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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 | . . 3 ⊢ (𝜑 → 𝑋 ≼ ω) | |
2 | meadjuni.dj | . . 3 ⊢ (𝜑 → Disj 𝑥 ∈ 𝑋 𝑥) | |
3 | 1, 2 | jca 507 | . 2 ⊢ (𝜑 → (𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥)) |
4 | meadjuni.x | . . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) | |
5 | meadjuni.s | . . . . 5 ⊢ 𝑆 = dom 𝑀 | |
6 | 4, 5 | syl6sseq 3870 | . . . 4 ⊢ (𝜑 → 𝑋 ⊆ dom 𝑀) |
7 | meadjuni.m | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ Meas) | |
8 | 7, 5 | dmmeasal 41607 | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
9 | 8, 4 | ssexd 5044 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ V) |
10 | elpwg 4387 | . . . . 5 ⊢ (𝑋 ∈ V → (𝑋 ∈ 𝒫 dom 𝑀 ↔ 𝑋 ⊆ dom 𝑀)) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ 𝒫 dom 𝑀 ↔ 𝑋 ⊆ dom 𝑀)) |
12 | 6, 11 | mpbird 249 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝒫 dom 𝑀) |
13 | ismea 41606 | . . . . 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 | 7, 13 | sylib 210 | . . . 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 491 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 dom 𝑀((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = (Σ^‘(𝑀 ↾ 𝑦)))) |
16 | breq1 4891 | . . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑦 ≼ ω ↔ 𝑋 ≼ ω)) | |
17 | disjeq1 4863 | . . . . . 6 ⊢ (𝑦 = 𝑋 → (Disj 𝑥 ∈ 𝑦 𝑥 ↔ Disj 𝑥 ∈ 𝑋 𝑥)) | |
18 | 16, 17 | anbi12d 624 | . . . . 5 ⊢ (𝑦 = 𝑋 → ((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) ↔ (𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥))) |
19 | unieq 4681 | . . . . . . 7 ⊢ (𝑦 = 𝑋 → ∪ 𝑦 = ∪ 𝑋) | |
20 | 19 | fveq2d 6452 | . . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑀‘∪ 𝑦) = (𝑀‘∪ 𝑋)) |
21 | reseq2 5639 | . . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝑀 ↾ 𝑦) = (𝑀 ↾ 𝑋)) | |
22 | 21 | fveq2d 6452 | . . . . . 6 ⊢ (𝑦 = 𝑋 → (Σ^‘(𝑀 ↾ 𝑦)) = (Σ^‘(𝑀 ↾ 𝑋))) |
23 | 20, 22 | eqeq12d 2793 | . . . . 5 ⊢ (𝑦 = 𝑋 → ((𝑀‘∪ 𝑦) = (Σ^‘(𝑀 ↾ 𝑦)) ↔ (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋)))) |
24 | 18, 23 | imbi12d 336 | . . . 4 ⊢ (𝑦 = 𝑋 → (((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = (Σ^‘(𝑀 ↾ 𝑦))) ↔ ((𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥) → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋))))) |
25 | 24 | rspcva 3509 | . . 3 ⊢ ((𝑋 ∈ 𝒫 dom 𝑀 ∧ ∀𝑦 ∈ 𝒫 dom 𝑀((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = (Σ^‘(𝑀 ↾ 𝑦)))) → ((𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥) → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋)))) |
26 | 12, 15, 25 | syl2anc 579 | . 2 ⊢ (𝜑 → ((𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥) → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋)))) |
27 | 3, 26 | mpd 15 | 1 ⊢ (𝜑 → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∀wral 3090 Vcvv 3398 ⊆ wss 3792 ∅c0 4141 𝒫 cpw 4379 ∪ cuni 4673 Disj wdisj 4856 class class class wbr 4888 dom cdm 5357 ↾ cres 5359 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 ωcom 7345 ≼ cdom 8241 0cc0 10274 +∞cpnf 10410 [,]cicc 12495 SAlgcsalg 41466 Σ^csumge0 41517 Meascmea 41604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-disj 4857 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-mea 41605 |
This theorem is referenced by: meadjun 41617 meadjiun 41621 |
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