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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumnul | Structured version Visualization version GIF version |
Description: Extended sum over the empty set. (Contributed by Thierry Arnoux, 19-Feb-2017.) |
Ref | Expression |
---|---|
esumnul | ⊢ Σ*𝑥 ∈ ∅𝐴 = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1798 | . . . 4 ⊢ Ⅎ𝑥⊤ | |
2 | nfcv 2891 | . . . 4 ⊢ Ⅎ𝑥∅ | |
3 | 0ex 5308 | . . . . 5 ⊢ ∅ ∈ V | |
4 | 3 | a1i 11 | . . . 4 ⊢ (⊤ → ∅ ∈ V) |
5 | ral0 4514 | . . . . . 6 ⊢ ∀𝑥 ∈ ∅ 𝐴 ∈ (0[,]+∞) | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (⊤ → ∀𝑥 ∈ ∅ 𝐴 ∈ (0[,]+∞)) |
7 | 6 | r19.21bi 3238 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ∅) → 𝐴 ∈ (0[,]+∞)) |
8 | pw0 4817 | . . . . . . . . . . . . 13 ⊢ 𝒫 ∅ = {∅} | |
9 | 8 | ineq1i 4206 | . . . . . . . . . . . 12 ⊢ (𝒫 ∅ ∩ Fin) = ({∅} ∩ Fin) |
10 | 0fin 9196 | . . . . . . . . . . . . 13 ⊢ ∅ ∈ Fin | |
11 | snssi 4813 | . . . . . . . . . . . . . 14 ⊢ (∅ ∈ Fin → {∅} ⊆ Fin) | |
12 | dfss2 3962 | . . . . . . . . . . . . . 14 ⊢ ({∅} ⊆ Fin ↔ ({∅} ∩ Fin) = {∅}) | |
13 | 11, 12 | sylib 217 | . . . . . . . . . . . . 13 ⊢ (∅ ∈ Fin → ({∅} ∩ Fin) = {∅}) |
14 | 10, 13 | ax-mp 5 | . . . . . . . . . . . 12 ⊢ ({∅} ∩ Fin) = {∅} |
15 | 9, 14 | eqtri 2753 | . . . . . . . . . . 11 ⊢ (𝒫 ∅ ∩ Fin) = {∅} |
16 | 15 | eleq2i 2817 | . . . . . . . . . 10 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↔ 𝑦 ∈ {∅}) |
17 | velsn 4646 | . . . . . . . . . 10 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅) | |
18 | 16, 17 | sylbb 218 | . . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → 𝑦 = ∅) |
19 | 18 | mpteq1d 5244 | . . . . . . . 8 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → (𝑥 ∈ 𝑦 ↦ 𝐴) = (𝑥 ∈ ∅ ↦ 𝐴)) |
20 | mpt0 6698 | . . . . . . . 8 ⊢ (𝑥 ∈ ∅ ↦ 𝐴) = ∅ | |
21 | 19, 20 | eqtrdi 2781 | . . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → (𝑥 ∈ 𝑦 ↦ 𝐴) = ∅) |
22 | 21 | oveq2d 7435 | . . . . . 6 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑥 ∈ 𝑦 ↦ 𝐴)) = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg ∅)) |
23 | xrge00 32831 | . . . . . . 7 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
24 | 23 | gsum0 18647 | . . . . . 6 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) Σg ∅) = 0 |
25 | 22, 24 | eqtrdi 2781 | . . . . 5 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑥 ∈ 𝑦 ↦ 𝐴)) = 0) |
26 | 25 | adantl 480 | . . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ (𝒫 ∅ ∩ Fin)) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑥 ∈ 𝑦 ↦ 𝐴)) = 0) |
27 | 1, 2, 4, 7, 26 | esumval 33796 | . . 3 ⊢ (⊤ → Σ*𝑥 ∈ ∅𝐴 = sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ*, < )) |
28 | 27 | mptru 1540 | . 2 ⊢ Σ*𝑥 ∈ ∅𝐴 = sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ*, < ) |
29 | fconstmpt 5740 | . . . . 5 ⊢ ((𝒫 ∅ ∩ Fin) × {0}) = (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) | |
