Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrvsum | Structured version Visualization version GIF version |
Description: An indexed sum of random variables is a random variable. (Contributed by Thierry Arnoux, 22-May-2017.) |
Ref | Expression |
---|---|
rrvsum.1 | ⊢ (𝜑 → 𝑃 ∈ Prob) |
rrvsum.2 | ⊢ (𝜑 → 𝑋:ℕ⟶(rRndVar‘𝑃)) |
rrvsum.3 | ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑆 = (seq1( ∘𝑓 + , 𝑋)‘𝑁)) |
Ref | Expression |
---|---|
rrvsum | ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑆 ∈ (rRndVar‘𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrvsum.3 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑆 = (seq1( ∘𝑓 + , 𝑋)‘𝑁)) | |
2 | fveq2 6191 | . . . . . 6 ⊢ (𝑘 = 1 → (seq1( ∘𝑓 + , 𝑋)‘𝑘) = (seq1( ∘𝑓 + , 𝑋)‘1)) | |
3 | 2 | eleq1d 2686 | . . . . 5 ⊢ (𝑘 = 1 → ((seq1( ∘𝑓 + , 𝑋)‘𝑘) ∈ (rRndVar‘𝑃) ↔ (seq1( ∘𝑓 + , 𝑋)‘1) ∈ (rRndVar‘𝑃))) |
4 | 3 | imbi2d 330 | . . . 4 ⊢ (𝑘 = 1 → ((𝜑 → (seq1( ∘𝑓 + , 𝑋)‘𝑘) ∈ (rRndVar‘𝑃)) ↔ (𝜑 → (seq1( ∘𝑓 + , 𝑋)‘1) ∈ (rRndVar‘𝑃)))) |
5 | fveq2 6191 | . . . . . 6 ⊢ (𝑘 = 𝑛 → (seq1( ∘𝑓 + , 𝑋)‘𝑘) = (seq1( ∘𝑓 + , 𝑋)‘𝑛)) | |
6 | 5 | eleq1d 2686 | . . . . 5 ⊢ (𝑘 = 𝑛 → ((seq1( ∘𝑓 + , 𝑋)‘𝑘) ∈ (rRndVar‘𝑃) ↔ (seq1( ∘𝑓 + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃))) |
7 | 6 | imbi2d 330 | . . . 4 ⊢ (𝑘 = 𝑛 → ((𝜑 → (seq1( ∘𝑓 + , 𝑋)‘𝑘) ∈ (rRndVar‘𝑃)) ↔ (𝜑 → (seq1( ∘𝑓 + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃)))) |
8 | fveq2 6191 | . . . . . 6 ⊢ (𝑘 = (𝑛 + 1) → (seq1( ∘𝑓 + , 𝑋)‘𝑘) = (seq1( ∘𝑓 + , 𝑋)‘(𝑛 + 1))) | |
9 | 8 | eleq1d 2686 | . . . . 5 ⊢ (𝑘 = (𝑛 + 1) → ((seq1( ∘𝑓 + , 𝑋)‘𝑘) ∈ (rRndVar‘𝑃) ↔ (seq1( ∘𝑓 + , 𝑋)‘(𝑛 + 1)) ∈ (rRndVar‘𝑃))) |
10 | 9 | imbi2d 330 | . . . 4 ⊢ (𝑘 = (𝑛 + 1) → ((𝜑 → (seq1( ∘𝑓 + , 𝑋)‘𝑘) ∈ (rRndVar‘𝑃)) ↔ (𝜑 → (seq1( ∘𝑓 + , 𝑋)‘(𝑛 + 1)) ∈ (rRndVar‘𝑃)))) |
11 | fveq2 6191 | . . . . . 6 ⊢ (𝑘 = 𝑁 → (seq1( ∘𝑓 + , 𝑋)‘𝑘) = (seq1( ∘𝑓 + , 𝑋)‘𝑁)) | |
12 | 11 | eleq1d 2686 | . . . . 5 ⊢ (𝑘 = 𝑁 → ((seq1( ∘𝑓 + , 𝑋)‘𝑘) ∈ (rRndVar‘𝑃) ↔ (seq1( ∘𝑓 + , 𝑋)‘𝑁) ∈ (rRndVar‘𝑃))) |
13 | 12 | imbi2d 330 | . . . 4 ⊢ (𝑘 = 𝑁 → ((𝜑 → (seq1( ∘𝑓 + , 𝑋)‘𝑘) ∈ (rRndVar‘𝑃)) ↔ (𝜑 → (seq1( ∘𝑓 + , 𝑋)‘𝑁) ∈ (rRndVar‘𝑃)))) |
14 | 1z 11407 | . . . . . 6 ⊢ 1 ∈ ℤ | |
15 | seq1 12814 | . . . . . 6 ⊢ (1 ∈ ℤ → (seq1( ∘𝑓 + , 𝑋)‘1) = (𝑋‘1)) | |
16 | 14, 15 | ax-mp 5 | . . . . 5 ⊢ (seq1( ∘𝑓 + , 𝑋)‘1) = (𝑋‘1) |
17 | 1nn 11031 | . . . . . 6 ⊢ 1 ∈ ℕ | |
18 | rrvsum.2 | . . . . . . 7 ⊢ (𝜑 → 𝑋:ℕ⟶(rRndVar‘𝑃)) | |
19 | 18 | ffvelrnda 6359 | . . . . . 6 ⊢ ((𝜑 ∧ 1 ∈ ℕ) → (𝑋‘1) ∈ (rRndVar‘𝑃)) |
20 | 17, 19 | mpan2 707 | . . . . 5 ⊢ (𝜑 → (𝑋‘1) ∈ (rRndVar‘𝑃)) |
21 | 16, 20 | syl5eqel 2705 | . . . 4 ⊢ (𝜑 → (seq1( ∘𝑓 + , 𝑋)‘1) ∈ (rRndVar‘𝑃)) |
22 | seqp1 12816 | . . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘1) → (seq1( ∘𝑓 + , 𝑋)‘(𝑛 + 1)) = ((seq1( ∘𝑓 + , 𝑋)‘𝑛) ∘𝑓 + (𝑋‘(𝑛 + 1)))) | |
23 | nnuz 11723 | . . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1) | |
24 | 22, 23 | eleq2s 2719 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (seq1( ∘𝑓 + , 𝑋)‘(𝑛 + 1)) = ((seq1( ∘𝑓 + , 𝑋)‘𝑛) ∘𝑓 + (𝑋‘(𝑛 + 1)))) |
