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 44080 | . . 3 ⊢ (𝜑 → (voln*‘𝑋) ∈ OutMeas) |
3 | eqid 2740 | . . 3 ⊢ ∪ dom (voln*‘𝑋) = ∪ dom (voln*‘𝑋) | |
4 | voncmpl.f | . . . 4 ⊢ (𝜑 → 𝐹 ⊆ 𝐸) | |
5 | 1 | dmvon 44113 | . . . . . . 7 ⊢ (𝜑 → dom (voln‘𝑋) = (CaraGen‘(voln*‘𝑋))) |
6 | eqid 2740 | . . . . . . . . 9 ⊢ (CaraGen‘(voln*‘𝑋)) = (CaraGen‘(voln*‘𝑋)) | |
7 | 6 | caragenss 44011 | . . . . . . . 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 3964 | . . . . . 6 ⊢ (𝜑 → dom (voln‘𝑋) ⊆ dom (voln*‘𝑋)) |
10 | voncmpl.e | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ dom (voln‘𝑋)) | |
11 | 9, 10 | sseldd 3927 | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ dom (voln*‘𝑋)) |
12 | elssuni 4877 | . . . . 5 ⊢ (𝐸 ∈ dom (voln*‘𝑋) → 𝐸 ⊆ ∪ dom (voln*‘𝑋)) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐸 ⊆ ∪ dom (voln*‘𝑋)) |
14 | 4, 13 | sstrd 3936 | . . 3 ⊢ (𝜑 → 𝐹 ⊆ ∪ dom (voln*‘𝑋)) |
15 | voncmpl.z | . . . . . . 7 ⊢ (𝜑 → ((voln‘𝑋)‘𝐸) = 0) | |
16 | 15 | eqcomd 2746 | . . . . . 6 ⊢ (𝜑 → 0 = ((voln‘𝑋)‘𝐸)) |
17 | 1 | vonval 44047 | . . . . . . 7 ⊢ (𝜑 → (voln‘𝑋) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋)))) |
18 | 17 | fveq1d 6771 | . . . . . 6 ⊢ (𝜑 → ((voln‘𝑋)‘𝐸) = (((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋)))‘𝐸)) |
19 | 16, 18 | eqtrd 2780 | . . . . 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 2781 | . . . . . . 7 ⊢ (𝜑 → (CaraGen‘(voln*‘𝑋)) = 𝑆) |
23 | 22 | reseq2d 5889 | . . . . . 6 ⊢ (𝜑 → ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))) = ((voln*‘𝑋) ↾ 𝑆)) |
24 | 23 | fveq1d 6771 | . . . . 5 ⊢ (𝜑 → (((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋)))‘𝐸) = (((voln*‘𝑋) ↾ 𝑆)‘𝐸)) |
25 | 10, 20 | eleqtrrdi 2852 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝑆) |
26 | fvres 6788 | . . . . . 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 2785 | . . . 4 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐸) = 0) |
29 | 2, 3, 13, 28, 4 | omess0 44041 | . . 3 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐹) = 0) |
30 | 2, 3, 14, 29, 6 | caragencmpl 44042 | . 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 1542 ∈ wcel 2110 ⊆ wss 3892 ∪ cuni 4845 dom cdm 5589 ↾ cres 5591 ‘cfv 6431 Fincfn 8714 0cc0 10870 OutMeascome 43996 CaraGenccaragen 43998 voln*covoln 44043 volncvoln 44045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-inf2 9375 ax-cc 10190 ax-ac2 10218 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 ax-pre-sup 10948 ax-addf 10949 ax-mulf 10950 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-disj 5045 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-isom 6440 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-of 7525 df-om 7705 df-1st 7822 df-2nd 7823 df-tpos 8031 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-2o 8287 df-er 8479 df-map 8598 df-pm 8599 df-ixp 8667 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-fi 9146 df-sup 9177 df-inf 9178 df-oi 9245 df-dju 9658 df-card 9696 df-acn 9699 df-ac 9871 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-dec 12435 df-uz 12580 df-q 12686 df-rp 12728 df-xneg 12845 df-xadd 12846 df-xmul 12847 df-ioo 13080 df-ico 13082 df-icc 13083 df-fz 13237 df-fzo 13380 df-fl 13508 df-seq 13718 df-exp 13779 df-hash 14041 df-cj 14806 df-re 14807 df-im 14808 df-sqrt 14942 df-abs 14943 df-clim 15193 df-rlim 15194 df-sum 15394 df-prod 15612 df-struct 16844 df-sets 16861 df-slot 16879 df-ndx 16891 df-base 16909 df-ress 16938 df-plusg 16971 df-mulr 16972 df-starv 16973 df-tset 16977 df-ple 16978 df-ds 16980 df-unif 16981 df-rest 17129 df-0g 17148 df-topgen 17150 df-mgm 18322 df-sgrp 18371 df-mnd 18382 df-grp 18576 df-minusg 18577 df-subg 18748 df-cmn 19384 df-abl 19385 df-mgp 19717 df-ur 19734 df-ring 19781 df-cring 19782 df-oppr 19858 df-dvdsr 19879 df-unit 19880 df-invr 19910 df-dvr 19921 df-drng 19989 df-psmet 20585 df-xmet 20586 df-met 20587 df-bl 20588 df-mopn 20589 df-cnfld 20594 df-top 22039 df-topon 22056 df-bases 22092 df-cmp 22534 df-ovol 24624 df-vol 24625 df-sumge0 43870 df-ome 43997 df-caragen 43999 df-ovoln 44044 df-voln 44046 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |