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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 2911 | . . 3 ⊢ Ⅎ𝑘 𝐴 ∈ 𝑉 |
3 | id 22 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉) | |
4 | 0e0iccpnf 13437 | . . . 4 ⊢ 0 ∈ (0[,]+∞) | |
5 | 4 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑘 ∈ 𝐴) → 0 ∈ (0[,]+∞)) |
6 | xrge0cmn 21292 | . . . . . 6 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
7 | cmnmnd 19713 | . . . . . 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 3470 | . . . . 5 ⊢ 𝑥 ∈ V | |
10 | xrge00 32678 | . . . . . 6 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
11 | 10 | gsumz 18757 | . . . . 5 ⊢ (((ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd ∧ 𝑥 ∈ V) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝑥 ↦ 0)) = 0) |
12 | 8, 9, 11 | mp2an 689 | . . . 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 33564 | . 2 ⊢ (𝐴 ∈ 𝑉 → Σ*𝑘 ∈ 𝐴0 = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ*, < )) |
15 | fconstmpt 5729 | . . . . . . 7 ⊢ ((𝒫 𝐴 ∩ Fin) × {0}) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) | |
16 | 15 | eqcomi 2733 | . . . . . 6 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = ((𝒫 𝐴 ∩ Fin) × {0}) |
17 | 0xr 11260 | . . . . . . . . 9 ⊢ 0 ∈ ℝ* | |
18 | 17 | rgenw 3057 | . . . . . . . 8 ⊢ ∀𝑥 ∈ (𝒫 𝐴 ∩ Fin)0 ∈ ℝ* |
19 | eqid 2724 | . . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) | |
20 | 19 | fnmpt 6681 | . . . . . . . 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 5345 | . . . . . . . . 9 ⊢ ∅ ∈ 𝒫 𝐴 | |
23 | 0fin 9168 | . . . . . . . . 9 ⊢ ∅ ∈ Fin | |
24 | elin 3957 | . . . . . . . . 9 ⊢ (∅ ∈ (𝒫 𝐴 ∩ Fin) ↔ (∅ ∈ 𝒫 𝐴 ∧ ∅ ∈ Fin)) | |
25 | 22, 23, 24 | mpbir2an 708 | . . . . . . . 8 ⊢ ∅ ∈ (𝒫 𝐴 ∩ Fin) |
26 | 25 | ne0ii 4330 | . . . . . . 7 ⊢ (𝒫 𝐴 ∩ Fin) ≠ ∅ |
27 | fconst5 7200 | . . . . . . 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 689 | . . . . . 6 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = ((𝒫 𝐴 ∩ Fin) × {0}) ↔ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = {0}) |
29 | 16, 28 | mpbi 229 | . . . . 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 9438 | . . 3 ⊢ (𝐴 ∈ 𝑉 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ*, < ) = sup({0}, ℝ*, < )) |
32 | xrltso 13121 | . . . 4 ⊢ < Or ℝ* | |
33 | supsn 9464 | . . . 4 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0) | |
34 | 32, 17, 33 | mp2an 689 | . . 3 ⊢ sup({0}, ℝ*, < ) = 0 |
35 | 31, 34 | eqtrdi 2780 | . 2 ⊢ (𝐴 ∈ 𝑉 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ*, < ) = 0) |
36 | 14, 35 | eqtrd 2764 | 1 ⊢ (𝐴 ∈ 𝑉 → Σ*𝑘 ∈ 𝐴0 = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Ⅎwnfc 2875 ≠ wne 2932 ∀wral 3053 Vcvv 3466 ∩ cin 3940 ∅c0 4315 𝒫 cpw 4595 {csn 4621 ↦ cmpt 5222 Or wor 5578 × cxp 5665 ran crn 5668 Fn wfn 6529 (class class class)co 7402 Fincfn 8936 supcsup 9432 0cc0 11107 +∞cpnf 11244 ℝ*cxr 11246 < clt 11247 [,]cicc 13328 ↾s cress 17178 Σg cgsu 17391 ℝ*𝑠cxrs 17451 Mndcmnd 18663 CMndccmn 19696 Σ*cesum 33545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-xadd 13094 df-ioo 13329 df-ioc 13330 df-ico 13331 df-icc 13332 df-fz 13486 df-fzo 13629 df-seq 13968 df-hash 14292 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-tset 17221 df-ple 17222 df-ds 17224 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-ordt 17452 df-xrs 17453 df-mre 17535 df-mrc 17536 df-acs 17538 df-ps 18527 df-tsr 18528 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18710 df-cntz 19229 df-cmn 19698 df-fbas 21231 df-fg 21232 df-top 22740 df-topon 22757 df-topsp 22779 df-bases 22793 df-ntr 22868 df-nei 22946 df-cn 23075 df-haus 23163 df-fil 23694 df-fm 23786 df-flim 23787 df-flf 23788 df-tsms 23975 df-esum 33546 |
This theorem is referenced by: esumpad 33573 esumrnmpt2 33586 measvunilem0 33731 ddemeas 33754 |
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