rrvadd - Mathbox for Thierry Arnoux

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Theorem rrvadd 31710

Description: The sum of two random variables is a random variable. (Contributed by Thierry Arnoux, 4-Jun-2017.)

**Hypotheses**
Ref	Expression
rrvadd.1	⊢ (𝜑 → 𝑃 ∈ Prob)
rrvadd.2	⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
rrvadd.3	⊢ (𝜑 → 𝑌 ∈ (rRndVar‘𝑃))

**Assertion**
Ref	Expression
rrvadd	⊢ (𝜑 → (𝑋 ∘_f + 𝑌) ∈ (rRndVar‘𝑃))

Proof of Theorem rrvadd Dummy variables 𝑎 𝑏 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfmpt1 5164	. . . 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 31702	. . . 4 ⊢ (𝜑 → 𝑋:∪ dom 𝑃⟶ℝ)
5		rrvadd.3	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (rRndVar‘𝑃))
6	2, 5	rrvvf 31702	. . . 4 ⊢ (𝜑 → 𝑌:∪ dom 𝑃⟶ℝ)
7	2	unveldomd 31673	. . . 4 ⊢ (𝜑 → ∪ dom 𝑃 ∈ dom 𝑃)
8		eqidd 2822	. . . 4 ⊢ (𝜑 → (𝑎 ∈ ∪ dom 𝑃 ↦ ⟨(𝑋‘𝑎), (𝑌‘𝑎)⟩) = (𝑎 ∈ ∪ dom 𝑃 ↦ ⟨(𝑋‘𝑎), (𝑌‘𝑎)⟩))
9		eqidd 2822	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)))
10	1, 4, 6, 7, 8, 9	ofoprabco 30409	. . 3 ⊢ (𝜑 → (𝑋 ∘_f + 𝑌) = ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ∘ (𝑎 ∈ ∪ dom 𝑃 ↦ ⟨(𝑋‘𝑎), (𝑌‘𝑎)⟩)))
11		domprobsiga 31669	. . . . 5 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra)
12	2, 11	syl 17	. . . 4 ⊢ (𝜑 → dom 𝑃 ∈ ∪ ran sigAlgebra)
13		brsigarn 31443	. . . . . 6 ⊢ 𝔅_ℝ ∈ (sigAlgebra‘ℝ)
14		elrnsiga 31385	. . . . . 6 ⊢ (𝔅_ℝ ∈ (sigAlgebra‘ℝ) → 𝔅_ℝ ∈ ∪ ran sigAlgebra)
15	13, 14	mp1i 13	. . . . 5 ⊢ (𝜑 → 𝔅_ℝ ∈ ∪ ran sigAlgebra)
16		sxsiga 31450	. . . . 5 ⊢ ((𝔅_ℝ ∈ ∪ ran sigAlgebra ∧ 𝔅_ℝ ∈ ∪ ran sigAlgebra) → (𝔅_ℝ ×_s 𝔅_ℝ) ∈ ∪ ran sigAlgebra)
17	15, 15, 16	syl2anc 586	. . . 4 ⊢ (𝜑 → (𝔅_ℝ ×_s 𝔅_ℝ) ∈ ∪ ran sigAlgebra)
18	2	rrvmbfm 31700	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅_ℝ)))
19	3, 18	mpbid 234	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM𝔅_ℝ))
20	2	rrvmbfm 31700	. . . . . 6 ⊢ (𝜑 → (𝑌 ∈ (rRndVar‘𝑃) ↔ 𝑌 ∈ (dom 𝑃MblFnM𝔅_ℝ)))
21	5, 20	mpbid 234	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (dom 𝑃MblFnM𝔅_ℝ))
22		fveq2 6670	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑋‘𝑎) = (𝑋‘𝑏))
23		fveq2 6670	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑌‘𝑎) = (𝑌‘𝑏))
24	22, 23	opeq12d 4811	. . . . . 6 ⊢ (𝑎 = 𝑏 → ⟨(𝑋‘𝑎), (𝑌‘𝑎)⟩ = ⟨(𝑋‘𝑏), (𝑌‘𝑏)⟩)
25	24	cbvmptv 5169	. . . . 5 ⊢ (𝑎 ∈ ∪ dom 𝑃 ↦ ⟨(𝑋‘𝑎), (𝑌‘𝑎)⟩) = (𝑏 ∈ ∪ dom 𝑃 ↦ ⟨(𝑋‘𝑏), (𝑌‘𝑏)⟩)
