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Mirrors > Home > MPE Home > Th. List > Mathboxes > probfinmeasbALTV | Structured version Visualization version GIF version |
Description: Alternate version of probfinmeasb 32395. (Contributed by Thierry Arnoux, 17-Dec-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
probfinmeasbALTV | ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆))) ∈ Prob) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | measdivcstALTV 32193 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆))) ∈ (measures‘𝑆)) | |
2 | ovex 7308 | . . . . . . 7 ⊢ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆)) ∈ V | |
3 | 2 | rgenw 3076 | . . . . . 6 ⊢ ∀𝑥 ∈ 𝑆 ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆)) ∈ V |
4 | dmmptg 6145 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝑆 ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆)) ∈ V → dom (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆))) = 𝑆) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ dom (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆))) = 𝑆 |
6 | 5 | fveq2i 6777 | . . . 4 ⊢ (measures‘dom (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆)))) = (measures‘𝑆) |
7 | 1, 6 | eleqtrrdi 2850 | . . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆))) ∈ (measures‘dom (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆))))) |
8 | measbasedom 32170 | . . 3 ⊢ ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆))) ∈ ∪ ran measures ↔ (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆))) ∈ (measures‘dom (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆))))) | |
9 | 7, 8 | sylibr 233 | . 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆))) ∈ ∪ ran measures) |
10 | 5 | unieqi 4852 | . . . 4 ⊢ ∪ dom (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆))) = ∪ 𝑆 |
11 | 10 | fveq2i 6777 | . . 3 ⊢ ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆)))‘∪ dom (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆)))) = ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆)))‘∪ 𝑆) |
12 | measbase 32165 | . . . . . . 7 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra) | |
13 | isrnsigau 32095 | . . . . . . . . 9 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))) | |
14 | 13 | simprd 496 | . . . . . . . 8 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))) |
15 | 14 | simp1d 1141 | . . . . . . 7 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∪ 𝑆 ∈ 𝑆) |
16 | 12, 15 | syl 17 | . . . . . 6 ⊢ (𝑀 ∈ (measures‘𝑆) → ∪ 𝑆 ∈ 𝑆) |
17 | id 22 | . . . . . . 7 ⊢ ((𝑀‘∪ 𝑆) ∈ ℝ+ → (𝑀‘∪ 𝑆) ∈ ℝ+) | |
18 | 17, 17 | rpxdivcld 31208 | . . . . . 6 ⊢ ((𝑀‘∪ 𝑆) ∈ ℝ+ → ((𝑀‘∪ 𝑆) /𝑒 (𝑀‘∪ 𝑆)) ∈ ℝ+) |
19 | 16, 18 | anim12i 613 | . . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → (∪ 𝑆 ∈ 𝑆 ∧ ((𝑀‘∪ 𝑆) /𝑒 (𝑀‘∪ 𝑆)) ∈ ℝ+)) |
20 | fveq2 6774 | . . . . . . 7 ⊢ (𝑥 = ∪ 𝑆 → (𝑀‘𝑥) = (𝑀‘∪ 𝑆)) | |
21 | 20 | oveq1d 7290 | . . . . . 6 ⊢ (𝑥 = ∪ 𝑆 → ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆)) = ((𝑀‘∪ 𝑆) /𝑒 (𝑀‘∪ 𝑆))) |
22 | eqid 2738 | . . . . . 6 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆))) = (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆))) | |
23 | 21, 22 | fvmptg 6873 | . . . . 5 ⊢ ((∪ 𝑆 ∈ 𝑆 ∧ ((𝑀‘∪ 𝑆) /𝑒 (𝑀‘∪ 𝑆)) ∈ ℝ+) → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆)))‘∪ 𝑆) = ((𝑀‘∪ 𝑆) /𝑒 (𝑀‘∪ 𝑆))) |
24 | 19, 23 | syl 17 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆)))‘∪ 𝑆) = ((𝑀‘∪ 𝑆) /𝑒 (𝑀‘∪ 𝑆))) |
25 | rpre 12738 | . . . . . 6 ⊢ ((𝑀‘∪ 𝑆) ∈ ℝ+ → (𝑀‘∪ 𝑆) ∈ ℝ) | |
26 | rpne0 12746 | . . . . . 6 ⊢ ((𝑀‘∪ 𝑆) ∈ ℝ+ → (𝑀‘∪ 𝑆) ≠ 0) | |
27 | xdivid 31202 | . . . . . 6 ⊢ (((𝑀‘∪ 𝑆) ∈ ℝ ∧ (𝑀‘∪ 𝑆) ≠ 0) → ((𝑀‘∪ 𝑆) /𝑒 (𝑀‘∪ 𝑆)) = 1) | |
28 | 25, 26, 27 | syl2anc 584 | . . . . 5 ⊢ ((𝑀‘∪ 𝑆) ∈ ℝ+ → ((𝑀‘∪ 𝑆) /𝑒 (𝑀‘∪ 𝑆)) = 1) |
29 | 28 | adantl 482 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → ((𝑀‘∪ 𝑆) /𝑒 (𝑀‘∪ 𝑆)) = 1) |
30 | 24, 29 | eqtrd 2778 | . . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆)))‘∪ 𝑆) = 1) |
31 | 11, 30 | eqtrid 2790 | . 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆)))‘∪ dom (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆)))) = 1) |
32 | elprob 32376 | . 2 ⊢ ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆))) ∈ Prob ↔ ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆))) ∈ ∪ ran measures ∧ ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆)))‘∪ dom (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆)))) = 1)) | |
33 | 9, 31, 32 | sylanbrc 583 | 1 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆))) ∈ Prob) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 Vcvv 3432 ∖ cdif 3884 ⊆ wss 3887 𝒫 cpw 4533 ∪ cuni 4839 class class class wbr 5074 ↦ cmpt 5157 dom cdm 5589 ran crn 5590 ‘cfv 6433 (class class class)co 7275 ωcom 7712 ≼ cdom 8731 ℝcr 10870 0cc0 10871 1c1 10872 ℝ+crp 12730 /𝑒 cxdiv 31191 sigAlgebracsiga 32076 measurescmeas 32163 Probcprb 32374 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-disj 5040 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 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 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-seq 13722 df-hash 14045 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-tset 16981 df-ple 16982 df-ds 16984 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-ordt 17212 df-xrs 17213 df-mre 17295 df-mrc 17296 df-acs 17298 df-ps 18284 df-tsr 18285 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-cntz 18923 df-cmn 19388 df-fbas 20594 df-fg 20595 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-ntr 22171 df-nei 22249 df-cn 22378 df-cnp 22379 df-haus 22466 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-tsms 23278 df-xdiv 31192 df-esum 31996 df-siga 32077 df-meas 32164 df-prob 32375 |
This theorem is referenced by: (None) |
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