25 | 24 | ad2antlr 763 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (seq1( ∘𝑓 + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃)) → (seq1( ∘𝑓 + , 𝑋)‘(𝑛 + 1)) = ((seq1( ∘𝑓 + , 𝑋)‘𝑛) ∘𝑓 + (𝑋‘(𝑛 + 1)))) |
26 | rrvsum.1 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ Prob) | |
27 | 26 | ad2antrr 762 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (seq1( ∘𝑓 + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃)) → 𝑃 ∈ Prob) |
28 | simpr 477 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (seq1( ∘𝑓 + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃)) → (seq1( ∘𝑓 + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃)) | |
29 | peano2nn 11032 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ) | |
30 | 18 | ffvelrnda 6359 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ ℕ) → (𝑋‘(𝑛 + 1)) ∈ (rRndVar‘𝑃)) |
31 | 29, 30 | sylan2 491 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑋‘(𝑛 + 1)) ∈ (rRndVar‘𝑃)) |
32 | 31 | adantr 481 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (seq1( ∘𝑓 + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃)) → (𝑋‘(𝑛 + 1)) ∈ (rRndVar‘𝑃)) |
33 | 27, 28, 32 | rrvadd 30514 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (seq1( ∘𝑓 + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃)) → ((seq1( ∘𝑓 + , 𝑋)‘𝑛) ∘𝑓 + (𝑋‘(𝑛 + 1))) ∈ (rRndVar‘𝑃)) |
34 | 25, 33 | eqeltrd 2701 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (seq1( ∘𝑓 + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃)) → (seq1( ∘𝑓 + , 𝑋)‘(𝑛 + 1)) ∈ (rRndVar‘𝑃)) |
35 | 34 | ex 450 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((seq1( ∘𝑓 + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃) → (seq1( ∘𝑓 + , 𝑋)‘(𝑛 + 1)) ∈ (rRndVar‘𝑃))) |
36 | 35 | expcom 451 | . . . . 5 ⊢ (𝑛 ∈ ℕ → (𝜑 → ((seq1( ∘𝑓 + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃) → (seq1( ∘𝑓 + , 𝑋)‘(𝑛 + 1)) ∈ (rRndVar‘𝑃)))) |
37 | 36 | a2d 29 | . . . 4 ⊢ (𝑛 ∈ ℕ → ((𝜑 → (seq1( ∘𝑓 + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃)) → (𝜑 → (seq1( ∘𝑓 + , 𝑋)‘(𝑛 + 1)) ∈ (rRndVar‘𝑃)))) |
38 | 4, 7, 10, 13, 21, 37 | nnind 11038 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝜑 → (seq1( ∘𝑓 + , 𝑋)‘𝑁) ∈ (rRndVar‘𝑃))) |
39 | 38 | impcom 446 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (seq1( ∘𝑓 + , 𝑋)‘𝑁) ∈ (rRndVar‘𝑃)) |
40 | 1, 39 | eqeltrd 2701 | 1 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑆 ∈ (rRndVar‘𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘𝑓 cof 6895 1c1 9937 + caddc 9939 ℕcn 11020 ℤcz 11377 ℤ≥cuz 11687 seqcseq 12801 Probcprb 30469 rRndVarcrrv 30502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-ac2 9285 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-ac 8939 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-refld 19951 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-fcls 21745 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-cfil 23053 df-cmet 23055 df-cms 23132 df-limc 23630 df-dv 23631 df-log 24303 df-cxp 24304 df-logb 24503 df-esum 30090 df-siga 30171 df-sigagen 30202 df-brsiga 30245 df-sx 30252 df-meas 30259 df-mbfm 30313 df-prob 30470 df-rrv 30503 |
This theorem is referenced by: (None) |
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