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| Mirrors > Home > MPE Home > Th. List > Mathboxes > voncmpl | Structured version Visualization version GIF version | ||
| Description: The Lebesgue measure is complete. See Definition 112Df of [Fremlin1] p. 19. This is an observation written after Definition 115E of [Fremlin1] p. 31. (Contributed by Glauco Siliprandi, 24-Dec-2020.) | 
| Ref | Expression | 
|---|---|
| voncmpl.x | ⊢ (𝜑 → 𝑋 ∈ Fin) | 
| voncmpl.s | ⊢ 𝑆 = dom (voln‘𝑋) | 
| voncmpl.e | ⊢ (𝜑 → 𝐸 ∈ dom (voln‘𝑋)) | 
| voncmpl.z | ⊢ (𝜑 → ((voln‘𝑋)‘𝐸) = 0) | 
| voncmpl.f | ⊢ (𝜑 → 𝐹 ⊆ 𝐸) | 
| Ref | Expression | 
|---|---|
| voncmpl | ⊢ (𝜑 → 𝐹 ∈ 𝑆) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | voncmpl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
| 2 | 1 | ovnome 46588 | . . 3 ⊢ (𝜑 → (voln*‘𝑋) ∈ OutMeas) | 
| 3 | eqid 2737 | . . 3 ⊢ ∪ dom (voln*‘𝑋) = ∪ dom (voln*‘𝑋) | |
| 4 | voncmpl.f | . . . 4 ⊢ (𝜑 → 𝐹 ⊆ 𝐸) | |
| 5 | 1 | dmvon 46621 | . . . . . . 7 ⊢ (𝜑 → dom (voln‘𝑋) = (CaraGen‘(voln*‘𝑋))) | 
| 6 | eqid 2737 | . . . . . . . . 9 ⊢ (CaraGen‘(voln*‘𝑋)) = (CaraGen‘(voln*‘𝑋)) | |
| 7 | 6 | caragenss 46519 | . . . . . . . 8 ⊢ ((voln*‘𝑋) ∈ OutMeas → (CaraGen‘(voln*‘𝑋)) ⊆ dom (voln*‘𝑋)) | 
| 8 | 2, 7 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (CaraGen‘(voln*‘𝑋)) ⊆ dom (voln*‘𝑋)) | 
| 9 | 5, 8 | eqsstrd 4018 | . . . . . 6 ⊢ (𝜑 → dom (voln‘𝑋) ⊆ dom (voln*‘𝑋)) | 
| 10 | voncmpl.e | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ dom (voln‘𝑋)) | |
| 11 | 9, 10 | sseldd 3984 | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ dom (voln*‘𝑋)) | 
| 12 | elssuni 4937 | . . . . 5 ⊢ (𝐸 ∈ dom (voln*‘𝑋) → 𝐸 ⊆ ∪ dom (voln*‘𝑋)) | |
| 13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐸 ⊆ ∪ dom (voln*‘𝑋)) | 
| 14 | 4, 13 | sstrd 3994 | . . 3 ⊢ (𝜑 → 𝐹 ⊆ ∪ dom (voln*‘𝑋)) | 
| 15 | voncmpl.z | . . . . . . 7 ⊢ (𝜑 → ((voln‘𝑋)‘𝐸) = 0) | |
| 16 | 15 | eqcomd 2743 | . . . . . 6 ⊢ (𝜑 → 0 = ((voln‘𝑋)‘𝐸)) | 
| 17 | 1 | vonval 46555 | . . . . . . 7 ⊢ (𝜑 → (voln‘𝑋) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋)))) | 
| 18 | 17 | fveq1d 6908 | . . . . . 6 ⊢ (𝜑 → ((voln‘𝑋)‘𝐸) = (((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋)))‘𝐸)) | 
| 19 | 16, 18 | eqtrd 2777 | . . . . 5 ⊢ (𝜑 → 0 = (((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋)))‘𝐸)) | 
| 20 | voncmpl.s | . . . . . . . . 9 ⊢ 𝑆 = dom (voln‘𝑋) | |
| 21 | 20 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 = dom (voln‘𝑋)) | 
| 22 | 21, 5 | eqtr2d 2778 | . . . . . . 7 ⊢ (𝜑 → (CaraGen‘(voln*‘𝑋)) = 𝑆) | 
| 23 | 22 | reseq2d 5997 | . . . . . 6 ⊢ (𝜑 → ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))) = ((voln*‘𝑋) ↾ 𝑆)) | 
| 24 | 23 | fveq1d 6908 | . . . . 5 ⊢ (𝜑 → (((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋)))‘𝐸) = (((voln*‘𝑋) ↾ 𝑆)‘𝐸)) | 
| 25 | 10, 20 | eleqtrrdi 2852 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝑆) | 
| 26 | fvres 6925 | . . . . . 6 ⊢ (𝐸 ∈ 𝑆 → (((voln*‘𝑋) ↾ 𝑆)‘𝐸) = ((voln*‘𝑋)‘𝐸)) | |
| 27 | 25, 26 | syl 17 | . . . . 5 ⊢ (𝜑 → (((voln*‘𝑋) ↾ 𝑆)‘𝐸) = ((voln*‘𝑋)‘𝐸)) | 
| 28 | 19, 24, 27 | 3eqtrrd 2782 | . . . 4 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐸) = 0) | 
| 29 | 2, 3, 13, 28, 4 | omess0 46549 | . . 3 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐹) = 0) | 
| 30 | 2, 3, 14, 29, 6 | caragencmpl 46550 | . 2 ⊢ (𝜑 → 𝐹 ∈ (CaraGen‘(voln*‘𝑋))) | 
| 31 | 30, 22 | eleqtrd 2843 | 1 ⊢ (𝜑 → 𝐹 ∈ 𝑆) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ∪ cuni 4907 dom cdm 5685 ↾ cres 5687 ‘cfv 6561 Fincfn 8985 0cc0 11155 OutMeascome 46504 CaraGenccaragen 46506 voln*covoln 46551 volncvoln 46553 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cc 10475 ax-ac2 10503 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-acn 9982 df-ac 10156 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 df-prod 15940 df-rest 17467 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-bases 22953 df-cmp 23395 df-ovol 25499 df-vol 25500 df-sumge0 46378 df-ome 46505 df-caragen 46507 df-ovoln 46552 df-voln 46554 | 
| This theorem is referenced by: (None) | 
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