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Mirrors > Home > MPE Home > Th. List > Mathboxes > caragencmpl | Structured version Visualization version GIF version |
Description: A measure built with the Caratheodory's construction is complete. See Definition 112Df of [Fremlin1] p. 19. This is Exercise 113Xa of [Fremlin1] p. 21 (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
caragencmpl.o | ⊢ (𝜑 → 𝑂 ∈ OutMeas) |
caragencmpl.x | ⊢ 𝑋 = ∪ dom 𝑂 |
caragencmpl.e | ⊢ (𝜑 → 𝐸 ⊆ 𝑋) |
caragencmpl.z | ⊢ (𝜑 → (𝑂‘𝐸) = 0) |
caragencmpl.s | ⊢ 𝑆 = (CaraGen‘𝑂) |
Ref | Expression |
---|---|
caragencmpl | ⊢ (𝜑 → 𝐸 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caragencmpl.o | . 2 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
2 | caragencmpl.x | . 2 ⊢ 𝑋 = ∪ dom 𝑂 | |
3 | caragencmpl.s | . 2 ⊢ 𝑆 = (CaraGen‘𝑂) | |
4 | caragencmpl.e | . . 3 ⊢ (𝜑 → 𝐸 ⊆ 𝑋) | |
5 | 1, 2 | unidmex 38242 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ V) |
6 | 5, 4 | ssexd 4733 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ V) |
7 | elpwg 4116 | . . . 4 ⊢ (𝐸 ∈ V → (𝐸 ∈ 𝒫 𝑋 ↔ 𝐸 ⊆ 𝑋)) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝜑 → (𝐸 ∈ 𝒫 𝑋 ↔ 𝐸 ⊆ 𝑋)) |
9 | 4, 8 | mpbird 246 | . 2 ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑋) |
10 | 1 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑂 ∈ OutMeas) |
11 | 4 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → 𝐸 ⊆ 𝑋) |
12 | caragencmpl.z | . . . . . . 7 ⊢ (𝜑 → (𝑂‘𝐸) = 0) | |
13 | 12 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘𝐸) = 0) |
14 | inss2 3796 | . . . . . . 7 ⊢ (𝑎 ∩ 𝐸) ⊆ 𝐸 | |
15 | 14 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑎 ∩ 𝐸) ⊆ 𝐸) |
16 | 10, 2, 11, 13, 15 | omess0 39424 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∩ 𝐸)) = 0) |
17 | 16 | oveq1d 6564 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (0 +𝑒 (𝑂‘(𝑎 ∖ 𝐸)))) |
18 | difssd 3700 | . . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∖ 𝐸) ⊆ 𝑎) | |
19 | elpwi 4117 | . . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝑎 ⊆ 𝑋) | |
20 | 18, 19 | sstrd 3578 | . . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∖ 𝐸) ⊆ 𝑋) |
21 | 20 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑎 ∖ 𝐸) ⊆ 𝑋) |
22 | 10, 2, 21 | omexrcl 39397 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∖ 𝐸)) ∈ ℝ*) |
23 | xaddid2 11947 | . . . . 5 ⊢ ((𝑂‘(𝑎 ∖ 𝐸)) ∈ ℝ* → (0 +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘(𝑎 ∖ 𝐸))) | |
24 | 22, 23 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (0 +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘(𝑎 ∖ 𝐸))) |
25 | 17, 24 | eqtrd 2644 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘(𝑎 ∖ 𝐸))) |
26 | 19 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑎 ⊆ 𝑋) |
27 | 18 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑎 ∖ 𝐸) ⊆ 𝑎) |
28 | 10, 2, 26, 27 | omessle 39388 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∖ 𝐸)) ≤ (𝑂‘𝑎)) |
29 | 25, 28 | eqbrtrd 4605 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) ≤ (𝑂‘𝑎)) |
30 | 1, 2, 3, 9, 29 | caragenel2d 39422 | 1 ⊢ (𝜑 → 𝐸 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 0cc0 9815 ℝ*cxr 9952 ≤ cle 9954 +𝑒 cxad 11820 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-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 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 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 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-reu 2903 df-rmo 2904 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-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 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-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 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-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-xadd 11823 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-sumge0 39256 df-ome 39380 df-caragen 39382 |
This theorem is referenced by: voncmpl 39511 |
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