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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 1784 | . . . 4 ⊢ Ⅎ𝑥⊤ | |
2 | nfcv 2947 | . . . 4 ⊢ Ⅎ𝑥∅ | |
3 | 0ex 5096 | . . . . 5 ⊢ ∅ ∈ V | |
4 | 3 | a1i 11 | . . . 4 ⊢ (⊤ → ∅ ∈ V) |
5 | ral0 4364 | . . . . . 6 ⊢ ∀𝑥 ∈ ∅ 𝐴 ∈ (0[,]+∞) | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (⊤ → ∀𝑥 ∈ ∅ 𝐴 ∈ (0[,]+∞)) |
7 | 6 | r19.21bi 3173 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ∅) → 𝐴 ∈ (0[,]+∞)) |
8 | pw0 4646 | . . . . . . . . . . . . 13 ⊢ 𝒫 ∅ = {∅} | |
9 | 8 | ineq1i 4100 | . . . . . . . . . . . 12 ⊢ (𝒫 ∅ ∩ Fin) = ({∅} ∩ Fin) |
10 | 0fin 8582 | . . . . . . . . . . . . 13 ⊢ ∅ ∈ Fin | |
11 | snssi 4642 | . . . . . . . . . . . . . 14 ⊢ (∅ ∈ Fin → {∅} ⊆ Fin) | |
12 | df-ss 3869 | . . . . . . . . . . . . . 14 ⊢ ({∅} ⊆ Fin ↔ ({∅} ∩ Fin) = {∅}) | |
13 | 11, 12 | sylib 219 | . . . . . . . . . . . . 13 ⊢ (∅ ∈ Fin → ({∅} ∩ Fin) = {∅}) |
14 | 10, 13 | ax-mp 5 | . . . . . . . . . . . 12 ⊢ ({∅} ∩ Fin) = {∅} |
15 | 9, 14 | eqtri 2817 | . . . . . . . . . . 11 ⊢ (𝒫 ∅ ∩ Fin) = {∅} |
16 | 15 | eleq2i 2872 | . . . . . . . . . 10 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↔ 𝑦 ∈ {∅}) |
17 | velsn 4482 | . . . . . . . . . 10 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅) | |
18 | 16, 17 | sylbb 220 | . . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → 𝑦 = ∅) |
19 | 18 | mpteq1d 5043 | . . . . . . . 8 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → (𝑥 ∈ 𝑦 ↦ 𝐴) = (𝑥 ∈ ∅ ↦ 𝐴)) |
20 | mpt0 6350 | . . . . . . . 8 ⊢ (𝑥 ∈ ∅ ↦ 𝐴) = ∅ | |
21 | 19, 20 | syl6eq 2845 | . . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → (𝑥 ∈ 𝑦 ↦ 𝐴) = ∅) |
22 | 21 | oveq2d 7023 | . . . . . 6 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑥 ∈ 𝑦 ↦ 𝐴)) = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg ∅)) |
23 | xrge00 30317 | . . . . . . 7 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
24 | 23 | gsum0 17705 | . . . . . 6 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) Σg ∅) = 0 |
25 | 22, 24 | syl6eq 2845 | . . . . 5 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑥 ∈ 𝑦 ↦ 𝐴)) = 0) |
26 | 25 | adantl 482 | . . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ (𝒫 ∅ ∩ Fin)) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑥 ∈ 𝑦 ↦ 𝐴)) = 0) |
27 | 1, 2, 4, 7, 26 | esumval 30878 | . . 3 ⊢ (⊤ → Σ*𝑥 ∈ ∅𝐴 = sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ*, < )) |
28 | 27 | mptru 1527 | . 2 ⊢ Σ*𝑥 ∈ ∅𝐴 = sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ*, < ) |
29 | fconstmpt 5492 | . . . . 5 ⊢ ((𝒫 ∅ ∩ Fin) × {0}) = (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) | |
30 | 29 | eqcomi 2802 | . . . 4 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = ((𝒫 ∅ ∩ Fin) × {0}) |
31 | 0xr 10523 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
32 | 31 | rgenw 3115 | . . . . . 6 ⊢ ∀𝑦 ∈ (𝒫 ∅ ∩ Fin)0 ∈ ℝ* |
33 | eqid 2793 | . . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) | |
