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Mirrors > Home > MPE Home > Th. List > Mathboxes > cntmeas | Structured version Visualization version GIF version |
Description: The Counting measure is a measure on any sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.) |
Ref | Expression |
---|---|
cntmeas | ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (♯ ↾ 𝑆) ∈ (measures‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashf2 31242 | . . . 4 ⊢ ♯:V⟶(0[,]+∞) | |
2 | ssv 3988 | . . . 4 ⊢ 𝑆 ⊆ V | |
3 | fssres 6537 | . . . 4 ⊢ ((♯:V⟶(0[,]+∞) ∧ 𝑆 ⊆ V) → (♯ ↾ 𝑆):𝑆⟶(0[,]+∞)) | |
4 | 1, 2, 3 | mp2an 688 | . . 3 ⊢ (♯ ↾ 𝑆):𝑆⟶(0[,]+∞) |
5 | 4 | a1i 11 | . 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (♯ ↾ 𝑆):𝑆⟶(0[,]+∞)) |
6 | 0elsiga 31272 | . . . 4 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∅ ∈ 𝑆) | |
7 | fvres 6682 | . . . 4 ⊢ (∅ ∈ 𝑆 → ((♯ ↾ 𝑆)‘∅) = (♯‘∅)) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ((♯ ↾ 𝑆)‘∅) = (♯‘∅)) |
9 | hash0 13716 | . . 3 ⊢ (♯‘∅) = 0 | |
10 | 8, 9 | syl6eq 2869 | . 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ((♯ ↾ 𝑆)‘∅) = 0) |
11 | vex 3495 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
12 | hasheuni 31243 | . . . . . . 7 ⊢ ((𝑥 ∈ V ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (♯‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(♯‘𝑦)) | |
13 | 11, 12 | mpan 686 | . . . . . 6 ⊢ (Disj 𝑦 ∈ 𝑥 𝑦 → (♯‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(♯‘𝑦)) |
14 | 13 | ad2antll 725 | . . . . 5 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑆) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (♯‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(♯‘𝑦)) |
15 | isrnsigau 31285 | . . . . . . . . . . 11 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))) | |
16 | 15 | simprd 496 | . . . . . . . . . 10 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) |
17 | 16 | simp3d 1136 | . . . . . . . . 9 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)) |
18 | fvres 6682 | . . . . . . . . . . 11 ⊢ (∪ 𝑥 ∈ 𝑆 → ((♯ ↾ 𝑆)‘∪ 𝑥) = (♯‘∪ 𝑥)) | |
19 | 18 | imim2i 16 | . . . . . . . . . 10 ⊢ ((𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆) → (𝑥 ≼ ω → ((♯ ↾ 𝑆)‘∪ 𝑥) = (♯‘∪ 𝑥))) |
20 | 19 | ralimi 3157 | . . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ((♯ ↾ 𝑆)‘∪ 𝑥) = (♯‘∪ 𝑥))) |
21 | 17, 20 | syl 17 | . . . . . . . 8 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ((♯ ↾ 𝑆)‘∪ 𝑥) = (♯‘∪ 𝑥))) |
22 | 21 | r19.21bi 3205 | . . . . . . 7 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑆) → (𝑥 ≼ ω → ((♯ ↾ 𝑆)‘∪ 𝑥) = (♯‘∪ 𝑥))) |
23 | 22 | imp 407 | . . . . . 6 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑆) ∧ 𝑥 ≼ ω) → ((♯ ↾ 𝑆)‘∪ 𝑥) = (♯‘∪ 𝑥)) |
24 | 23 | adantrr 713 | . . . . 5 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑆) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → ((♯ ↾ 𝑆)‘∪ 𝑥) = (♯‘∪ 𝑥)) |
25 | elpwi 4547 | . . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 𝑆 → 𝑥 ⊆ 𝑆) | |
26 | 25 | sseld 3963 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 𝑆 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑆)) |
27 | fvres 6682 | . . . . . . . . 9 ⊢ (𝑦 ∈ 𝑆 → ((♯ ↾ 𝑆)‘𝑦) = (♯‘𝑦)) | |
28 | 26, 27 | syl6 35 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝑆 → (𝑦 ∈ 𝑥 → ((♯ ↾ 𝑆)‘𝑦) = (♯‘𝑦))) |
29 | 28 | imp 407 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 𝑆 ∧ 𝑦 ∈ 𝑥) → ((♯ ↾ 𝑆)‘𝑦) = (♯‘𝑦)) |
30 | 29 | esumeq2dv 31196 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝑆 → Σ*𝑦 ∈ 𝑥((♯ ↾ 𝑆)‘𝑦) = Σ*𝑦 ∈ 𝑥(♯‘𝑦)) |
31 | 30 | ad2antlr 723 | . . . . 5 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑆) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → Σ*𝑦 ∈ 𝑥((♯ ↾ 𝑆)‘𝑦) = Σ*𝑦 ∈ 𝑥(♯‘𝑦)) |
32 | 14, 24, 31 | 3eqtr4d 2863 | . . . 4 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑆) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → ((♯ ↾ 𝑆)‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥((♯ ↾ 𝑆)‘𝑦)) |
33 | 32 | ex 413 | . . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑆) → ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ((♯ ↾ 𝑆)‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥((♯ ↾ 𝑆)‘𝑦))) |
34 | 33 | ralrimiva 3179 | . 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ((♯ ↾ 𝑆)‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥((♯ ↾ 𝑆)‘𝑦))) |
35 | ismeas 31357 | . 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ((♯ ↾ 𝑆) ∈ (measures‘𝑆) ↔ ((♯ ↾ 𝑆):𝑆⟶(0[,]+∞) ∧ ((♯ ↾ 𝑆)‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ((♯ ↾ 𝑆)‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥((♯ ↾ 𝑆)‘𝑦))))) | |
36 | 5, 10, 34, 35 | mpbir3and 1334 | 1 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (♯ ↾ 𝑆) ∈ (measures‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 ∖ cdif 3930 ⊆ wss 3933 ∅c0 4288 𝒫 cpw 4535 ∪ cuni 4830 Disj wdisj 5022 class class class wbr 5057 ran crn 5549 ↾ cres 5550 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ωcom 7569 ≼ cdom 8495 0cc0 10525 +∞cpnf 10660 [,]cicc 12729 ♯chash 13678 Σ*cesum 31185 sigAlgebracsiga 31266 measurescmeas 31353 |
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-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-iin 4913 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-supp 7820 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-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 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-xnn0 11956 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-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-fac 13622 df-bc 13651 df-hash 13679 df-shft 14414 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-limsup 14816 df-clim 14833 df-rlim 14834 df-sum 15031 df-ef 15409 df-sin 15411 df-cos 15412 df-pi 15414 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-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-ordt 16762 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-ps 17798 df-tsr 17799 df-plusf 17839 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-cntz 18385 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-subrg 19462 df-abv 19517 df-lmod 19565 df-scaf 19566 df-sra 19873 df-rgmod 19874 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-fbas 20470 df-fg 20471 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-lp 21672 df-perf 21673 df-cn 21763 df-cnp 21764 df-haus 21851 df-tx 22098 df-hmeo 22291 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-tmd 22608 df-tgp 22609 df-tsms 22662 df-trg 22695 df-xms 22857 df-ms 22858 df-tms 22859 df-nm 23119 df-ngp 23120 df-nrg 23122 df-nlm 23123 df-ii 23412 df-cncf 23413 df-limc 24391 df-dv 24392 df-log 25067 df-esum 31186 df-siga 31267 df-meas 31354 |
This theorem is referenced by: pwcntmeas 31385 |
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