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Mathbox for Glauco Siliprandi |
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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 46099 | . . 3 ⊢ (𝜑 → (voln*‘𝑋) ∈ OutMeas) |
3 | eqid 2725 | . . 3 ⊢ ∪ dom (voln*‘𝑋) = ∪ dom (voln*‘𝑋) | |
4 | voncmpl.f | . . . 4 ⊢ (𝜑 → 𝐹 ⊆ 𝐸) | |
5 | 1 | dmvon 46132 | . . . . . . 7 ⊢ (𝜑 → dom (voln‘𝑋) = (CaraGen‘(voln*‘𝑋))) |
6 | eqid 2725 | . . . . . . . . 9 ⊢ (CaraGen‘(voln*‘𝑋)) = (CaraGen‘(voln*‘𝑋)) | |
7 | 6 | caragenss 46030 | . . . . . . . 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 4015 | . . . . . 6 ⊢ (𝜑 → dom (voln‘𝑋) ⊆ dom (voln*‘𝑋)) |
10 | voncmpl.e | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ dom (voln‘𝑋)) | |
11 | 9, 10 | sseldd 3977 | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ dom (voln*‘𝑋)) |
12 | elssuni 4941 | . . . . 5 ⊢ (𝐸 ∈ dom (voln*‘𝑋) → 𝐸 ⊆ ∪ dom (voln*‘𝑋)) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐸 ⊆ ∪ dom (voln*‘𝑋)) |
14 | 4, 13 | sstrd 3987 | . . 3 ⊢ (𝜑 → 𝐹 ⊆ ∪ dom (voln*‘𝑋)) |
15 | voncmpl.z | . . . . . . 7 ⊢ (𝜑 → ((voln‘𝑋)‘𝐸) = 0) | |
16 | 15 | eqcomd 2731 | . . . . . 6 ⊢ (𝜑 → 0 = ((voln‘𝑋)‘𝐸)) |
17 | 1 | vonval 46066 | . . . . . . 7 ⊢ (𝜑 → (voln‘𝑋) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋)))) |
18 | 17 | fveq1d 6898 | . . . . . 6 ⊢ (𝜑 → ((voln‘𝑋)‘𝐸) = (((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋)))‘𝐸)) |
19 | 16, 18 | eqtrd 2765 | . . . . 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 2766 | . . . . . . 7 ⊢ (𝜑 → (CaraGen‘(voln*‘𝑋)) = 𝑆) |
23 | 22 | reseq2d 5985 | . . . . . 6 ⊢ (𝜑 → ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))) = ((voln*‘𝑋) ↾ 𝑆)) |
24 | 23 | fveq1d 6898 | . . . . 5 ⊢ (𝜑 → (((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋)))‘𝐸) = (((voln*‘𝑋) ↾ 𝑆)‘𝐸)) |
25 | 10, 20 | eleqtrrdi 2836 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝑆) |
26 | fvres 6915 | . . . . . 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 2770 | . . . 4 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐸) = 0) |
29 | 2, 3, 13, 28, 4 | omess0 46060 | . . 3 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐹) = 0) |
30 | 2, 3, 14, 29, 6 | caragencmpl 46061 | . 2 ⊢ (𝜑 → 𝐹 ∈ (CaraGen‘(voln*‘𝑋))) |
31 | 30, 22 | eleqtrd 2827 | 1 ⊢ (𝜑 → 𝐹 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 ∪ cuni 4909 dom cdm 5678 ↾ cres 5680 ‘cfv 6549 Fincfn 8964 0cc0 11140 OutMeascome 46015 CaraGenccaragen 46017 voln*covoln 46062 volncvoln 46064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cc 10460 ax-ac2 10488 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 ax-addf 11219 ax-mulf 11220 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fi 9436 df-sup 9467 df-inf 9468 df-oi 9535 df-dju 9926 df-card 9964 df-acn 9967 df-ac 10141 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-ioo 13363 df-ico 13365 df-icc 13366 df-fz 13520 df-fzo 13663 df-fl 13793 df-seq 14003 df-exp 14063 df-hash 14326 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-clim 15468 df-rlim 15469 df-sum 15669 df-prod 15886 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-rest 17407 df-0g 17426 df-topgen 17428 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18901 df-minusg 18902 df-subg 19086 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-cring 20188 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-dvr 20352 df-drng 20638 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-cnfld 21297 df-top 22840 df-topon 22857 df-bases 22893 df-cmp 23335 df-ovol 25437 df-vol 25438 df-sumge0 45889 df-ome 46016 df-caragen 46018 df-ovoln 46063 df-voln 46065 |
This theorem is referenced by: (None) |
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