| 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 5146 | . . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ≼ ω ↔ 𝑋 ≼ ω)) | |
| 4 | disjeq1 5117 | . . . . 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 4918 | . . . . . 6 ⊢ (𝑦 = 𝑋 → ∪ 𝑦 = ∪ 𝑋) | |
| 7 | 6 | fveq2d 6910 | . . . . 5 ⊢ (𝑦 = 𝑋 → (𝑀‘∪ 𝑦) = (𝑀‘∪ 𝑋)) |
| 8 | reseq2 5992 | . . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑀 ↾ 𝑦) = (𝑀 ↾ 𝑋)) | |
| 9 | 8 | fveq2d 6910 | . . . . 5 ⊢ (𝑦 = 𝑋 → (Σ^‘(𝑀 ↾ 𝑦)) = (Σ^‘(𝑀 ↾ 𝑋))) |
| 10 | 7, 9 | eqeq12d 2753 | . . . 4 ⊢ (𝑦 = 𝑋 → ((𝑀‘∪ 𝑦) = (Σ^‘(𝑀 ↾ 𝑦)) ↔ (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋)))) |
| 11 | 5, 10 | imbi12d 344 | . . 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 46466 | . . . . 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 218 | . . . 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 495 | . . 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 46467 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
| 18 | meadjuni.x | . . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) | |
| 19 | 17, 18 | ssexd 5324 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ V) |
| 20 | 18, 16 | sseqtrdi 4024 | . . . 4 ⊢ (𝜑 → 𝑋 ⊆ dom 𝑀) |
| 21 | 19, 20 | elpwd 4606 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝒫 dom 𝑀) |
| 22 | 11, 15, 21 | rspcdva 3623 | . 2 ⊢ (𝜑 → ((𝑋 ≼ ω ∧ Disj 𝑥 ∈ 𝑋 𝑥) → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋)))) |
| 23 | 1, 2, 22 | mp2and 699 | 1 ⊢ (𝜑 → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 ∪ cuni 4907 Disj wdisj 5110 class class class wbr 5143 dom cdm 5685 ↾ cres 5687 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ωcom 7887 ≼ cdom 8983 0cc0 11155 +∞cpnf 11292 [,]cicc 13390 SAlgcsalg 46323 Σ^csumge0 46377 Meascmea 46464 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-mea 46465 |
| This theorem is referenced by: meadjun 46477 meadjiun 46481 |
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