![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 6493 | . . . . . 6 ⊢ (𝑘 = 1 → (seq1( ∘𝑓 + , 𝑋)‘𝑘) = (seq1( ∘𝑓 + , 𝑋)‘1)) | |
3 | 2 | eleq1d 2844 | . . . . 5 ⊢ (𝑘 = 1 → ((seq1( ∘𝑓 + , 𝑋)‘𝑘) ∈ (rRndVar‘𝑃) ↔ (seq1( ∘𝑓 + , 𝑋)‘1) ∈ (rRndVar‘𝑃))) |
4 | 3 | imbi2d 333 | . . . 4 ⊢ (𝑘 = 1 → ((𝜑 → (seq1( ∘𝑓 + , 𝑋)‘𝑘) ∈ (rRndVar‘𝑃)) ↔ (𝜑 → (seq1( ∘𝑓 + , 𝑋)‘1) ∈ (rRndVar‘𝑃)))) |
5 | fveq2 6493 | . . . . . 6 ⊢ (𝑘 = 𝑛 → (seq1( ∘𝑓 + , 𝑋)‘𝑘) = (seq1( ∘𝑓 + , 𝑋)‘𝑛)) | |
6 | 5 | eleq1d 2844 | . . . . 5 ⊢ (𝑘 = 𝑛 → ((seq1( ∘𝑓 + , 𝑋)‘𝑘) ∈ (rRndVar‘𝑃) ↔ (seq1( ∘𝑓 + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃))) |
7 | 6 | imbi2d 333 | . . . 4 ⊢ (𝑘 = 𝑛 → ((𝜑 → (seq1( ∘𝑓 + , 𝑋)‘𝑘) ∈ (rRndVar‘𝑃)) ↔ (𝜑 → (seq1( ∘𝑓 + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃)))) |
8 | fveq2 6493 | . . . . . 6 ⊢ (𝑘 = (𝑛 + 1) → (seq1( ∘𝑓 + , 𝑋)‘𝑘) = (seq1( ∘𝑓 + , 𝑋)‘(𝑛 + 1))) | |
9 | 8 | eleq1d 2844 | . . . . 5 ⊢ (𝑘 = (𝑛 + 1) → ((seq1( ∘𝑓 + , 𝑋)‘𝑘) ∈ (rRndVar‘𝑃) ↔ (seq1( ∘𝑓 + , 𝑋)‘(𝑛 + 1)) ∈ (rRndVar‘𝑃))) |
10 | 9 | imbi2d 333 | . . . 4 ⊢ (𝑘 = (𝑛 + 1) → ((𝜑 → (seq1( ∘𝑓 + , 𝑋)‘𝑘) ∈ (rRndVar‘𝑃)) ↔ (𝜑 → (seq1( ∘𝑓 + , 𝑋)‘(𝑛 + 1)) ∈ (rRndVar‘𝑃)))) |
11 | fveq2 6493 | . . . . . 6 ⊢ (𝑘 = 𝑁 → (seq1( ∘𝑓 + , 𝑋)‘𝑘) = (seq1( ∘𝑓 + , 𝑋)‘𝑁)) | |
12 | 11 | eleq1d 2844 | . . . . 5 ⊢ (𝑘 = 𝑁 → ((seq1( ∘𝑓 + , 𝑋)‘𝑘) ∈ (rRndVar‘𝑃) ↔ (seq1( ∘𝑓 + , 𝑋)‘𝑁) ∈ (rRndVar‘𝑃))) |
13 | 12 | imbi2d 333 | . . . 4 ⊢ (𝑘 = 𝑁 → ((𝜑 → (seq1( ∘𝑓 + , 𝑋)‘𝑘) ∈ (rRndVar‘𝑃)) ↔ (𝜑 → (seq1( ∘𝑓 + , 𝑋)‘𝑁) ∈ (rRndVar‘𝑃)))) |
14 | 1z 11819 | . . . . . 6 ⊢ 1 ∈ ℤ | |
15 | seq1 13191 | . . . . . 6 ⊢ (1 ∈ ℤ → (seq1( ∘𝑓 + , 𝑋)‘1) = (𝑋‘1)) | |
16 | 14, 15 | ax-mp 5 | . . . . 5 ⊢ (seq1( ∘𝑓 + , 𝑋)‘1) = (𝑋‘1) |
17 | 1nn 11446 | . . . . . 6 ⊢ 1 ∈ ℕ | |
18 | rrvsum.2 | . . . . . . 7 ⊢ (𝜑 → 𝑋:ℕ⟶(rRndVar‘𝑃)) | |
19 | 18 | ffvelrnda 6670 | . . . . . 6 ⊢ ((𝜑 ∧ 1 ∈ ℕ) → (𝑋‘1) ∈ (rRndVar‘𝑃)) |
20 | 17, 19 | mpan2 678 | . . . . 5 ⊢ (𝜑 → (𝑋‘1) ∈ (rRndVar‘𝑃)) |
21 | 16, 20 | syl5eqel 2864 | . . . 4 ⊢ (𝜑 → (seq1( ∘𝑓 + , 𝑋)‘1) ∈ (rRndVar‘𝑃)) |
22 | seqp1 13193 | . . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘1) → (seq1( ∘𝑓 + , 𝑋)‘(𝑛 + 1)) = ((seq1( ∘𝑓 + , 𝑋)‘𝑛) ∘𝑓 + (𝑋‘(𝑛 + 1)))) | |
23 | nnuz 12089 | . . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1) | |
24 | 22, 23 | eleq2s 2878 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (seq1( ∘𝑓 + , 𝑋)‘(𝑛 + 1)) = ((seq1( ∘𝑓 + , 𝑋)‘𝑛) ∘𝑓 + (𝑋‘(𝑛 + 1)))) |
25 | 24 | ad2antlr 714 | . . . . . . . 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 713 | . . . . . . . . 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 11447 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ) | |
30 | 18 | ffvelrnda 6670 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ ℕ) → (𝑋‘(𝑛 + 1)) ∈ (rRndVar‘𝑃)) |
31 | 29, 30 | sylan2 583 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑋‘(𝑛 + 1)) ∈ (rRndVar‘𝑃)) |
32 | 31 | adantr 473 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (seq1( ∘𝑓 + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃)) → (𝑋‘(𝑛 + 1)) ∈ (rRndVar‘𝑃)) |
33 | 27, 28, 32 | rrvadd 31356 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (seq1( ∘𝑓 + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃)) → ((seq1( ∘𝑓 + , 𝑋)‘𝑛) ∘𝑓 + (𝑋‘(𝑛 + 1))) ∈ (rRndVar‘𝑃)) |
34 | 25, 33 | eqeltrd 2860 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (seq1( ∘𝑓 + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃)) → (seq1( ∘𝑓 + , 𝑋)‘(𝑛 + 1)) ∈ (rRndVar‘𝑃)) |
