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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > probfinmeasb | Structured version Visualization version GIF version |
Description: Build a probability measure from a finite measure. (Contributed by Thierry Arnoux, 31-Jan-2017.) |
Ref | Expression |
---|---|
probfinmeasb | ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → (𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆)) ∈ Prob) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | measdivcst 30886 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → (𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆)) ∈ (measures‘𝑆)) | |
2 | measfn 30865 | . . . . . . . 8 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀 Fn 𝑆) | |
3 | 2 | adantr 474 | . . . . . . 7 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → 𝑀 Fn 𝑆) |
4 | measbase 30858 | . . . . . . . 8 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra) | |
5 | 4 | adantr 474 | . . . . . . 7 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → 𝑆 ∈ ∪ ran sigAlgebra) |
6 | simpr 479 | . . . . . . 7 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → (𝑀‘∪ 𝑆) ∈ ℝ+) | |
7 | 3, 5, 6 | ofcfn 30760 | . . . . . 6 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → (𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆)) Fn 𝑆) |
8 | fndm 6235 | . . . . . 6 ⊢ ((𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆)) Fn 𝑆 → dom (𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆)) = 𝑆) | |
9 | 7, 8 | syl 17 | . . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → dom (𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆)) = 𝑆) |
10 | 9 | fveq2d 6450 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → (measures‘dom (𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆))) = (measures‘𝑆)) |
11 | 1, 10 | eleqtrrd 2862 | . . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → (𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆)) ∈ (measures‘dom (𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆)))) |
12 | measbasedom 30863 | . . 3 ⊢ ((𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆)) ∈ ∪ ran measures ↔ (𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆)) ∈ (measures‘dom (𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆)))) | |
13 | 11, 12 | sylibr 226 | . 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → (𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆)) ∈ ∪ ran measures) |
14 | 9 | unieqd 4681 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → ∪ dom (𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆)) = ∪ 𝑆) |
15 | 14 | fveq2d 6450 | . . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → ((𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆))‘∪ dom (𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆))) = ((𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆))‘∪ 𝑆)) |
16 | unielsiga 30789 | . . . . 5 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∪ 𝑆 ∈ 𝑆) | |
17 | 5, 16 | syl 17 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → ∪ 𝑆 ∈ 𝑆) |
18 | eqidd 2779 | . . . . 5 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) ∧ ∪ 𝑆 ∈ 𝑆) → (𝑀‘∪ 𝑆) = (𝑀‘∪ 𝑆)) | |
19 | 3, 5, 6, 18 | ofcval 30759 | . . . 4 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) ∧ ∪ 𝑆 ∈ 𝑆) → ((𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆))‘∪ 𝑆) = ((𝑀‘∪ 𝑆) /𝑒 (𝑀‘∪ 𝑆))) |
20 | 17, 19 | mpdan 677 | . . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → ((𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆))‘∪ 𝑆) = ((𝑀‘∪ 𝑆) /𝑒 (𝑀‘∪ 𝑆))) |
21 | rpre 12145 | . . . . 5 ⊢ ((𝑀‘∪ 𝑆) ∈ ℝ+ → (𝑀‘∪ 𝑆) ∈ ℝ) | |
22 | rpne0 12155 | . . . . 5 ⊢ ((𝑀‘∪ 𝑆) ∈ ℝ+ → (𝑀‘∪ 𝑆) ≠ 0) | |
23 | xdivid 30198 | . . . . 5 ⊢ (((𝑀‘∪ 𝑆) ∈ ℝ ∧ (𝑀‘∪ 𝑆) ≠ 0) → ((𝑀‘∪ 𝑆) /𝑒 (𝑀‘∪ 𝑆)) = 1) | |
24 | 21, 22, 23 | syl2anc 579 | . . . 4 ⊢ ((𝑀‘∪ 𝑆) ∈ ℝ+ → ((𝑀‘∪ 𝑆) /𝑒 (𝑀‘∪ 𝑆)) = 1) |
25 | 24 | adantl 475 | . . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → ((𝑀‘∪ 𝑆) /𝑒 (𝑀‘∪ 𝑆)) = 1) |
26 | 15, 20, 25 | 3eqtrd 2818 | . 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → ((𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆))‘∪ dom (𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆))) = 1) |
27 | elprob 31070 | . 2 ⊢ ((𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆)) ∈ Prob ↔ ((𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆)) ∈ ∪ ran measures ∧ ((𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆))‘∪ dom (𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆))) = 1)) | |
28 | 13, 26, 27 | sylanbrc 578 | 1 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → (𝑀∘𝑓/𝑐 /𝑒 (𝑀‘∪ 𝑆)) ∈ Prob) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∪ cuni 4671 dom cdm 5355 ran crn 5356 Fn wfn 6130 ‘cfv 6135 (class class class)co 6922 ℝcr 10271 0cc0 10272 1c1 10273 ℝ+crp 12137 /𝑒 cxdiv 30187 ∘𝑓/𝑐cofc 30755 sigAlgebracsiga 30768 measurescmeas 30856 Probcprb 31068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-disj 4855 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-ioc 12492 df-ico 12493 df-icc 12494 df-fz 12644 df-fzo 12785 df-seq 13120 df-hash 13436 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-tset 16357 df-ple 16358 df-ds 16360 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-ordt 16547 df-xrs 16548 df-mre 16632 df-mrc 16633 df-acs 16635 df-ps 17586 df-tsr 17587 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-submnd 17722 df-cntz 18133 df-cmn 18581 df-fbas 20139 df-fg 20140 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-ntr 21232 df-nei 21310 df-cn 21439 df-cnp 21440 df-haus 21527 df-fil 22058 df-fm 22150 df-flim 22151 df-flf 22152 df-tsms 22338 df-xdiv 30188 df-esum 30688 df-ofc 30756 df-siga 30769 df-meas 30857 df-prob 31069 |
This theorem is referenced by: coinflipprob 31140 |
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