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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esum0 | Structured version Visualization version GIF version |
Description: Extended sum of zero. (Contributed by Thierry Arnoux, 3-Mar-2017.) |
Ref | Expression |
---|---|
esum0.k | ⊢ Ⅎ𝑘𝐴 |
Ref | Expression |
---|---|
esum0 | ⊢ (𝐴 ∈ 𝑉 → Σ*𝑘 ∈ 𝐴0 = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | esum0.k | . . . 4 ⊢ Ⅎ𝑘𝐴 | |
2 | 1 | nfel1 2971 | . . 3 ⊢ Ⅎ𝑘 𝐴 ∈ 𝑉 |
3 | id 22 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉) | |
4 | 0e0iccpnf 12837 | . . . 4 ⊢ 0 ∈ (0[,]+∞) | |
5 | 4 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑘 ∈ 𝐴) → 0 ∈ (0[,]+∞)) |
6 | xrge0cmn 20133 | . . . . . 6 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
7 | cmnmnd 18914 | . . . . . 6 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd) | |
8 | 6, 7 | ax-mp 5 | . . . . 5 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd |
9 | vex 3444 | . . . . 5 ⊢ 𝑥 ∈ V | |
10 | xrge00 30720 | . . . . . 6 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
11 | 10 | gsumz 17992 | . . . . 5 ⊢ (((ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd ∧ 𝑥 ∈ V) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝑥 ↦ 0)) = 0) |
12 | 8, 9, 11 | mp2an 691 | . . . 4 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝑥 ↦ 0)) = 0 |
13 | 12 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝑥 ↦ 0)) = 0) |
14 | 2, 1, 3, 5, 13 | esumval 31415 | . 2 ⊢ (𝐴 ∈ 𝑉 → Σ*𝑘 ∈ 𝐴0 = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ*, < )) |
15 | fconstmpt 5578 | . . . . . . 7 ⊢ ((𝒫 𝐴 ∩ Fin) × {0}) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) | |
16 | 15 | eqcomi 2807 | . . . . . 6 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = ((𝒫 𝐴 ∩ Fin) × {0}) |
17 | 0xr 10677 | . . . . . . . . 9 ⊢ 0 ∈ ℝ* | |
18 | 17 | rgenw 3118 | . . . . . . . 8 ⊢ ∀𝑥 ∈ (𝒫 𝐴 ∩ Fin)0 ∈ ℝ* |
19 | eqid 2798 | . . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) | |
20 | 19 | fnmpt 6460 | . . . . . . . 8 ⊢ (∀𝑥 ∈ (𝒫 𝐴 ∩ Fin)0 ∈ ℝ* → (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) Fn (𝒫 𝐴 ∩ Fin)) |
21 | 18, 20 | ax-mp 5 | . . . . . . 7 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) Fn (𝒫 𝐴 ∩ Fin) |
22 | 0elpw 5221 | . . . . . . . . 9 ⊢ ∅ ∈ 𝒫 𝐴 | |
23 | 0fin 8730 | . . . . . . . . 9 ⊢ ∅ ∈ Fin | |
24 | elin 3897 | . . . . . . . . 9 ⊢ (∅ ∈ (𝒫 𝐴 ∩ Fin) ↔ (∅ ∈ 𝒫 𝐴 ∧ ∅ ∈ Fin)) | |
25 | 22, 23, 24 | mpbir2an 710 | . . . . . . . 8 ⊢ ∅ ∈ (𝒫 𝐴 ∩ Fin) |
26 | 25 | ne0ii 4253 | . . . . . . 7 ⊢ (𝒫 𝐴 ∩ Fin) ≠ ∅ |
27 | fconst5 6945 | . . . . . . 7 ⊢ (((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) Fn (𝒫 𝐴 ∩ Fin) ∧ (𝒫 𝐴 ∩ Fin) ≠ ∅) → ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = ((𝒫 𝐴 ∩ Fin) × {0}) ↔ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = {0})) | |
28 | 21, 26, 27 | mp2an 691 | . . . . . 6 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = ((𝒫 𝐴 ∩ Fin) × {0}) ↔ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = {0}) |
29 | 16, 28 | mpbi 233 | . . . . 5 ⊢ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = {0} |
30 | 29 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = {0}) |
31 | 30 | supeq1d 8894 | . . 3 ⊢ (𝐴 ∈ 𝑉 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ*, < ) = sup({0}, ℝ*, < )) |
32 | xrltso 12522 | . . . 4 ⊢ < Or ℝ* | |
33 | supsn 8920 | . . . 4 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0) | |
34 | 32, 17, 33 | mp2an 691 | . . 3 ⊢ sup({0}, ℝ*, < ) = 0 |
35 | 31, 34 | eqtrdi 2849 | . 2 ⊢ (𝐴 ∈ 𝑉 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ*, < ) = 0) |
36 | 14, 35 | eqtrd 2833 | 1 ⊢ (𝐴 ∈ 𝑉 → Σ*𝑘 ∈ 𝐴0 = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Ⅎwnfc 2936 ≠ wne 2987 ∀wral 3106 Vcvv 3441 ∩ cin 3880 ∅c0 4243 𝒫 cpw 4497 {csn 4525 ↦ cmpt 5110 Or wor 5437 × cxp 5517 ran crn 5520 Fn wfn 6319 (class class class)co 7135 Fincfn 8492 supcsup 8888 0cc0 10526 +∞cpnf 10661 ℝ*cxr 10663 < clt 10664 [,]cicc 12729 ↾s cress 16476 Σg cgsu 16706 ℝ*𝑠cxrs 16765 Mndcmnd 17903 CMndccmn 18898 Σ*cesum 31396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-xadd 12496 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-tset 16576 df-ple 16577 df-ds 16579 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-ordt 16766 df-xrs 16767 df-mre 16849 df-mrc 16850 df-acs 16852 df-ps 17802 df-tsr 17803 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-cntz 18439 df-cmn 18900 df-fbas 20088 df-fg 20089 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-ntr 21625 df-nei 21703 df-cn 21832 df-haus 21920 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-tsms 22732 df-esum 31397 |
This theorem is referenced by: esumpad 31424 esumrnmpt2 31437 measvunilem0 31582 ddemeas 31605 |
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