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 5166 | . . . 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 31704 | . . . 4 ⊢ (𝜑 → 𝑋:∪ dom 𝑃⟶ℝ) |
5 | rrvadd.3 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (rRndVar‘𝑃)) | |
6 | 2, 5 | rrvvf 31704 | . . . 4 ⊢ (𝜑 → 𝑌:∪ dom 𝑃⟶ℝ) |
7 | 2 | unveldomd 31675 | . . . 4 ⊢ (𝜑 → ∪ dom 𝑃 ∈ dom 𝑃) |
8 | eqidd 2824 | . . . 4 ⊢ (𝜑 → (𝑎 ∈ ∪ dom 𝑃 ↦ 〈(𝑋‘𝑎), (𝑌‘𝑎)〉) = (𝑎 ∈ ∪ dom 𝑃 ↦ 〈(𝑋‘𝑎), (𝑌‘𝑎)〉)) | |
9 | eqidd 2824 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦))) | |
10 | 1, 4, 6, 7, 8, 9 | ofoprabco 30411 | . . 3 ⊢ (𝜑 → (𝑋 ∘f + 𝑌) = ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ∘ (𝑎 ∈ ∪ dom 𝑃 ↦ 〈(𝑋‘𝑎), (𝑌‘𝑎)〉))) |
11 | domprobsiga 31671 | . . . . 5 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra) | |
12 | 2, 11 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝑃 ∈ ∪ ran sigAlgebra) |
13 | brsigarn 31445 | . . . . . 6 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ) | |
14 | elrnsiga 31387 | . . . . . 6 ⊢ (𝔅ℝ ∈ (sigAlgebra‘ℝ) → 𝔅ℝ ∈ ∪ ran sigAlgebra) | |
15 | 13, 14 | mp1i 13 | . . . . 5 ⊢ (𝜑 → 𝔅ℝ ∈ ∪ ran sigAlgebra) |
16 | sxsiga 31452 | . . . . 5 ⊢ ((𝔅ℝ ∈ ∪ ran sigAlgebra ∧ 𝔅ℝ ∈ ∪ ran sigAlgebra) → (𝔅ℝ ×s 𝔅ℝ) ∈ ∪ ran sigAlgebra) | |
17 | 15, 15, 16 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝔅ℝ ×s 𝔅ℝ) ∈ ∪ ran sigAlgebra) |
18 | 2 | rrvmbfm 31702 | . . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ))) |
19 | 3, 18 | mpbid 234 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ)) |
20 | 2 | rrvmbfm 31702 | . . . . . 6 ⊢ (𝜑 → (𝑌 ∈ (rRndVar‘𝑃) ↔ 𝑌 ∈ (dom 𝑃MblFnM𝔅ℝ))) |
21 | 5, 20 | mpbid 234 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (dom 𝑃MblFnM𝔅ℝ)) |
22 | fveq2 6672 | . . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑋‘𝑎) = (𝑋‘𝑏)) | |
23 | fveq2 6672 | . . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑌‘𝑎) = (𝑌‘𝑏)) | |
24 | 22, 23 | opeq12d 4813 | . . . . . 6 ⊢ (𝑎 = 𝑏 → 〈(𝑋‘𝑎), (𝑌‘𝑎)〉 = 〈(𝑋‘𝑏), (𝑌‘𝑏)〉) |
25 | 24 | cbvmptv 5171 | . . . . 5 ⊢ (𝑎 ∈ ∪ dom 𝑃 ↦ 〈(𝑋‘𝑎), (𝑌‘𝑎)〉) = (𝑏 ∈ ∪ dom 𝑃 ↦ 〈(𝑋‘𝑏), (𝑌‘𝑏)〉) |
26 | 12, 15, 15, 19, 21, 25 | mbfmco2 31525 | . . . 4 ⊢ (𝜑 → (𝑎 ∈ ∪ dom 𝑃 ↦ 〈(𝑋‘𝑎), (𝑌‘𝑎)〉) ∈ (dom 𝑃MblFnM(𝔅ℝ ×s 𝔅ℝ))) |
27 | eqid 2823 | . . . . . . 7 ⊢ (topGen‘ran (,)) = (topGen‘ran (,)) | |
28 | 27 | raddcn 31174 | . . . . . 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 31550 | . . . . . 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 31443 | . . . . . 6 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
33 | 32 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝔅ℝ = (sigaGen‘(topGen‘ran (,)))) |
34 | 29, 31, 33 | cnmbfm 31523 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ∈ ((𝔅ℝ ×s 𝔅ℝ)MblFnM𝔅ℝ)) |
35 | 12, 17, 15, 26, 34 | mbfmco 31524 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ∘ (𝑎 ∈ ∪ dom 𝑃 ↦ 〈(𝑋‘𝑎), (𝑌‘𝑎)〉)) ∈ (dom 𝑃MblFnM𝔅ℝ)) |
36 | 10, 35 | eqeltrd 2915 | . 2 ⊢ (𝜑 → (𝑋 ∘f + 𝑌) ∈ (dom 𝑃MblFnM𝔅ℝ)) |
37 | 2 | rrvmbfm 31702 | . 2 ⊢ (𝜑 → ((𝑋 ∘f + 𝑌) ∈ (rRndVar‘𝑃) ↔ (𝑋 ∘f + 𝑌) ∈ (dom 𝑃MblFnM𝔅ℝ))) |
38 | 36, 37 | mpbird 259 | 1 ⊢ (𝜑 → (𝑋 ∘f + 𝑌) ∈ (rRndVar‘𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 〈cop 4575 ∪ cuni 4840 ↦ cmpt 5148 dom cdm 5557 ran crn 5558 ∘ ccom 5561 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 ∘f cof 7409 ℝcr 10538 + caddc 10542 (,)cioo 12741 topGenctg 16713 Cn ccn 21834 ×t ctx 22170 sigAlgebracsiga 31369 sigaGencsigagen 31399 𝔅ℝcbrsiga 31442 ×s csx 31449 MblFnMcmbfm 31510 Probcprb 31667 rRndVarcrrv 31700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-ac2 9887 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-omul 8109 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-dju 9332 df-card 9370 df-acn 9373 df-ac 9544 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14428 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-ef 15423 df-sin 15425 df-cos 15426 df-pi 15428 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-refld 20751 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-cmp 21997 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-fcls 22551 df-xms 22932 df-ms 22933 df-tms 22934 df-cncf 23488 df-cfil 23860 df-cmet 23862 df-cms 23940 df-limc 24466 df-dv 24467 df-log 25142 df-cxp 25143 df-logb 25345 df-esum 31289 df-siga 31370 df-sigagen 31400 df-brsiga 31443 df-sx 31450 df-meas 31457 df-mbfm 31511 df-prob 31668 df-rrv 31701 |
This theorem is referenced by: rrvsum 31714 |
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