26	12, 15, 15, 19, 21, 25	mbfmco2 31523	. . . 4 ⊢ (𝜑 → (𝑎 ∈ ∪ dom 𝑃 ↦ ⟨(𝑋‘𝑎), (𝑌‘𝑎)⟩) ∈ (dom 𝑃MblFnM(𝔅_ℝ ×_s 𝔅_ℝ)))
27		eqid 2821	. . . . . . 7 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
28	27	raddcn 31172	. . . . . 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 31548	. . . . . 6 ⊢ (𝔅_ℝ ×_s 𝔅_ℝ) = (sigaGen‘((topGen‘ran (,)) ×_t (topGen‘ran (,))))
31	30	a1i 11	. . . . 5 ⊢ (𝜑 → (𝔅_ℝ ×_s 𝔅_ℝ) = (sigaGen‘((topGen‘ran (,)) ×_t (topGen‘ran (,)))))
32		df-brsiga 31441	. . . . . 6 ⊢ 𝔅_ℝ = (sigaGen‘(topGen‘ran (,)))
33	32	a1i 11	. . . . 5 ⊢ (𝜑 → 𝔅_ℝ = (sigaGen‘(topGen‘ran (,))))
34	29, 31, 33	cnmbfm 31521	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ∈ ((𝔅_ℝ ×_s 𝔅_ℝ)MblFnM𝔅_ℝ))
35	12, 17, 15, 26, 34	mbfmco 31522	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ∘ (𝑎 ∈ ∪ dom 𝑃 ↦ ⟨(𝑋‘𝑎), (𝑌‘𝑎)⟩)) ∈ (dom 𝑃MblFnM𝔅_ℝ))
36	10, 35	eqeltrd 2913	. 2 ⊢ (𝜑 → (𝑋 ∘_f + 𝑌) ∈ (dom 𝑃MblFnM𝔅_ℝ))
37	2	rrvmbfm 31700	. 2 ⊢ (𝜑 → ((𝑋 ∘_f + 𝑌) ∈ (rRndVar‘𝑃) ↔ (𝑋 ∘_f + 𝑌) ∈ (dom 𝑃MblFnM𝔅_ℝ)))
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 4573 ∪ cuni 4838 ↦ cmpt 5146 dom cdm 5555 ran crn 5556 ∘ ccom 5559 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 ∘_f cof 7407 ℝcr 10536 + caddc 10540 (,)cioo 12739 topGenctg 16711 Cn ccn 21832 ×_t ctx 22168 sigAlgebracsiga 31367 sigaGencsigagen 31397 𝔅_ℝcbrsiga 31440 ×_s csx 31447 MblFnMcmbfm 31508 Probcprb 31665 rRndVarcrrv 31698

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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-ac2 9885 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-omul 8107 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-dju 9330 df-card 9368 df-acn 9371 df-ac 9542 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-sin 15423 df-cos 15424 df-pi 15426 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-refld 20749 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-cmp 21995 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-fcls 22549 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-cfil 23858 df-cmet 23860 df-cms 23938 df-limc 24464 df-dv 24465 df-log 25140 df-cxp 25141 df-logb 25343 df-esum 31287 df-siga 31368 df-sigagen 31398 df-brsiga 31441 df-sx 31448 df-meas 31455 df-mbfm 31509 df-prob 31666 df-rrv 31699

This theorem is referenced by: rrvsum 31712

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