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 6670 | . . . . . 6 ⊢ (𝑘 = 1 → (seq1( ∘f + , 𝑋)‘𝑘) = (seq1( ∘f + , 𝑋)‘1)) | |
3 | 2 | eleq1d 2897 | . . . . 5 ⊢ (𝑘 = 1 → ((seq1( ∘f + , 𝑋)‘𝑘) ∈ (rRndVar‘𝑃) ↔ (seq1( ∘f + , 𝑋)‘1) ∈ (rRndVar‘𝑃))) |
4 | 3 | imbi2d 343 | . . . 4 ⊢ (𝑘 = 1 → ((𝜑 → (seq1( ∘f + , 𝑋)‘𝑘) ∈ (rRndVar‘𝑃)) ↔ (𝜑 → (seq1( ∘f + , 𝑋)‘1) ∈ (rRndVar‘𝑃)))) |
5 | fveq2 6670 | . . . . . 6 ⊢ (𝑘 = 𝑛 → (seq1( ∘f + , 𝑋)‘𝑘) = (seq1( ∘f + , 𝑋)‘𝑛)) | |
6 | 5 | eleq1d 2897 | . . . . 5 ⊢ (𝑘 = 𝑛 → ((seq1( ∘f + , 𝑋)‘𝑘) ∈ (rRndVar‘𝑃) ↔ (seq1( ∘f + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃))) |
7 | 6 | imbi2d 343 | . . . 4 ⊢ (𝑘 = 𝑛 → ((𝜑 → (seq1( ∘f + , 𝑋)‘𝑘) ∈ (rRndVar‘𝑃)) ↔ (𝜑 → (seq1( ∘f + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃)))) |
8 | fveq2 6670 | . . . . . 6 ⊢ (𝑘 = (𝑛 + 1) → (seq1( ∘f + , 𝑋)‘𝑘) = (seq1( ∘f + , 𝑋)‘(𝑛 + 1))) | |
9 | 8 | eleq1d 2897 | . . . . 5 ⊢ (𝑘 = (𝑛 + 1) → ((seq1( ∘f + , 𝑋)‘𝑘) ∈ (rRndVar‘𝑃) ↔ (seq1( ∘f + , 𝑋)‘(𝑛 + 1)) ∈ (rRndVar‘𝑃))) |
10 | 9 | imbi2d 343 | . . . 4 ⊢ (𝑘 = (𝑛 + 1) → ((𝜑 → (seq1( ∘f + , 𝑋)‘𝑘) ∈ (rRndVar‘𝑃)) ↔ (𝜑 → (seq1( ∘f + , 𝑋)‘(𝑛 + 1)) ∈ (rRndVar‘𝑃)))) |
11 | fveq2 6670 | . . . . . 6 ⊢ (𝑘 = 𝑁 → (seq1( ∘f + , 𝑋)‘𝑘) = (seq1( ∘f + , 𝑋)‘𝑁)) | |
12 | 11 | eleq1d 2897 | . . . . 5 ⊢ (𝑘 = 𝑁 → ((seq1( ∘f + , 𝑋)‘𝑘) ∈ (rRndVar‘𝑃) ↔ (seq1( ∘f + , 𝑋)‘𝑁) ∈ (rRndVar‘𝑃))) |
13 | 12 | imbi2d 343 | . . . 4 ⊢ (𝑘 = 𝑁 → ((𝜑 → (seq1( ∘f + , 𝑋)‘𝑘) ∈ (rRndVar‘𝑃)) ↔ (𝜑 → (seq1( ∘f + , 𝑋)‘𝑁) ∈ (rRndVar‘𝑃)))) |
14 | 1z 12013 | . . . . . 6 ⊢ 1 ∈ ℤ | |
15 | seq1 13383 | . . . . . 6 ⊢ (1 ∈ ℤ → (seq1( ∘f + , 𝑋)‘1) = (𝑋‘1)) | |
16 | 14, 15 | ax-mp 5 | . . . . 5 ⊢ (seq1( ∘f + , 𝑋)‘1) = (𝑋‘1) |
17 | 1nn 11649 | . . . . . 6 ⊢ 1 ∈ ℕ | |
18 | rrvsum.2 | . . . . . . 7 ⊢ (𝜑 → 𝑋:ℕ⟶(rRndVar‘𝑃)) | |
19 | 18 | ffvelrnda 6851 | . . . . . 6 ⊢ ((𝜑 ∧ 1 ∈ ℕ) → (𝑋‘1) ∈ (rRndVar‘𝑃)) |
20 | 17, 19 | mpan2 689 | . . . . 5 ⊢ (𝜑 → (𝑋‘1) ∈ (rRndVar‘𝑃)) |
21 | 16, 20 | eqeltrid 2917 | . . . 4 ⊢ (𝜑 → (seq1( ∘f + , 𝑋)‘1) ∈ (rRndVar‘𝑃)) |
22 | seqp1 13385 | . . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘1) → (seq1( ∘f + , 𝑋)‘(𝑛 + 1)) = ((seq1( ∘f + , 𝑋)‘𝑛) ∘f + (𝑋‘(𝑛 + 1)))) | |
23 | nnuz 12282 | . . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1) | |
24 | 22, 23 | eleq2s 2931 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (seq1( ∘f + , 𝑋)‘(𝑛 + 1)) = ((seq1( ∘f + , 𝑋)‘𝑛) ∘f + (𝑋‘(𝑛 + 1)))) |
25 | 24 | ad2antlr 725 | . . . . . . . 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 724 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (seq1( ∘f + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃)) → 𝑃 ∈ Prob) |
28 | simpr 487 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (seq1( ∘f + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃)) → (seq1( ∘f + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃)) | |
29 | peano2nn 11650 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ) | |
30 | 18 | ffvelrnda 6851 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ ℕ) → (𝑋‘(𝑛 + 1)) ∈ (rRndVar‘𝑃)) |
31 | 29, 30 | sylan2 594 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑋‘(𝑛 + 1)) ∈ (rRndVar‘𝑃)) |
32 | 31 | adantr 483 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (seq1( ∘f + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃)) → (𝑋‘(𝑛 + 1)) ∈ (rRndVar‘𝑃)) |
33 | 27, 28, 32 | rrvadd 31710 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (seq1( ∘f + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃)) → ((seq1( ∘f + , 𝑋)‘𝑛) ∘f + (𝑋‘(𝑛 + 1))) ∈ (rRndVar‘𝑃)) |
34 | 25, 33 | eqeltrd 2913 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (seq1( ∘f + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃)) → (seq1( ∘f + , 𝑋)‘(𝑛 + 1)) ∈ (rRndVar‘𝑃)) |
35 | 34 | ex 415 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((seq1( ∘f + , 𝑋)‘𝑛) ∈ (rRndVar‘𝑃) → (seq1( ∘f + , 𝑋)‘(𝑛 + 1)) ∈ (rRndVar‘𝑃))) |
36 | 35 | expcom 416 | . . . . 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 11656 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝜑 → (seq1( ∘f + , 𝑋)‘𝑁) ∈ (rRndVar‘𝑃))) |
39 | 38 | impcom 410 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (seq1( ∘f + , 𝑋)‘𝑁) ∈ (rRndVar‘𝑃)) |
40 | 1, 39 | eqeltrd 2913 | 1 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑆 ∈ (rRndVar‘𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ∘f cof 7407 1c1 10538 + caddc 10540 ℕcn 11638 ℤcz 11982 ℤ≥cuz 12244 seqcseq 13370 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: (None) |
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