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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 7924 | . . . . 5 ⊢ (𝑂 ∈ OutMeas → dom 𝑂 ∈ V) | |
| 5 | uniexg 7761 | . . . . 5 ⊢ (dom 𝑂 ∈ V → ∪ dom 𝑂 ∈ V) | |
| 6 | 1, 4, 5 | 3syl 18 | . . . 4 ⊢ (𝜑 → ∪ dom 𝑂 ∈ V) | 
| 7 | 2, 6 | eqeltrid 2844 | . . 3 ⊢ (𝜑 → 𝑋 ∈ V) | 
| 8 | pwidg 4619 | . . 3 ⊢ (𝑋 ∈ V → 𝑋 ∈ 𝒫 𝑋) | |
| 9 | 7, 8 | syl 17 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑋) | 
| 10 | elpwi 4606 | . . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝑎 ⊆ 𝑋) | |
| 11 | dfss2 3968 | . . . . . . . 8 ⊢ (𝑎 ⊆ 𝑋 ↔ (𝑎 ∩ 𝑋) = 𝑎) | |
| 12 | 11 | biimpi 216 | . . . . . . 7 ⊢ (𝑎 ⊆ 𝑋 → (𝑎 ∩ 𝑋) = 𝑎) | 
| 13 | 10, 12 | syl 17 | . . . . . 6 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∩ 𝑋) = 𝑎) | 
| 14 | 13 | fveq2d 6909 | . . . . 5 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑂‘(𝑎 ∩ 𝑋)) = (𝑂‘𝑎)) | 
| 15 | 14 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∩ 𝑋)) = (𝑂‘𝑎)) | 
| 16 | ssdif0 4365 | . . . . . . . 8 ⊢ (𝑎 ⊆ 𝑋 ↔ (𝑎 ∖ 𝑋) = ∅) | |
| 17 | 10, 16 | sylib 218 | . . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∖ 𝑋) = ∅) | 
| 18 | 17 | fveq2d 6909 | . . . . . 6 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑂‘(𝑎 ∖ 𝑋)) = (𝑂‘∅)) | 
| 19 | 18 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∖ 𝑋)) = (𝑂‘∅)) | 
| 20 | 1 | ome0 46517 | . . . . . 6 ⊢ (𝜑 → (𝑂‘∅) = 0) | 
| 21 | 20 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘∅) = 0) | 
| 22 | 19, 21 | eqtrd 2776 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∖ 𝑋)) = 0) | 
| 23 | 15, 22 | oveq12d 7450 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝑋)) +𝑒 (𝑂‘(𝑎 ∖ 𝑋))) = ((𝑂‘𝑎) +𝑒 0)) | 
| 24 | iccssxr 13471 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
| 25 | 1 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑂 ∈ OutMeas) | 
| 26 | 10 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑎 ⊆ 𝑋) | 
| 27 | 25, 2, 26 | omecl 46523 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘𝑎) ∈ (0[,]+∞)) | 
| 28 | 24, 27 | sselid 3980 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘𝑎) ∈ ℝ*) | 
| 29 | 28 | xaddridd 13286 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘𝑎) +𝑒 0) = (𝑂‘𝑎)) | 
| 30 | eqidd 2737 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘𝑎) = (𝑂‘𝑎)) | |
| 31 | 23, 29, 30 | 3eqtrd 2780 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝑋)) +𝑒 (𝑂‘(𝑎 ∖ 𝑋))) = (𝑂‘𝑎)) | 
| 32 | 1, 2, 3, 9, 31 | carageneld 46522 | 1 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∖ cdif 3947 ∩ cin 3949 ⊆ wss 3950 ∅c0 4332 𝒫 cpw 4599 ∪ cuni 4906 dom cdm 5684 ‘cfv 6560 (class class class)co 7432 0cc0 11156 +∞cpnf 11293 ℝ*cxr 11295 +𝑒 cxad 13153 [,]cicc 13391 OutMeascome 46509 CaraGenccaragen 46511 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-xadd 13156 df-icc 13395 df-ome 46510 df-caragen 46512 | 
| This theorem is referenced by: caragenuni 46531 rrnmbl 46634 | 
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