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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 6989 | . . . . 5 ⊢ (𝑂 ∈ OutMeas → dom 𝑂 ∈ V) | |
5 | uniexg 6853 | . . . . 5 ⊢ (dom 𝑂 ∈ V → ∪ dom 𝑂 ∈ V) | |
6 | 1, 4, 5 | 3syl 18 | . . . 4 ⊢ (𝜑 → ∪ dom 𝑂 ∈ V) |
7 | 2, 6 | syl5eqel 2692 | . . 3 ⊢ (𝜑 → 𝑋 ∈ V) |
8 | pwidg 4121 | . . 3 ⊢ (𝑋 ∈ V → 𝑋 ∈ 𝒫 𝑋) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑋) |
10 | elpwi 4117 | . . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝑎 ⊆ 𝑋) | |
11 | df-ss 3554 | . . . . . . . 8 ⊢ (𝑎 ⊆ 𝑋 ↔ (𝑎 ∩ 𝑋) = 𝑎) | |
12 | 11 | biimpi 205 | . . . . . . 7 ⊢ (𝑎 ⊆ 𝑋 → (𝑎 ∩ 𝑋) = 𝑎) |
13 | 10, 12 | syl 17 | . . . . . 6 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∩ 𝑋) = 𝑎) |
14 | 13 | fveq2d 6107 | . . . . 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 3896 | . . . . . . . 8 ⊢ (𝑎 ⊆ 𝑋 ↔ (𝑎 ∖ 𝑋) = ∅) | |
17 | 10, 16 | sylib 207 | . . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∖ 𝑋) = ∅) |
18 | 17 | fveq2d 6107 | . . . . . 6 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑂‘(𝑎 ∖ 𝑋)) = (𝑂‘∅)) |
19 | 18 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∖ 𝑋)) = (𝑂‘∅)) |
20 | 1 | ome0 39387 | . . . . . 6 ⊢ (𝜑 → (𝑂‘∅) = 0) |
21 | 20 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘∅) = 0) |
22 | 19, 21 | eqtrd 2644 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∖ 𝑋)) = 0) |
23 | 15, 22 | oveq12d 6567 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝑋)) +𝑒 (𝑂‘(𝑎 ∖ 𝑋))) = ((𝑂‘𝑎) +𝑒 0)) |
24 | iccssxr 12127 | . . . . 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 39393 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘𝑎) ∈ (0[,]+∞)) |
28 | 24, 27 | sseldi 3566 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘𝑎) ∈ ℝ*) |
29 | 28 | xaddid1d 11948 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘𝑎) +𝑒 0) = (𝑂‘𝑎)) |
30 | eqidd 2611 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘𝑎) = (𝑂‘𝑎)) | |
31 | 23, 29, 30 | 3eqtrd 2648 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝑋)) +𝑒 (𝑂‘(𝑎 ∖ 𝑋))) = (𝑂‘𝑎)) |
32 | 1, 2, 3, 9, 31 | carageneld 39392 | 1 ⊢ (𝜑 → 𝑋 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 ∪ cuni 4372 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 0cc0 9815 +∞cpnf 9950 ℝ*cxr 9952 +𝑒 cxad 11820 [,]cicc 12049 OutMeascome 39379 CaraGenccaragen 39381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-xadd 11823 df-icc 12053 df-ome 39380 df-caragen 39382 |
This theorem is referenced by: caragenuni 39401 rrnmbl 39504 |
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