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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrvadd | Structured version Visualization version GIF version |
Description: The sum of two random variables is a random variable. (Contributed by Thierry Arnoux, 4-Jun-2017.) |
Ref | Expression |
---|---|
rrvadd.1 | ⊢ (𝜑 → 𝑃 ∈ Prob) |
rrvadd.2 | ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) |
rrvadd.3 | ⊢ (𝜑 → 𝑌 ∈ (rRndVar‘𝑃)) |
Ref | Expression |
---|---|
rrvadd | ⊢ (𝜑 → (𝑋 ∘f + 𝑌) ∈ (rRndVar‘𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfmpt1 5246 | . . . 4 ⊢ Ⅎ𝑎(𝑎 ∈ ∪ dom 𝑃 ↦ 〈(𝑋‘𝑎), (𝑌‘𝑎)〉) | |
2 | rrvadd.1 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ Prob) | |
3 | rrvadd.2 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) | |
4 | 2, 3 | rrvvf 33932 | . . . 4 ⊢ (𝜑 → 𝑋:∪ dom 𝑃⟶ℝ) |
5 | rrvadd.3 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (rRndVar‘𝑃)) | |
6 | 2, 5 | rrvvf 33932 | . . . 4 ⊢ (𝜑 → 𝑌:∪ dom 𝑃⟶ℝ) |
7 | 2 | unveldomd 33903 | . . . 4 ⊢ (𝜑 → ∪ dom 𝑃 ∈ dom 𝑃) |
8 | eqidd 2725 | . . . 4 ⊢ (𝜑 → (𝑎 ∈ ∪ dom 𝑃 ↦ 〈(𝑋‘𝑎), (𝑌‘𝑎)〉) = (𝑎 ∈ ∪ dom 𝑃 ↦ 〈(𝑋‘𝑎), (𝑌‘𝑎)〉)) | |
9 | eqidd 2725 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦))) | |
10 | 1, 4, 6, 7, 8, 9 | ofoprabco 32358 | . . 3 ⊢ (𝜑 → (𝑋 ∘f + 𝑌) = ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ∘ (𝑎 ∈ ∪ dom 𝑃 ↦ 〈(𝑋‘𝑎), (𝑌‘𝑎)〉))) |
11 | domprobsiga 33899 | . . . . 5 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra) | |
12 | 2, 11 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝑃 ∈ ∪ ran sigAlgebra) |
13 | brsigarn 33671 | . . . . . 6 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ) | |
14 | elrnsiga 33613 | . . . . . 6 ⊢ (𝔅ℝ ∈ (sigAlgebra‘ℝ) → 𝔅ℝ ∈ ∪ ran sigAlgebra) | |
15 | 13, 14 | mp1i 13 | . . . . 5 ⊢ (𝜑 → 𝔅ℝ ∈ ∪ ran sigAlgebra) |
16 | sxsiga 33678 | . . . . 5 ⊢ ((𝔅ℝ ∈ ∪ ran sigAlgebra ∧ 𝔅ℝ ∈ ∪ ran sigAlgebra) → (𝔅ℝ ×s 𝔅ℝ) ∈ ∪ ran sigAlgebra) | |
17 | 15, 15, 16 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (𝔅ℝ ×s 𝔅ℝ) ∈ ∪ ran sigAlgebra) |
18 | 2 | rrvmbfm 33930 | . . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ))) |
19 | 3, 18 | mpbid 231 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ)) |
20 | 2 | rrvmbfm 33930 | . . . . . 6 ⊢ (𝜑 → (𝑌 ∈ (rRndVar‘𝑃) ↔ 𝑌 ∈ (dom 𝑃MblFnM𝔅ℝ))) |
21 | 5, 20 | mpbid 231 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (dom 𝑃MblFnM𝔅ℝ)) |
22 | fveq2 6881 | . . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑋‘𝑎) = (𝑋‘𝑏)) | |
23 | fveq2 6881 | . . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑌‘𝑎) = (𝑌‘𝑏)) | |
24 | 22, 23 | opeq12d 4873 | . . . . . 6 ⊢ (𝑎 = 𝑏 → 〈(𝑋‘𝑎), (𝑌‘𝑎)〉 = 〈(𝑋‘𝑏), (𝑌‘𝑏)〉) |
25 | 24 | cbvmptv 5251 | . . . . 5 ⊢ (𝑎 ∈ ∪ dom 𝑃 ↦ 〈(𝑋‘𝑎), (𝑌‘𝑎)〉) = (𝑏 ∈ ∪ dom 𝑃 ↦ 〈(𝑋‘𝑏), (𝑌‘𝑏)〉) |
26 | 12, 15, 15, 19, 21, 25 | mbfmco2 33753 | . . . 4 ⊢ (𝜑 → (𝑎 ∈ ∪ dom 𝑃 ↦ 〈(𝑋‘𝑎), (𝑌‘𝑎)〉) ∈ (dom 𝑃MblFnM(𝔅ℝ ×s 𝔅ℝ))) |
27 | eqid 2724 | . . . . . . 7 ⊢ (topGen‘ran (,)) = (topGen‘ran (,)) | |
