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 2779 | . . 3 ⊢ Ⅎ𝑘 𝐴 ∈ 𝑉 |
3 | id 22 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉) | |
4 | 0e0iccpnf 12283 | . . . 4 ⊢ 0 ∈ (0[,]+∞) | |
5 | 4 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑘 ∈ 𝐴) → 0 ∈ (0[,]+∞)) |
6 | xrge0cmn 19788 | . . . . . 6 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
7 | cmnmnd 18208 | . . . . . 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 3203 | . . . . 5 ⊢ 𝑥 ∈ V | |
10 | xrge00 29686 | . . . . . 6 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
11 | 10 | gsumz 17374 | . . . . 5 ⊢ (((ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd ∧ 𝑥 ∈ V) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝑥 ↦ 0)) = 0) |
12 | 8, 9, 11 | mp2an 708 | . . . 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 30108 | . 2 ⊢ (𝐴 ∈ 𝑉 → Σ*𝑘 ∈ 𝐴0 = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ*, < )) |
15 | fconstmpt 5163 | . . . . . . 7 ⊢ ((𝒫 𝐴 ∩ Fin) × {0}) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) | |
16 | 15 | eqcomi 2631 | . . . . . 6 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = ((𝒫 𝐴 ∩ Fin) × {0}) |
17 | 0xr 10086 | . . . . . . . . 9 ⊢ 0 ∈ ℝ* | |
18 | 17 | rgenw 2924 | . . . . . . . 8 ⊢ ∀𝑥 ∈ (𝒫 𝐴 ∩ Fin)0 ∈ ℝ* |
19 | eqid 2622 | . . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) | |
20 | 19 | fnmpt 6020 | . . . . . . . 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 4834 | . . . . . . . . 9 ⊢ ∅ ∈ 𝒫 𝐴 | |
23 | 0fin 8188 | . . . . . . . . 9 ⊢ ∅ ∈ Fin | |
24 | elin 3796 | . . . . . . . . 9 ⊢ (∅ ∈ (𝒫 𝐴 ∩ Fin) ↔ (∅ ∈ 𝒫 𝐴 ∧ ∅ ∈ Fin)) | |
25 | 22, 23, 24 | mpbir2an 955 | . . . . . . . 8 ⊢ ∅ ∈ (𝒫 𝐴 ∩ Fin) |
26 | 25 | ne0ii 3923 | . . . . . . 7 ⊢ (𝒫 𝐴 ∩ Fin) ≠ ∅ |
27 | fconst5 6471 | . . . . . . 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 708 | . . . . . 6 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = ((𝒫 𝐴 ∩ Fin) × {0}) ↔ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = {0}) |
29 | 16, 28 | mpbi 220 | . . . . 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 8352 | . . 3 ⊢ (𝐴 ∈ 𝑉 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ*, < ) = sup({0}, ℝ*, < )) |
32 | xrltso 11974 | . . . 4 ⊢ < Or ℝ* | |
33 | supsn 8378 | . . . 4 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0) | |
34 | 32, 17, 33 | mp2an 708 | . . 3 ⊢ sup({0}, ℝ*, < ) = 0 |
35 | 31, 34 | syl6eq 2672 | . 2 ⊢ (𝐴 ∈ 𝑉 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ*, < ) = 0) |
36 | 14, 35 | eqtrd 2656 | 1 ⊢ (𝐴 ∈ 𝑉 → Σ*𝑘 ∈ 𝐴0 = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Ⅎwnfc 2751 ≠ wne 2794 ∀wral 2912 Vcvv 3200 ∩ cin 3573 ∅c0 3915 𝒫 cpw 4158 {csn 4177 ↦ cmpt 4729 Or wor 5034 × cxp 5112 ran crn 5115 Fn wfn 5883 (class class class)co 6650 Fincfn 7955 supcsup 8346 0cc0 9936 +∞cpnf 10071 ℝ*cxr 10073 < clt 10074 [,]cicc 12178 ↾s cress 15858 Σg cgsu 16101 ℝ*𝑠cxrs 16160 Mndcmnd 17294 CMndccmn 18193 Σ*cesum 30089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-xadd 11947 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-tset 15960 df-ple 15961 df-ds 15964 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-ordt 16161 df-xrs 16162 df-mre 16246 df-mrc 16247 df-acs 16249 df-ps 17200 df-tsr 17201 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-cntz 17750 df-cmn 18195 df-fbas 19743 df-fg 19744 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-ntr 20824 df-nei 20902 df-cn 21031 df-haus 21119 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-tsms 21930 df-esum 30090 |
This theorem is referenced by: esumpad 30117 esumrnmpt2 30130 measvunilem0 30276 ddemeas 30299 |
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