30 | 29 | eqcomi 2734 | . . . 4 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = ((𝒫 ∅ ∩ Fin) × {0}) |
31 | 0xr 11293 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
32 | 31 | rgenw 3054 | . . . . . 6 ⊢ ∀𝑦 ∈ (𝒫 ∅ ∩ Fin)0 ∈ ℝ* |
33 | eqid 2725 | . . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) | |
34 | 33 | fnmpt 6696 | . . . . . 6 ⊢ (∀𝑦 ∈ (𝒫 ∅ ∩ Fin)0 ∈ ℝ* → (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) Fn (𝒫 ∅ ∩ Fin)) |
35 | 32, 34 | ax-mp 5 | . . . . 5 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) Fn (𝒫 ∅ ∩ Fin) |
36 | 3 | snnz 4782 | . . . . . 6 ⊢ {∅} ≠ ∅ |
37 | 15, 36 | eqnetri 3000 | . . . . 5 ⊢ (𝒫 ∅ ∩ Fin) ≠ ∅ |
38 | fconst5 7218 | . . . . 5 ⊢ (((𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) Fn (𝒫 ∅ ∩ Fin) ∧ (𝒫 ∅ ∩ Fin) ≠ ∅) → ((𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = ((𝒫 ∅ ∩ Fin) × {0}) ↔ ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = {0})) | |
39 | 35, 37, 38 | mp2an 690 | . . . 4 ⊢ ((𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = ((𝒫 ∅ ∩ Fin) × {0}) ↔ ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = {0}) |
40 | 30, 39 | mpbi 229 | . . 3 ⊢ ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = {0} |
41 | 40 | supeq1i 9472 | . 2 ⊢ sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ*, < ) = sup({0}, ℝ*, < ) |
42 | xrltso 13155 | . . 3 ⊢ < Or ℝ* | |
43 | supsn 9497 | . . 3 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0) | |
44 | 42, 31, 43 | mp2an 690 | . 2 ⊢ sup({0}, ℝ*, < ) = 0 |
45 | 28, 41, 44 | 3eqtri 2757 | 1 ⊢ Σ*𝑥 ∈ ∅𝐴 = 0 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ≠ wne 2929 ∀wral 3050 Vcvv 3461 ∩ cin 3943 ⊆ wss 3944 ∅c0 4322 𝒫 cpw 4604 {csn 4630 ↦ cmpt 5232 Or wor 5589 × cxp 5676 ran crn 5679 Fn wfn 6544 (class class class)co 7419 Fincfn 8964 supcsup 9465 0cc0 11140 +∞cpnf 11277 ℝ*cxr 11279 < clt 11280 [,]cicc 13362 ↾s cress 17212 Σg cgsu 17425 ℝ*𝑠cxrs 17485 Σ*cesum 33777 |
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 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-fi 9436 df-sup 9467 df-inf 9468 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-xadd 13128 df-ioo 13363 df-ioc 13364 df-ico 13365 df-icc 13366 df-fz 13520 df-fzo 13663 df-seq 14003 df-hash 14326 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-tset 17255 df-ple 17256 df-ds 17258 df-rest 17407 df-topn 17408 df-0g 17426 df-gsum 17427 df-topgen 17428 df-ordt 17486 df-xrs 17487 df-mre 17569 df-mrc 17570 df-acs 17572 df-ps 18561 df-tsr 18562 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-cntz 19280 df-cmn 19749 df-fbas 21293 df-fg 21294 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22893 df-ntr 22968 df-nei 23046 df-cn 23175 df-haus 23263 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-tsms 24075 df-esum 33778 |
This theorem is referenced by: esumrnmpt2 33818 esum2dlem 33842 ddemeas 33986 carsgclctunlem1 34068 |
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