Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumdivc | Structured version Visualization version GIF version |
Description: An extended sum divided by a constant. (Contributed by Thierry Arnoux, 6-Jul-2017.) |
Ref | Expression |
---|---|
esumdivc.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
esumdivc.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
esumdivc.c | ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
Ref | Expression |
---|---|
esumdivc | ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴𝐵 /𝑒 𝐶) = Σ*𝑘 ∈ 𝐴(𝐵 /𝑒 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | esumdivc.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | esumdivc.b | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
3 | 1red 10644 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℝ) | |
4 | esumdivc.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
5 | 4 | rpred 12434 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
6 | 4 | rpne0d 12439 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 0) |
7 | rexdiv 30604 | . . . . 5 ⊢ ((1 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐶 ≠ 0) → (1 /𝑒 𝐶) = (1 / 𝐶)) | |
8 | 3, 5, 6, 7 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → (1 /𝑒 𝐶) = (1 / 𝐶)) |
9 | ioorp 12817 | . . . . . 6 ⊢ (0(,)+∞) = ℝ+ | |
10 | ioossico 12829 | . . . . . 6 ⊢ (0(,)+∞) ⊆ (0[,)+∞) | |
11 | 9, 10 | eqsstrri 4004 | . . . . 5 ⊢ ℝ+ ⊆ (0[,)+∞) |
12 | 4 | rpreccld 12444 | . . . . 5 ⊢ (𝜑 → (1 / 𝐶) ∈ ℝ+) |
13 | 11, 12 | sseldi 3967 | . . . 4 ⊢ (𝜑 → (1 / 𝐶) ∈ (0[,)+∞)) |
14 | 8, 13 | eqeltrd 2915 | . . 3 ⊢ (𝜑 → (1 /𝑒 𝐶) ∈ (0[,)+∞)) |
15 | 1, 2, 14 | esummulc1 31342 | . 2 ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴𝐵 ·e (1 /𝑒 𝐶)) = Σ*𝑘 ∈ 𝐴(𝐵 ·e (1 /𝑒 𝐶))) |
16 | iccssxr 12822 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
17 | 2 | ralrimiva 3184 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) |
18 | nfcv 2979 | . . . . . 6 ⊢ Ⅎ𝑘𝐴 | |
19 | 18 | esumcl 31291 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
20 | 1, 17, 19 | syl2anc 586 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
21 | 16, 20 | sseldi 3967 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ*) |
22 | xdivrec 30605 | . . 3 ⊢ ((Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ ∧ 𝐶 ≠ 0) → (Σ*𝑘 ∈ 𝐴𝐵 /𝑒 𝐶) = (Σ*𝑘 ∈ 𝐴𝐵 ·e (1 /𝑒 𝐶))) | |
23 | 21, 5, 6, 22 | syl3anc 1367 | . 2 ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴𝐵 /𝑒 𝐶) = (Σ*𝑘 ∈ 𝐴𝐵 ·e (1 /𝑒 𝐶))) |
24 | 16, 2 | sseldi 3967 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
25 | 5 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ) |
26 | 6 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ≠ 0) |
27 | xdivrec 30605 | . . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ ∧ 𝐶 ≠ 0) → (𝐵 /𝑒 𝐶) = (𝐵 ·e (1 /𝑒 𝐶))) | |
28 | 24, 25, 26, 27 | syl3anc 1367 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 /𝑒 𝐶) = (𝐵 ·e (1 /𝑒 𝐶))) |
29 | 28 | esumeq2dv 31299 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴(𝐵 /𝑒 𝐶) = Σ*𝑘 ∈ 𝐴(𝐵 ·e (1 /𝑒 𝐶))) |
30 | 15, 23, 29 | 3eqtr4d 2868 | 1 ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴𝐵 /𝑒 𝐶) = Σ*𝑘 ∈ 𝐴(𝐵 /𝑒 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 (class class class)co 7158 ℝcr 10538 0cc0 10539 1c1 10540 +∞cpnf 10674 ℝ*cxr 10676 / cdiv 11299 ℝ+crp 12392 ·e cxmu 12509 (,)cioo 12741 [,)cico 12743 [,]cicc 12744 /𝑒 cxdiv 30595 Σ*cesum 31288 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-tset 16586 df-ple 16587 df-ds 16589 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-ordt 16776 df-xrs 16777 df-mre 16859 df-mrc 16860 df-acs 16862 df-ps 17812 df-tsr 17813 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-cntz 18449 df-cmn 18910 df-fbas 20544 df-fg 20545 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-ntr 21630 df-nei 21708 df-cn 21837 df-cnp 21838 df-haus 21925 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-tsms 22737 df-xdiv 30596 df-esum 31289 |
This theorem is referenced by: measdivcst 31485 measdivcstALTV 31486 |
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