)) = (Σ^‘(𝑀 ↾ 𝑋))) |
10 | 7, 9 | eqeq12d 2744 | . . . 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 45886 | . . . . 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 45887 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
18 | meadjuni.x | . . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) | |
19 | 17, 18 | ssexd 5328 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ V) |
20 | 18, 16 | sseqtrdi 4032 | . . . 4 ⊢ (𝜑 → 𝑋 ⊆ dom 𝑀) |
21 | 19, 20 | elpwd 4612 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝒫 dom 𝑀) |
22 | 11, 15, 21 | rspcdva 3612 | . 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 1533 ∈ wcel 2098 ∀wral 3058 Vcvv 3473 ⊆ wss 3949 ∅c0 4326 𝒫 cpw 4606 ∪ cuni 4912 Disj wdisj 5117 class class class wbr 5152 dom cdm 5682 ↾ cres 5684 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 ωcom 7878 ≼ cdom 8970 0cc0 11148 +∞cpnf 11285 [,]cicc 13369 SAlgcsalg 45743 Σ^csumge0 45797 Meascmea 45884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-disj 5118 df-br 5153 df-opab 5215 df-mpt 5236 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 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-mea 45885 |
This theorem is referenced by: meadjun 45897 meadjiun 45901 |
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