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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 11222 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℝ) | |
4 | esumdivc.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
5 | 4 | rpred 13023 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
6 | 4 | rpne0d 13028 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 0) |
7 | rexdiv 32526 | . . . . 5 ⊢ ((1 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐶 ≠ 0) → (1 /𝑒 𝐶) = (1 / 𝐶)) | |
8 | 3, 5, 6, 7 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → (1 /𝑒 𝐶) = (1 / 𝐶)) |
9 | ioorp 13409 | . . . . . 6 ⊢ (0(,)+∞) = ℝ+ | |
10 | ioossico 13422 | . . . . . 6 ⊢ (0(,)+∞) ⊆ (0[,)+∞) | |
11 | 9, 10 | eqsstrri 4017 | . . . . 5 ⊢ ℝ+ ⊆ (0[,)+∞) |
12 | 4 | rpreccld 13033 | . . . . 5 ⊢ (𝜑 → (1 / 𝐶) ∈ ℝ+) |
13 | 11, 12 | sselid 3980 | . . . 4 ⊢ (𝜑 → (1 / 𝐶) ∈ (0[,)+∞)) |
14 | 8, 13 | eqeltrd 2832 | . . 3 ⊢ (𝜑 → (1 /𝑒 𝐶) ∈ (0[,)+∞)) |
15 | 1, 2, 14 | esummulc1 33544 | . 2 ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴𝐵 ·e (1 /𝑒 𝐶)) = Σ*𝑘 ∈ 𝐴(𝐵 ·e (1 /𝑒 𝐶))) |
16 | iccssxr 13414 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
17 | 2 | ralrimiva 3145 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) |
18 | nfcv 2902 | . . . . . 6 ⊢ Ⅎ𝑘𝐴 | |
19 | 18 | esumcl 33493 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
20 | 1, 17, 19 | syl2anc 583 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
21 | 16, 20 | sselid 3980 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ*) |
22 | xdivrec 32527 | . . 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 1370 | . 2 ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴𝐵 /𝑒 𝐶) = (Σ*𝑘 ∈ 𝐴𝐵 ·e (1 /𝑒 𝐶))) |
24 | 16, 2 | sselid 3980 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
25 | 5 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ) |
26 | 6 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ≠ 0) |
27 | xdivrec 32527 | . . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ ∧ 𝐶 ≠ 0) → (𝐵 /𝑒 𝐶) = (𝐵 ·e (1 /𝑒 𝐶))) | |
28 | 24, 25, 26, 27 | syl3anc 1370 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 /𝑒 𝐶) = (𝐵 ·e (1 /𝑒 𝐶))) |
29 | 28 | esumeq2dv 33501 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴(𝐵 /𝑒 𝐶) = Σ*𝑘 ∈ 𝐴(𝐵 ·e (1 /𝑒 𝐶))) |
30 | 15, 23, 29 | 3eqtr4d 2781 | 1 ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴𝐵 /𝑒 𝐶) = Σ*𝑘 ∈ 𝐴(𝐵 /𝑒 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 (class class class)co 7412 ℝcr 11115 0cc0 11116 1c1 11117 +∞cpnf 11252 ℝ*cxr 11254 / cdiv 11878 ℝ+crp 12981 ·e cxmu 13098 (,)cioo 13331 [,)cico 13333 [,]cicc 13334 /𝑒 cxdiv 32517 Σ*cesum 33490 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-fi 9412 df-sup 9443 df-inf 9444 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ioo 13335 df-ioc 13336 df-ico 13337 df-icc 13338 df-fz 13492 df-fzo 13635 df-seq 13974 df-hash 14298 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-tset 17223 df-ple 17224 df-ds 17226 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-ordt 17454 df-xrs 17455 df-mre 17537 df-mrc 17538 df-acs 17540 df-ps 18529 df-tsr 18530 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-submnd 18712 df-cntz 19229 df-cmn 19698 df-fbas 21230 df-fg 21231 df-top 22716 df-topon 22733 df-topsp 22755 df-bases 22769 df-ntr 22844 df-nei 22922 df-cn 23051 df-cnp 23052 df-haus 23139 df-fil 23670 df-fm 23762 df-flim 23763 df-flf 23764 df-tsms 23951 df-xdiv 32518 df-esum 33491 |
This theorem is referenced by: measdivcst 33687 measdivcstALTV 33688 |
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