28 | 27 | raddcn 33398 | . . . . . 6 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ∈ (((topGen‘ran (,)) ×t (topGen‘ran (,))) Cn (topGen‘ran (,))) |
29 | 28 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ∈ (((topGen‘ran (,)) ×t (topGen‘ran (,))) Cn (topGen‘ran (,)))) |
30 | 27 | sxbrsiga 33778 | . . . . . 6 ⊢ (𝔅ℝ ×s 𝔅ℝ) = (sigaGen‘((topGen‘ran (,)) ×t (topGen‘ran (,)))) |
31 | 30 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝔅ℝ ×s 𝔅ℝ) = (sigaGen‘((topGen‘ran (,)) ×t (topGen‘ran (,))))) |
32 | df-brsiga 33669 | . . . . . 6 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
33 | 32 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝔅ℝ = (sigaGen‘(topGen‘ran (,)))) |
34 | 29, 31, 33 | cnmbfm 33751 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ∈ ((𝔅ℝ ×s 𝔅ℝ)MblFnM𝔅ℝ)) |
35 | 12, 17, 15, 26, 34 | mbfmco 33752 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ∘ (𝑎 ∈ ∪ dom 𝑃 ↦ 〈(𝑋‘𝑎), (𝑌‘𝑎)〉)) ∈ (dom 𝑃MblFnM𝔅ℝ)) |
36 | 10, 35 | eqeltrd 2825 | . 2 ⊢ (𝜑 → (𝑋 ∘f + 𝑌) ∈ (dom 𝑃MblFnM𝔅ℝ)) |
37 | 2 | rrvmbfm 33930 | . 2 ⊢ (𝜑 → ((𝑋 ∘f + 𝑌) ∈ (rRndVar‘𝑃) ↔ (𝑋 ∘f + 𝑌) ∈ (dom 𝑃MblFnM𝔅ℝ))) |
38 | 36, 37 | mpbird 257 | 1 ⊢ (𝜑 → (𝑋 ∘f + 𝑌) ∈ (rRndVar‘𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 〈cop 4626 ∪ cuni 4899 ↦ cmpt 5221 dom cdm 5666 ran crn 5667 ∘ ccom 5670 ‘cfv 6533 (class class class)co 7401 ∈ cmpo 7403 ∘f cof 7661 ℝcr 11105 + caddc 11109 (,)cioo 13321 topGenctg 17382 Cn ccn 23050 ×t ctx 23386 sigAlgebracsiga 33595 sigaGencsigagen 33625 𝔅ℝcbrsiga 33668 ×s csx 33675 MblFnMcmbfm 33736 Probcprb 33895 rRndVarcrrv 33928 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 ax-ac2 10454 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-omul 8466 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-acn 9933 df-ac 10107 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ioc 13326 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15011 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-limsup 15412 df-clim 15429 df-rlim 15430 df-sum 15630 df-ef 16008 df-sin 16010 df-cos 16011 df-pi 16013 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18704 df-mulg 18986 df-cntz 19223 df-cmn 19692 df-psmet 21220 df-xmet 21221 df-met 21222 df-bl 21223 df-mopn 21224 df-fbas 21225 df-fg 21226 df-cnfld 21229 df-refld 21466 df-top 22718 df-topon 22735 df-topsp 22757 df-bases 22771 df-cld 22845 df-ntr 22846 df-cls 22847 df-nei 22924 df-lp 22962 df-perf 22963 df-cn 23053 df-cnp 23054 df-haus 23141 df-cmp 23213 df-tx 23388 df-hmeo 23581 df-fil 23672 df-fm 23764 df-flim 23765 df-flf 23766 df-fcls 23767 df-xms 24148 df-ms 24149 df-tms 24150 df-cncf 24720 df-cfil 25105 df-cmet 25107 df-cms 25185 df-limc 25717 df-dv 25718 df-log 26407 df-cxp 26408 df-logb 26613 df-esum 33515 df-siga 33596 df-sigagen 33626 df-brsiga 33669 df-sx 33676 df-meas 33683 df-mbfm 33737 df-prob 33896 df-rrv 33929 |
This theorem is referenced by: rrvsum 33942 |
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