rrvadd - Mathbox for Thierry Arnoux

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Theorem rrvadd 34609

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 5197	. . . 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 34601	. . . 4 ⊢ (𝜑 → 𝑋:∪ dom 𝑃⟶ℝ)
5		rrvadd.3	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (rRndVar‘𝑃))
6	2, 5	rrvvf 34601	. . . 4 ⊢ (𝜑 → 𝑌:∪ dom 𝑃⟶ℝ)
7	2	unveldomd 34572	. . . 4 ⊢ (𝜑 → ∪ dom 𝑃 ∈ dom 𝑃)
8		eqidd 2737	. . . 4 ⊢ (𝜑 → (𝑎 ∈ ∪ dom 𝑃 ↦ ⟨(𝑋‘𝑎), (𝑌‘𝑎)⟩) = (𝑎 ∈ ∪ dom 𝑃 ↦ ⟨(𝑋‘𝑎), (𝑌‘𝑎)⟩))
9		eqidd 2737	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)))
10	1, 4, 6, 7, 8, 9	ofoprabco 32742	. . 3 ⊢ (𝜑 → (𝑋 ∘_f + 𝑌) = ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ∘ (𝑎 ∈ ∪ dom 𝑃 ↦ ⟨(𝑋‘𝑎), (𝑌‘𝑎)⟩)))
11		domprobsiga 34568	. . . . 5 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra)
12	2, 11	syl 17	. . . 4 ⊢ (𝜑 → dom 𝑃 ∈ ∪ ran sigAlgebra)
13		brsigarn 34341	. . . . . 6 ⊢ 𝔅_ℝ ∈ (sigAlgebra‘ℝ)
14		elrnsiga 34283	. . . . . 6 ⊢ (𝔅_ℝ ∈ (sigAlgebra‘ℝ) → 𝔅_ℝ ∈ ∪ ran sigAlgebra)
15	13, 14	mp1i 13	. . . . 5 ⊢ (𝜑 → 𝔅_ℝ ∈ ∪ ran sigAlgebra)
16		sxsiga 34348	. . . . 5 ⊢ ((𝔅_ℝ ∈ ∪ ran sigAlgebra ∧ 𝔅_ℝ ∈ ∪ ran sigAlgebra) → (𝔅_ℝ ×_s 𝔅_ℝ) ∈ ∪ ran sigAlgebra)
17	15, 15, 16	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝔅_ℝ ×_s 𝔅_ℝ) ∈ ∪ ran sigAlgebra)
18	2	rrvmbfm 34599	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅_ℝ)))
19	3, 18	mpbid 232	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM𝔅_ℝ))
20	2	rrvmbfm 34599	. . . . . 6 ⊢ (𝜑 → (𝑌 ∈ (rRndVar‘𝑃) ↔ 𝑌 ∈ (dom 𝑃MblFnM𝔅_ℝ)))
21	5, 20	mpbid 232	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (dom 𝑃MblFnM𝔅_ℝ))
22		fveq2 6834	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑋‘𝑎) = (𝑋‘𝑏))
23		fveq2 6834	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑌‘𝑎) = (𝑌‘𝑏))
24	22, 23	opeq12d 4837	. . . . . 6 ⊢ (𝑎 = 𝑏 → ⟨(𝑋‘𝑎), (𝑌‘𝑎)⟩ = ⟨(𝑋‘𝑏), (𝑌‘𝑏)⟩)
25	24	cbvmptv 5202	. . . . 5 ⊢ (𝑎 ∈ ∪ dom 𝑃 ↦ ⟨(𝑋‘𝑎), (𝑌‘𝑎)⟩) = (𝑏 ∈ ∪ dom 𝑃 ↦ ⟨(𝑋‘𝑏), (𝑌‘𝑏)⟩)
26	12, 15, 15, 19, 21, 25	mbfmco2 34422	. . . 4 ⊢ (𝜑 → (𝑎 ∈ ∪ dom 𝑃 ↦ ⟨(𝑋‘𝑎), (𝑌‘𝑎)⟩) ∈ (dom 𝑃MblFnM(𝔅_ℝ ×_s 𝔅_ℝ)))
27		eqid 2736	. . . . . . 7 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
28	27	raddcn 34086	. . . . . 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 34447	. . . . . 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 34339	. . . . . 6 ⊢ 𝔅_ℝ = (sigaGen‘(topGen‘ran (,)))
33	32	a1i 11	. . . . 5 ⊢ (𝜑 → 𝔅_ℝ = (sigaGen‘(topGen‘ran (,))))
34	29, 31, 33	cnmbfm 34420	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ∈ ((𝔅_ℝ ×_s 𝔅_ℝ)MblFnM𝔅_ℝ))
35	12, 17, 15, 26, 34	mbfmco 34421	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ∘ (𝑎 ∈ ∪ dom 𝑃 ↦ ⟨(𝑋‘𝑎), (𝑌‘𝑎)⟩)) ∈ (dom 𝑃MblFnM𝔅_ℝ))
36	10, 35	eqeltrd 2836	. 2 ⊢ (𝜑 → (𝑋 ∘_f + 𝑌) ∈ (dom 𝑃MblFnM𝔅_ℝ))
37	2	rrvmbfm 34599	. 2 ⊢ (𝜑 → ((𝑋 ∘_f + 𝑌) ∈ (rRndVar‘𝑃) ↔ (𝑋 ∘_f + 𝑌) ∈ (dom 𝑃MblFnM𝔅_ℝ)))
38	36, 37	mpbird 257	1 ⊢ (𝜑 → (𝑋 ∘_f + 𝑌) ∈ (rRndVar‘𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2113 ⟨cop 4586 ∪ cuni 4863 ↦ cmpt 5179 dom cdm 5624 ran crn 5625 ∘ ccom 5628 ‘cfv 6492 (class class class)co 7358 ∈ cmpo 7360 ∘_f cof 7620 ℝcr 11025 + caddc 11029 (,)cioo 13261 topGenctg 17357 Cn ccn 23168 ×_t ctx 23504 sigAlgebracsiga 34265 sigaGencsigagen 34295 𝔅_ℝcbrsiga 34338 ×_s csx 34345 MblFnMcmbfm 34406 Probcprb 34564 rRndVarcrrv 34597

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-ac2 10373 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-omul 8402 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9813 df-card 9851 df-acn 9854 df-ac 10026 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ioc 13266 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-fac 14197 df-bc 14226 df-hash 14254 df-shft 14990 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-limsup 15394 df-clim 15411 df-rlim 15412 df-sum 15610 df-ef 15990 df-sin 15992 df-cos 15993 df-pi 15995 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-refld 21560 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cn 23171 df-cnp 23172 df-haus 23259 df-cmp 23331 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-fcls 23885 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-cfil 25211 df-cmet 25213 df-cms 25291 df-limc 25823 df-dv 25824 df-log 26521 df-cxp 26522 df-logb 26731 df-esum 34185 df-siga 34266 df-sigagen 34296 df-brsiga 34339 df-sx 34346 df-meas 34353 df-mbfm 34407 df-prob 34565 df-rrv 34598

This theorem is referenced by: rrvsum 34611

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