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 4675 | . . . 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 29833 | . . . 4 ⊢ (𝜑 → 𝑋:∪ dom 𝑃⟶ℝ) |
5 | rrvadd.3 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (rRndVar‘𝑃)) | |
6 | 2, 5 | rrvvf 29833 | . . . 4 ⊢ (𝜑 → 𝑌:∪ dom 𝑃⟶ℝ) |
7 | 2 | unveldomd 29804 | . . . 4 ⊢ (𝜑 → ∪ dom 𝑃 ∈ dom 𝑃) |
8 | eqidd 2611 | . . . 4 ⊢ (𝜑 → (𝑎 ∈ ∪ dom 𝑃 ↦ 〈(𝑋‘𝑎), (𝑌‘𝑎)〉) = (𝑎 ∈ ∪ dom 𝑃 ↦ 〈(𝑋‘𝑎), (𝑌‘𝑎)〉)) | |
9 | eqidd 2611 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦))) | |
10 | 1, 4, 6, 7, 8, 9 | ofoprabco 28847 | . . 3 ⊢ (𝜑 → (𝑋 ∘𝑓 + 𝑌) = ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ∘ (𝑎 ∈ ∪ dom 𝑃 ↦ 〈(𝑋‘𝑎), (𝑌‘𝑎)〉))) |
11 | domprobsiga 29800 | . . . . 5 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra) | |
12 | 2, 11 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝑃 ∈ ∪ ran sigAlgebra) |
13 | brsigarn 29574 | . . . . . 6 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ) | |
14 | elrnsiga 29516 | . . . . . 6 ⊢ (𝔅ℝ ∈ (sigAlgebra‘ℝ) → 𝔅ℝ ∈ ∪ ran sigAlgebra) | |
15 | 13, 14 | mp1i 13 | . . . . 5 ⊢ (𝜑 → 𝔅ℝ ∈ ∪ ran sigAlgebra) |
16 | sxsiga 29581 | . . . . 5 ⊢ ((𝔅ℝ ∈ ∪ ran sigAlgebra ∧ 𝔅ℝ ∈ ∪ ran sigAlgebra) → (𝔅ℝ ×s 𝔅ℝ) ∈ ∪ ran sigAlgebra) | |
17 | 15, 15, 16 | syl2anc 691 | . . . 4 ⊢ (𝜑 → (𝔅ℝ ×s 𝔅ℝ) ∈ ∪ ran sigAlgebra) |
18 | 2 | rrvmbfm 29831 | . . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ))) |
19 | 3, 18 | mpbid 221 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ)) |
20 | 2 | rrvmbfm 29831 | . . . . . 6 ⊢ (𝜑 → (𝑌 ∈ (rRndVar‘𝑃) ↔ 𝑌 ∈ (dom 𝑃MblFnM𝔅ℝ))) |
21 | 5, 20 | mpbid 221 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (dom 𝑃MblFnM𝔅ℝ)) |
22 | fveq2 6103 | . . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑋‘𝑎) = (𝑋‘𝑏)) | |
23 | fveq2 6103 | . . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑌‘𝑎) = (𝑌‘𝑏)) | |
24 | 22, 23 | opeq12d 4348 | . . . . . 6 ⊢ (𝑎 = 𝑏 → 〈(𝑋‘𝑎), (𝑌‘𝑎)〉 = 〈(𝑋‘𝑏), (𝑌‘𝑏)〉) |
25 | 24 | cbvmptv 4678 | . . . . 5 ⊢ (𝑎 ∈ ∪ dom 𝑃 ↦ 〈(𝑋‘𝑎), (𝑌‘𝑎)〉) = (𝑏 ∈ ∪ dom 𝑃 ↦ 〈(𝑋‘𝑏), (𝑌‘𝑏)〉) |
26 | 12, 15, 15, 19, 21, 25 | mbfmco2 29654 | . . . 4 ⊢ (𝜑 → (𝑎 ∈ ∪ dom 𝑃 ↦ 〈(𝑋‘𝑎), (𝑌‘𝑎)〉) ∈ (dom 𝑃MblFnM(𝔅ℝ ×s 𝔅ℝ))) |
27 | eqid 2610 | . . . . . . 7 ⊢ (topGen‘ran (,)) = (topGen‘ran (,)) | |
28 | 27 | raddcn 29303 | . . . . . 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 29679 | . . . . . 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 29572 | . . . . . 6 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
33 | 32 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝔅ℝ = (sigaGen‘(topGen‘ran (,)))) |
34 | 29, 31, 33 | cnmbfm 29652 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ∈ ((𝔅ℝ ×s 𝔅ℝ)MblFnM𝔅ℝ)) |
35 | 12, 17, 15, 26, 34 | mbfmco 29653 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ∘ (𝑎 ∈ ∪ dom 𝑃 ↦ 〈(𝑋‘𝑎), (𝑌‘𝑎)〉)) ∈ (dom 𝑃MblFnM𝔅ℝ)) |
36 | 10, 35 | eqeltrd 2688 | . 2 ⊢ (𝜑 → (𝑋 ∘𝑓 + 𝑌) ∈ (dom 𝑃MblFnM𝔅ℝ)) |
37 | 2 | rrvmbfm 29831 | . 2 ⊢ (𝜑 → ((𝑋 ∘𝑓 + 𝑌) ∈ (rRndVar‘𝑃) ↔ (𝑋 ∘𝑓 + 𝑌) ∈ (dom 𝑃MblFnM𝔅ℝ))) |
38 | 36, 37 | mpbird 246 | 1 ⊢ (𝜑 → (𝑋 ∘𝑓 + 𝑌) ∈ (rRndVar‘𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 〈cop 4131 ∪ cuni 4372 ↦ cmpt 4643 dom cdm 5038 ran crn 5039 ∘ ccom 5042 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ∘𝑓 cof 6793 ℝcr 9814 + caddc 9818 (,)cioo 12046 topGenctg 15921 Cn ccn 20838 ×t ctx 21173 sigAlgebracsiga 29497 sigaGencsigagen 29528 𝔅ℝcbrsiga 29571 ×s csx 29578 MblFnMcmbfm 29639 Probcprb 29796 rRndVarcrrv 29829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-ac2 9168 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-ac 8822 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-refld 19770 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-cmp 21000 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-fcls 21555 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-cfil 22861 df-cmet 22863 df-cms 22940 df-limc 23436 df-dv 23437 df-log 24107 df-cxp 24108 df-logb 24303 df-esum 29417 df-siga 29498 df-sigagen 29529 df-brsiga 29572 df-sx 29579 df-meas 29586 df-mbfm 29640 df-prob 29797 df-rrv 29830 |
This theorem is referenced by: rrvsum 29843 |
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