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Mirrors > Home > MPE Home > Th. List > Mathboxes > probfinmeasbALTV | Structured version Visualization version GIF version |
Description: Alternate version of probfinmeasb 34410. (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 34206 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆))) ∈ (measures‘𝑆)) | |
2 | ovex 7464 | . . . . . . 7 ⊢ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆)) ∈ V | |
3 | 2 | rgenw 3063 | . . . . . 6 ⊢ ∀𝑥 ∈ 𝑆 ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆)) ∈ V |
4 | dmmptg 6264 | . . . . . 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 6910 | . . . 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 34183 | . . 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 234 | . 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆))) ∈ ∪ ran measures) |
10 | 5 | unieqi 4924 | . . . 4 ⊢ ∪ dom (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆))) = ∪ 𝑆 |
11 | 10 | fveq2i 6910 | . . 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 34178 | . . . . . . 7 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra) | |
13 | isrnsigau 34108 | . . . . . . . . 9 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))) | |
14 | 13 | simprd 495 | . . . . . . . 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 32901 | . . . . . 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 6907 | . . . . . . 7 ⊢ (𝑥 = ∪ 𝑆 → (𝑀‘𝑥) = (𝑀‘∪ 𝑆)) | |
21 | 20 | oveq1d 7446 | . . . . . 6 ⊢ (𝑥 = ∪ 𝑆 → ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆)) = ((𝑀‘∪ 𝑆) /𝑒 (𝑀‘∪ 𝑆))) |
22 | eqid 2735 | . . . . . 6 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆))) = (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆))) | |
23 | 21, 22 | fvmptg 7014 | . . . . 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 13041 | . . . . . 6 ⊢ ((𝑀‘∪ 𝑆) ∈ ℝ+ → (𝑀‘∪ 𝑆) ∈ ℝ) | |
26 | rpne0 13049 | . . . . . 6 ⊢ ((𝑀‘∪ 𝑆) ∈ ℝ+ → (𝑀‘∪ 𝑆) ≠ 0) | |
27 | xdivid 32895 | . . . . . 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 481 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → ((𝑀‘∪ 𝑆) /𝑒 (𝑀‘∪ 𝑆)) = 1) |
30 | 24, 29 | eqtrd 2775 | . . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆)))‘∪ 𝑆) = 1) |
31 | 11, 30 | eqtrid 2787 | . 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 34391 | . 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 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 Vcvv 3478 ∖ cdif 3960 ⊆ wss 3963 𝒫 cpw 4605 ∪ cuni 4912 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 ran crn 5690 ‘cfv 6563 (class class class)co 7431 ωcom 7887 ≼ cdom 8982 ℝcr 11152 0cc0 11153 1c1 11154 ℝ+crp 13032 /𝑒 cxdiv 32884 sigAlgebracsiga 34089 measurescmeas 34176 Probcprb 34389 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-tset 17317 df-ple 17318 df-ds 17320 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-ordt 17548 df-xrs 17549 df-mre 17631 df-mrc 17632 df-acs 17634 df-ps 18624 df-tsr 18625 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-cntz 19348 df-cmn 19815 df-fbas 21379 df-fg 21380 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-ntr 23044 df-nei 23122 df-cn 23251 df-cnp 23252 df-haus 23339 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-tsms 24151 df-xdiv 32885 df-esum 34009 df-siga 34090 df-meas 34177 df-prob 34390 |
This theorem is referenced by: (None) |
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