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Mathbox for Glauco Siliprandi |
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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 7363 | . . . . 5 ⊢ (𝑂 ∈ OutMeas → dom 𝑂 ∈ V) | |
5 | uniexg 7220 | . . . . 5 ⊢ (dom 𝑂 ∈ V → ∪ dom 𝑂 ∈ V) | |
6 | 1, 4, 5 | 3syl 18 | . . . 4 ⊢ (𝜑 → ∪ dom 𝑂 ∈ V) |
7 | 2, 6 | syl5eqel 2910 | . . 3 ⊢ (𝜑 → 𝑋 ∈ V) |
8 | pwidg 4395 | . . 3 ⊢ (𝑋 ∈ V → 𝑋 ∈ 𝒫 𝑋) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑋) |
10 | elpwi 4390 | . . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝑎 ⊆ 𝑋) | |
11 | df-ss 3812 | . . . . . . . 8 ⊢ (𝑎 ⊆ 𝑋 ↔ (𝑎 ∩ 𝑋) = 𝑎) | |
12 | 11 | biimpi 208 | . . . . . . 7 ⊢ (𝑎 ⊆ 𝑋 → (𝑎 ∩ 𝑋) = 𝑎) |
13 | 10, 12 | syl 17 | . . . . . 6 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∩ 𝑋) = 𝑎) |
14 | 13 | fveq2d 6441 | . . . . 5 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑂‘(𝑎 ∩ 𝑋)) = (𝑂‘𝑎)) |
15 | 14 | adantl 475 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∩ 𝑋)) = (𝑂‘𝑎)) |
16 | ssdif0 4173 | . . . . . . . 8 ⊢ (𝑎 ⊆ 𝑋 ↔ (𝑎 ∖ 𝑋) = ∅) | |
17 | 10, 16 | sylib 210 | . . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∖ 𝑋) = ∅) |
18 | 17 | fveq2d 6441 | . . . . . 6 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑂‘(𝑎 ∖ 𝑋)) = (𝑂‘∅)) |
19 | 18 | adantl 475 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∖ 𝑋)) = (𝑂‘∅)) |
20 | 1 | ome0 41503 | . . . . . 6 ⊢ (𝜑 → (𝑂‘∅) = 0) |
21 | 20 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘∅) = 0) |
22 | 19, 21 | eqtrd 2861 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∖ 𝑋)) = 0) |
23 | 15, 22 | oveq12d 6928 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝑋)) +𝑒 (𝑂‘(𝑎 ∖ 𝑋))) = ((𝑂‘𝑎) +𝑒 0)) |
24 | iccssxr 12551 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
25 | 1 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑂 ∈ OutMeas) |
26 | 10 | adantl 475 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑎 ⊆ 𝑋) |
27 | 25, 2, 26 | omecl 41509 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘𝑎) ∈ (0[,]+∞)) |
28 | 24, 27 | sseldi 3825 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘𝑎) ∈ ℝ*) |
29 | 28 | xaddid1d 12369 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘𝑎) +𝑒 0) = (𝑂‘𝑎)) |
30 | eqidd 2826 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘𝑎) = (𝑂‘𝑎)) | |
31 | 23, 29, 30 | 3eqtrd 2865 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝑋)) +𝑒 (𝑂‘(𝑎 ∖ 𝑋))) = (𝑂‘𝑎)) |
32 | 1, 2, 3, 9, 31 | carageneld 41508 | 1 ⊢ (𝜑 → 𝑋 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ∖ cdif 3795 ∩ cin 3797 ⊆ wss 3798 ∅c0 4146 𝒫 cpw 4380 ∪ cuni 4660 dom cdm 5346 ‘cfv 6127 (class class class)co 6910 0cc0 10259 +∞cpnf 10395 ℝ*cxr 10397 +𝑒 cxad 12237 [,]cicc 12473 OutMeascome 41495 CaraGenccaragen 41497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-po 5265 df-so 5266 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-1st 7433 df-2nd 7434 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-xadd 12240 df-icc 12477 df-ome 41496 df-caragen 41498 |
This theorem is referenced by: caragenuni 41517 rrnmbl 41620 |
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