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Mathbox for Glauco Siliprandi |
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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 40851 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ V) |
6 | 5, 4 | ssexd 5119 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ V) |
7 | elpwg 4461 | . . . 4 ⊢ (𝐸 ∈ V → (𝐸 ∈ 𝒫 𝑋 ↔ 𝐸 ⊆ 𝑋)) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝜑 → (𝐸 ∈ 𝒫 𝑋 ↔ 𝐸 ⊆ 𝑋)) |
9 | 4, 8 | mpbird 258 | . 2 ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑋) |
10 | 1 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑂 ∈ OutMeas) |
11 | 4 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → 𝐸 ⊆ 𝑋) |
12 | caragencmpl.z | . . . . . . 7 ⊢ (𝜑 → (𝑂‘𝐸) = 0) | |
13 | 12 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘𝐸) = 0) |
14 | inss2 4126 | . . . . . . 7 ⊢ (𝑎 ∩ 𝐸) ⊆ 𝐸 | |
15 | 14 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑎 ∩ 𝐸) ⊆ 𝐸) |
16 | 10, 2, 11, 13, 15 | omess0 42358 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∩ 𝐸)) = 0) |
17 | 16 | oveq1d 7031 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (0 +𝑒 (𝑂‘(𝑎 ∖ 𝐸)))) |
18 | difssd 4030 | . . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∖ 𝐸) ⊆ 𝑎) | |
19 | elpwi 4463 | . . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝑎 ⊆ 𝑋) | |
20 | 18, 19 | sstrd 3899 | . . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∖ 𝐸) ⊆ 𝑋) |
21 | 20 | adantl 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑎 ∖ 𝐸) ⊆ 𝑋) |
22 | 10, 2, 21 | omexrcl 42331 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∖ 𝐸)) ∈ ℝ*) |
23 | xaddid2 12485 | . . . . 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 2831 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘(𝑎 ∖ 𝐸))) |
26 | 19 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑎 ⊆ 𝑋) |
27 | 18 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑎 ∖ 𝐸) ⊆ 𝑎) |
28 | 10, 2, 26, 27 | omessle 42322 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∖ 𝐸)) ≤ (𝑂‘𝑎)) |
29 | 25, 28 | eqbrtrd 4984 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) ≤ (𝑂‘𝑎)) |
30 | 1, 2, 3, 9, 29 | caragenel2d 42356 | 1 ⊢ (𝜑 → 𝐸 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2081 Vcvv 3437 ∖ cdif 3856 ∩ cin 3858 ⊆ wss 3859 𝒫 cpw 4453 ∪ cuni 4745 dom cdm 5443 ‘cfv 6225 (class class class)co 7016 0cc0 10383 ℝ*cxr 10520 ≤ cle 10522 +𝑒 cxad 12355 OutMeascome 42313 CaraGenccaragen 42315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-inf2 8950 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-oadd 7957 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-sup 8752 df-oi 8820 df-card 9214 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-n0 11746 df-z 11830 df-uz 12094 df-rp 12240 df-xadd 12358 df-ico 12594 df-icc 12595 df-fz 12743 df-fzo 12884 df-seq 13220 df-exp 13280 df-hash 13541 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-clim 14679 df-sum 14877 df-sumge0 42187 df-ome 42314 df-caragen 42316 |
This theorem is referenced by: voncmpl 42445 |
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