34 | 33 | fnmpt 6348 | . . . . . 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 4612 | . . . . . 6 ⊢ {∅} ≠ ∅ |
37 | 15, 36 | eqnetri 3052 | . . . . 5 ⊢ (𝒫 ∅ ∩ Fin) ≠ ∅ |
38 | fconst5 6826 | . . . . 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 688 | . . . 4 ⊢ ((𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = ((𝒫 ∅ ∩ Fin) × {0}) ↔ ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = {0}) |
40 | 30, 39 | mpbi 231 | . . 3 ⊢ ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = {0} |
41 | 40 | supeq1i 8747 | . 2 ⊢ sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ*, < ) = sup({0}, ℝ*, < ) |
42 | xrltso 12373 | . . 3 ⊢ < Or ℝ* | |
43 | supsn 8772 | . . 3 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0) | |
44 | 42, 31, 43 | mp2an 688 | . 2 ⊢ sup({0}, ℝ*, < ) = 0 |
45 | 28, 41, 44 | 3eqtri 2821 | 1 ⊢ Σ*𝑥 ∈ ∅𝐴 = 0 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1520 ⊤wtru 1521 ∈ wcel 2079 ≠ wne 2982 ∀wral 3103 Vcvv 3432 ∩ cin 3853 ⊆ wss 3854 ∅c0 4206 𝒫 cpw 4447 {csn 4466 ↦ cmpt 5035 Or wor 5353 × cxp 5433 ran crn 5436 Fn wfn 6212 (class class class)co 7007 Fincfn 8347 supcsup 8740 0cc0 10372 +∞cpnf 10507 ℝ*cxr 10509 < clt 10510 [,]cicc 12580 ↾s cress 16301 Σg cgsu 16531 ℝ*𝑠cxrs 16590 Σ*cesum 30859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 ax-pre-sup 10450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-fal 1533 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-iin 4822 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-se 5395 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-isom 6226 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-of 7258 df-om 7428 df-1st 7536 df-2nd 7537 df-supp 7673 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-oadd 7948 df-er 8130 df-map 8249 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-fsupp 8670 df-fi 8711 df-sup 8742 df-inf 8743 df-oi 8810 df-card 9203 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-nn 11476 df-2 11537 df-3 11538 df-4 11539 df-5 11540 df-6 11541 df-7 11542 df-8 11543 df-9 11544 df-n0 11735 df-z 11819 df-dec 11937 df-uz 12083 df-q 12187 df-xadd 12347 df-ioo 12581 df-ioc 12582 df-ico 12583 df-icc 12584 df-fz 12732 df-fzo 12873 df-seq 13208 df-hash 13529 df-struct 16302 df-ndx 16303 df-slot 16304 df-base 16306 df-sets 16307 df-ress 16308 df-plusg 16395 df-mulr 16396 df-tset 16401 df-ple 16402 df-ds 16404 df-rest 16513 df-topn 16514 df-0g 16532 df-gsum 16533 df-topgen 16534 df-ordt 16591 df-xrs 16592 df-mre 16674 df-mrc 16675 df-acs 16677 df-ps 17627 df-tsr 17628 df-mgm 17669 df-sgrp 17711 df-mnd 17722 df-submnd 17763 df-cntz 18176 df-cmn 18623 df-fbas 20212 df-fg 20213 df-top 21174 df-topon 21191 df-topsp 21213 df-bases 21226 df-ntr 21300 df-nei 21378 df-cn 21507 df-haus 21595 df-fil 22126 df-fm 22218 df-flim 22219 df-flf 22220 df-tsms 22406 df-esum 30860 |
This theorem is referenced by: esumrnmpt2 30900 esum2dlem 30924 ddemeas 31068 carsgclctunlem1 31148 |
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