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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 7594 | . . . . 5 ⊢ (𝑂 ∈ OutMeas → dom 𝑂 ∈ V) | |
5 | uniexg 7446 | . . . . 5 ⊢ (dom 𝑂 ∈ V → ∪ dom 𝑂 ∈ V) | |
6 | 1, 4, 5 | 3syl 18 | . . . 4 ⊢ (𝜑 → ∪ dom 𝑂 ∈ V) |
7 | 2, 6 | eqeltrid 2894 | . . 3 ⊢ (𝜑 → 𝑋 ∈ V) |
8 | pwidg 4519 | . . 3 ⊢ (𝑋 ∈ V → 𝑋 ∈ 𝒫 𝑋) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑋) |
10 | elpwi 4506 | . . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝑎 ⊆ 𝑋) | |
11 | df-ss 3898 | . . . . . . . 8 ⊢ (𝑎 ⊆ 𝑋 ↔ (𝑎 ∩ 𝑋) = 𝑎) | |
12 | 11 | biimpi 219 | . . . . . . 7 ⊢ (𝑎 ⊆ 𝑋 → (𝑎 ∩ 𝑋) = 𝑎) |
13 | 10, 12 | syl 17 | . . . . . 6 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∩ 𝑋) = 𝑎) |
14 | 13 | fveq2d 6649 | . . . . 5 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑂‘(𝑎 ∩ 𝑋)) = (𝑂‘𝑎)) |
15 | 14 | adantl 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∩ 𝑋)) = (𝑂‘𝑎)) |
16 | ssdif0 4277 | . . . . . . . 8 ⊢ (𝑎 ⊆ 𝑋 ↔ (𝑎 ∖ 𝑋) = ∅) | |
17 | 10, 16 | sylib 221 | . . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∖ 𝑋) = ∅) |
18 | 17 | fveq2d 6649 | . . . . . 6 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑂‘(𝑎 ∖ 𝑋)) = (𝑂‘∅)) |
19 | 18 | adantl 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∖ 𝑋)) = (𝑂‘∅)) |
20 | 1 | ome0 43136 | . . . . . 6 ⊢ (𝜑 → (𝑂‘∅) = 0) |
21 | 20 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘∅) = 0) |
22 | 19, 21 | eqtrd 2833 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∖ 𝑋)) = 0) |
23 | 15, 22 | oveq12d 7153 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝑋)) +𝑒 (𝑂‘(𝑎 ∖ 𝑋))) = ((𝑂‘𝑎) +𝑒 0)) |
24 | iccssxr 12808 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
25 | 1 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑂 ∈ OutMeas) |
26 | 10 | adantl 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑎 ⊆ 𝑋) |
27 | 25, 2, 26 | omecl 43142 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘𝑎) ∈ (0[,]+∞)) |
28 | 24, 27 | sseldi 3913 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘𝑎) ∈ ℝ*) |
29 | 28 | xaddid1d 12624 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘𝑎) +𝑒 0) = (𝑂‘𝑎)) |
30 | eqidd 2799 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘𝑎) = (𝑂‘𝑎)) | |
31 | 23, 29, 30 | 3eqtrd 2837 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝑋)) +𝑒 (𝑂‘(𝑎 ∖ 𝑋))) = (𝑂‘𝑎)) |
32 | 1, 2, 3, 9, 31 | carageneld 43141 | 1 ⊢ (𝜑 → 𝑋 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∖ cdif 3878 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 𝒫 cpw 4497 ∪ cuni 4800 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 0cc0 10526 +∞cpnf 10661 ℝ*cxr 10663 +𝑒 cxad 12493 [,]cicc 12729 OutMeascome 43128 CaraGenccaragen 43130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-xadd 12496 df-icc 12733 df-ome 43129 df-caragen 43131 |
This theorem is referenced by: caragenuni 43150 rrnmbl 43253 |
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