35 | 34 | ex 405 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((seq1( ∘𝑓 + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃) → (seq1( ∘𝑓 + , 𝑋)‘(𝑛 + 1)) ∈ (rRndVar‘𝑃))) |
36 | 35 | expcom 406 | . . . . 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 11453 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝜑 → (seq1( ∘𝑓 + , 𝑋)‘𝑁) ∈ (rRndVar‘𝑃))) |
39 | 38 | impcom 399 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (seq1( ∘𝑓 + , 𝑋)‘𝑁) ∈ (rRndVar‘𝑃)) |
40 | 1, 39 | eqeltrd 2860 | 1 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑆 ∈ (rRndVar‘𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ⟶wf 6178 ‘cfv 6182 (class class class)co 6970 ∘𝑓 cof 7219 1c1 10330 + caddc 10332 ℕcn 11433 ℤcz 11787 ℤ≥cuz 12052 seqcseq 13178 Probcprb 31311 rRndVarcrrv 31344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-inf2 8892 ax-ac2 9677 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 ax-pre-sup 10407 ax-addf 10408 ax-mulf 10409 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-se 5361 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-om 7391 df-1st 7495 df-2nd 7496 df-supp 7628 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-2o 7900 df-oadd 7903 df-omul 7904 df-er 8083 df-map 8202 df-pm 8203 df-ixp 8254 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-fsupp 8623 df-fi 8664 df-sup 8695 df-inf 8696 df-oi 8763 df-dju 9118 df-card 9156 df-acn 9159 df-ac 9330 df-cda 9382 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-div 11093 df-nn 11434 df-2 11497 df-3 11498 df-4 11499 df-5 11500 df-6 11501 df-7 11502 df-8 11503 df-9 11504 df-n0 11702 df-z 11788 df-dec 11906 df-uz 12053 df-q 12157 df-rp 12199 df-xneg 12318 df-xadd 12319 df-xmul 12320 df-ioo 12552 df-ioc 12553 df-ico 12554 df-icc 12555 df-fz 12703 df-fzo 12844 df-fl 12971 df-mod 13047 df-seq 13179 df-exp 13239 df-fac 13443 df-bc 13472 df-hash 13500 df-shft 14281 df-cj 14313 df-re 14314 df-im 14315 df-sqrt 14449 df-abs 14450 df-limsup 14683 df-clim 14700 df-rlim 14701 df-sum 14898 df-ef 15275 df-sin 15277 df-cos 15278 df-pi 15280 df-struct 16335 df-ndx 16336 df-slot 16337 df-base 16339 df-sets 16340 df-ress 16341 df-plusg 16428 df-mulr 16429 df-starv 16430 df-sca 16431 df-vsca 16432 df-ip 16433 df-tset 16434 df-ple 16435 df-ds 16437 df-unif 16438 df-hom 16439 df-cco 16440 df-rest 16546 df-topn 16547 df-0g 16565 df-gsum 16566 df-topgen 16567 df-pt 16568 df-prds 16571 df-xrs 16625 df-qtop 16630 df-imas 16631 df-xps 16633 df-mre 16709 df-mrc 16710 df-acs 16712 df-mgm 17704 df-sgrp 17746 df-mnd 17757 df-submnd 17798 df-mulg 18006 df-cntz 18212 df-cmn 18662 df-psmet 20233 df-xmet 20234 df-met 20235 df-bl 20236 df-mopn 20237 df-fbas 20238 df-fg 20239 df-cnfld 20242 df-refld 20445 df-top 21200 df-topon 21217 df-topsp 21239 df-bases 21252 df-cld 21325 df-ntr 21326 df-cls 21327 df-nei 21404 df-lp 21442 df-perf 21443 df-cn 21533 df-cnp 21534 df-haus 21621 df-cmp 21693 df-tx 21868 df-hmeo 22061 df-fil 22152 df-fm 22244 df-flim 22245 df-flf 22246 df-fcls 22247 df-xms 22627 df-ms 22628 df-tms 22629 df-cncf 23183 df-cfil 23555 df-cmet 23557 df-cms 23635 df-limc 24161 df-dv 24162 df-log 24835 df-cxp 24836 df-logb 25038 df-esum 30931 df-siga 31012 df-sigagen 31043 df-brsiga 31086 df-sx 31093 df-meas 31100 df-mbfm 31154 df-prob 31312 df-rrv 31345 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |