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Mirrors > Home > MPE Home > Th. List > Mathboxes > caragenunidm | Structured version Visualization version GIF version |
Description: The base set of an outer measure belongs to the sigma-algebra generated by the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
caragenunidm.o | ⊢ (𝜑 → 𝑂 ∈ OutMeas) |
caragenunidm.x | ⊢ 𝑋 = ∪ dom 𝑂 |
caragenunidm.s | ⊢ 𝑆 = (CaraGen‘𝑂) |
Ref | Expression |
---|---|
caragenunidm | ⊢ (𝜑 → 𝑋 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caragenunidm.o | . 2 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
2 | caragenunidm.x | . 2 ⊢ 𝑋 = ∪ dom 𝑂 | |
3 | caragenunidm.s | . 2 ⊢ 𝑆 = (CaraGen‘𝑂) | |
4 | dmexg 7607 | . . . . 5 ⊢ (𝑂 ∈ OutMeas → dom 𝑂 ∈ V) | |
5 | uniexg 7460 | . . . . 5 ⊢ (dom 𝑂 ∈ V → ∪ dom 𝑂 ∈ V) | |
6 | 1, 4, 5 | 3syl 18 | . . . 4 ⊢ (𝜑 → ∪ dom 𝑂 ∈ V) |
7 | 2, 6 | eqeltrid 2917 | . . 3 ⊢ (𝜑 → 𝑋 ∈ V) |
8 | pwidg 4555 | . . 3 ⊢ (𝑋 ∈ V → 𝑋 ∈ 𝒫 𝑋) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑋) |
10 | elpwi 4550 | . . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝑎 ⊆ 𝑋) | |
11 | df-ss 3951 | . . . . . . . 8 ⊢ (𝑎 ⊆ 𝑋 ↔ (𝑎 ∩ 𝑋) = 𝑎) | |
12 | 11 | biimpi 218 | . . . . . . 7 ⊢ (𝑎 ⊆ 𝑋 → (𝑎 ∩ 𝑋) = 𝑎) |
13 | 10, 12 | syl 17 | . . . . . 6 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∩ 𝑋) = 𝑎) |
14 | 13 | fveq2d 6668 | . . . . 5 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑂‘(𝑎 ∩ 𝑋)) = (𝑂‘𝑎)) |
15 | 14 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∩ 𝑋)) = (𝑂‘𝑎)) |
16 | ssdif0 4322 | . . . . . . . 8 ⊢ (𝑎 ⊆ 𝑋 ↔ (𝑎 ∖ 𝑋) = ∅) | |
17 | 10, 16 | sylib 220 | . . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∖ 𝑋) = ∅) |
18 | 17 | fveq2d 6668 | . . . . . 6 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑂‘(𝑎 ∖ 𝑋)) = (𝑂‘∅)) |
19 | 18 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∖ 𝑋)) = (𝑂‘∅)) |
20 | 1 | ome0 42773 | . . . . . 6 ⊢ (𝜑 → (𝑂‘∅) = 0) |
21 | 20 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘∅) = 0) |
22 | 19, 21 | eqtrd 2856 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∖ 𝑋)) = 0) |
23 | 15, 22 | oveq12d 7168 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝑋)) +𝑒 (𝑂‘(𝑎 ∖ 𝑋))) = ((𝑂‘𝑎) +𝑒 0)) |
24 | iccssxr 12813 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
25 | 1 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑂 ∈ OutMeas) |
26 | 10 | adantl 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑎 ⊆ 𝑋) |
27 | 25, 2, 26 | omecl 42779 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘𝑎) ∈ (0[,]+∞)) |
28 | 24, 27 | sseldi 3964 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘𝑎) ∈ ℝ*) |
29 | 28 | xaddid1d 12630 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘𝑎) +𝑒 0) = (𝑂‘𝑎)) |
30 | eqidd 2822 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘𝑎) = (𝑂‘𝑎)) | |
31 | 23, 29, 30 | 3eqtrd 2860 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝑋)) +𝑒 (𝑂‘(𝑎 ∖ 𝑋))) = (𝑂‘𝑎)) |
32 | 1, 2, 3, 9, 31 | carageneld 42778 | 1 ⊢ (𝜑 → 𝑋 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∖ cdif 3932 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 𝒫 cpw 4538 ∪ cuni 4831 dom cdm 5549 ‘cfv 6349 (class class class)co 7150 0cc0 10531 +∞cpnf 10666 ℝ*cxr 10668 +𝑒 cxad 12499 [,]cicc 12735 OutMeascome 42765 CaraGenccaragen 42767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-xadd 12502 df-icc 12739 df-ome 42766 df-caragen 42768 |
This theorem is referenced by: caragenuni 42787 rrnmbl 42890 |
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