Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 42732 | . . 3 ⊢ (𝜑 → (voln*‘𝑋) ∈ OutMeas) |
3 | eqid 2818 | . . 3 ⊢ ∪ dom (voln*‘𝑋) = ∪ dom (voln*‘𝑋) | |
4 | voncmpl.f | . . . 4 ⊢ (𝜑 → 𝐹 ⊆ 𝐸) | |
5 | 1 | dmvon 42765 | . . . . . . 7 ⊢ (𝜑 → dom (voln‘𝑋) = (CaraGen‘(voln*‘𝑋))) |
6 | eqid 2818 | . . . . . . . . 9 ⊢ (CaraGen‘(voln*‘𝑋)) = (CaraGen‘(voln*‘𝑋)) | |
7 | 6 | caragenss 42663 | . . . . . . . 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 4002 | . . . . . 6 ⊢ (𝜑 → dom (voln‘𝑋) ⊆ dom (voln*‘𝑋)) |
10 | voncmpl.e | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ dom (voln‘𝑋)) | |
11 | 9, 10 | sseldd 3965 | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ dom (voln*‘𝑋)) |
12 | elssuni 4859 | . . . . 5 ⊢ (𝐸 ∈ dom (voln*‘𝑋) → 𝐸 ⊆ ∪ dom (voln*‘𝑋)) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐸 ⊆ ∪ dom (voln*‘𝑋)) |
14 | 4, 13 | sstrd 3974 | . . 3 ⊢ (𝜑 → 𝐹 ⊆ ∪ dom (voln*‘𝑋)) |
15 | voncmpl.z | . . . . . . 7 ⊢ (𝜑 → ((voln‘𝑋)‘𝐸) = 0) | |
16 | 15 | eqcomd 2824 | . . . . . 6 ⊢ (𝜑 → 0 = ((voln‘𝑋)‘𝐸)) |
17 | 1 | vonval 42699 | . . . . . . 7 ⊢ (𝜑 → (voln‘𝑋) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋)))) |
18 | 17 | fveq1d 6665 | . . . . . 6 ⊢ (𝜑 → ((voln‘𝑋)‘𝐸) = (((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋)))‘𝐸)) |
19 | 16, 18 | eqtrd 2853 | . . . . 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 2854 | . . . . . . 7 ⊢ (𝜑 → (CaraGen‘(voln*‘𝑋)) = 𝑆) |
23 | 22 | reseq2d 5846 | . . . . . 6 ⊢ (𝜑 → ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))) = ((voln*‘𝑋) ↾ 𝑆)) |
24 | 23 | fveq1d 6665 | . . . . 5 ⊢ (𝜑 → (((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋)))‘𝐸) = (((voln*‘𝑋) ↾ 𝑆)‘𝐸)) |
25 | 10, 20 | eleqtrrdi 2921 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝑆) |
26 | fvres 6682 | . . . . . 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 2858 | . . . 4 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐸) = 0) |
29 | 2, 3, 13, 28, 4 | omess0 42693 | . . 3 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐹) = 0) |
30 | 2, 3, 14, 29, 6 | caragencmpl 42694 | . 2 ⊢ (𝜑 → 𝐹 ∈ (CaraGen‘(voln*‘𝑋))) |
31 | 30, 22 | eleqtrd 2912 | 1 ⊢ (𝜑 → 𝐹 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 ∪ cuni 4830 dom cdm 5548 ↾ cres 5550 ‘cfv 6348 Fincfn 8497 0cc0 10525 OutMeascome 42648 CaraGenccaragen 42650 voln*covoln 42695 volncvoln 42697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cc 9845 ax-ac2 9873 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-disj 5023 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-dju 9318 df-card 9356 df-acn 9359 df-ac 9530 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-rlim 14834 df-sum 15031 df-prod 15248 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-rest 16684 df-0g 16703 df-topgen 16705 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-subg 18214 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-drng 19433 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-cnfld 20474 df-top 21430 df-topon 21447 df-bases 21482 df-cmp 21923 df-ovol 23992 df-vol 23993 df-sumge0 42522 df-ome 42649 df-caragen 42651 df-ovoln 42696 df-voln 42698 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |