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 1801 | . . . 4 ⊢ Ⅎ𝑥⊤ | |
2 | nfcv 2977 | . . . 4 ⊢ Ⅎ𝑥∅ | |
3 | 0ex 5210 | . . . . 5 ⊢ ∅ ∈ V | |
4 | 3 | a1i 11 | . . . 4 ⊢ (⊤ → ∅ ∈ V) |
5 | ral0 4455 | . . . . . 6 ⊢ ∀𝑥 ∈ ∅ 𝐴 ∈ (0[,]+∞) | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (⊤ → ∀𝑥 ∈ ∅ 𝐴 ∈ (0[,]+∞)) |
7 | 6 | r19.21bi 3208 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ∅) → 𝐴 ∈ (0[,]+∞)) |
8 | pw0 4744 | . . . . . . . . . . . . 13 ⊢ 𝒫 ∅ = {∅} | |
9 | 8 | ineq1i 4184 | . . . . . . . . . . . 12 ⊢ (𝒫 ∅ ∩ Fin) = ({∅} ∩ Fin) |
10 | 0fin 8745 | . . . . . . . . . . . . 13 ⊢ ∅ ∈ Fin | |
11 | snssi 4740 | . . . . . . . . . . . . . 14 ⊢ (∅ ∈ Fin → {∅} ⊆ Fin) | |
12 | df-ss 3951 | . . . . . . . . . . . . . 14 ⊢ ({∅} ⊆ Fin ↔ ({∅} ∩ Fin) = {∅}) | |
13 | 11, 12 | sylib 220 | . . . . . . . . . . . . 13 ⊢ (∅ ∈ Fin → ({∅} ∩ Fin) = {∅}) |
14 | 10, 13 | ax-mp 5 | . . . . . . . . . . . 12 ⊢ ({∅} ∩ Fin) = {∅} |
15 | 9, 14 | eqtri 2844 | . . . . . . . . . . 11 ⊢ (𝒫 ∅ ∩ Fin) = {∅} |
16 | 15 | eleq2i 2904 | . . . . . . . . . 10 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↔ 𝑦 ∈ {∅}) |
17 | velsn 4582 | . . . . . . . . . 10 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅) | |
18 | 16, 17 | sylbb 221 | . . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → 𝑦 = ∅) |
19 | 18 | mpteq1d 5154 | . . . . . . . 8 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → (𝑥 ∈ 𝑦 ↦ 𝐴) = (𝑥 ∈ ∅ ↦ 𝐴)) |
20 | mpt0 6489 | . . . . . . . 8 ⊢ (𝑥 ∈ ∅ ↦ 𝐴) = ∅ | |
21 | 19, 20 | syl6eq 2872 | . . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → (𝑥 ∈ 𝑦 ↦ 𝐴) = ∅) |
22 | 21 | oveq2d 7171 | . . . . . 6 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑥 ∈ 𝑦 ↦ 𝐴)) = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg ∅)) |
23 | xrge00 30673 | . . . . . . 7 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
24 | 23 | gsum0 17893 | . . . . . 6 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) Σg ∅) = 0 |
25 | 22, 24 | syl6eq 2872 | . . . . 5 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑥 ∈ 𝑦 ↦ 𝐴)) = 0) |
26 | 25 | adantl 484 | . . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ (𝒫 ∅ ∩ Fin)) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑥 ∈ 𝑦 ↦ 𝐴)) = 0) |
27 | 1, 2, 4, 7, 26 | esumval 31305 | . . 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 5613 | . . . . 5 ⊢ ((𝒫 ∅ ∩ Fin) × {0}) = (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) | |
30 | 29 | eqcomi 2830 | . . . 4 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = ((𝒫 ∅ ∩ Fin) × {0}) |
31 | 0xr 10687 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
32 | 31 | rgenw 3150 | . . . . . 6 ⊢ ∀𝑦 ∈ (𝒫 ∅ ∩ Fin)0 ∈ ℝ* |
33 | eqid 2821 | . . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) | |
34 | 33 | fnmpt 6487 | . . . . . 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 4710 | . . . . . 6 ⊢ {∅} ≠ ∅ |
37 | 15, 36 | eqnetri 3086 | . . . . 5 ⊢ (𝒫 ∅ ∩ Fin) ≠ ∅ |
38 | fconst5 6967 | . . . . 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 232 | . . 3 ⊢ ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = {0} |
41 | 40 | supeq1i 8910 | . 2 ⊢ sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ*, < ) = sup({0}, ℝ*, < ) |
42 | xrltso 12533 | . . 3 ⊢ < Or ℝ* | |
43 | supsn 8935 | . . 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 2848 | 1 ⊢ Σ*𝑥 ∈ ∅𝐴 = 0 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 ⊤wtru 1534 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 𝒫 cpw 4538 {csn 4566 ↦ cmpt 5145 Or wor 5472 × cxp 5552 ran crn 5555 Fn wfn 6349 (class class class)co 7155 Fincfn 8508 supcsup 8903 0cc0 10536 +∞cpnf 10671 ℝ*cxr 10673 < clt 10674 [,]cicc 12740 ↾s cress 16483 Σg cgsu 16713 ℝ*𝑠cxrs 16772 Σ*cesum 31286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-fi 8874 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-xadd 12507 df-ioo 12741 df-ioc 12742 df-ico 12743 df-icc 12744 df-fz 12892 df-fzo 13033 df-seq 13369 df-hash 13690 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-tset 16583 df-ple 16584 df-ds 16586 df-rest 16695 df-topn 16696 df-0g 16714 df-gsum 16715 df-topgen 16716 df-ordt 16773 df-xrs 16774 df-mre 16856 df-mrc 16857 df-acs 16859 df-ps 17809 df-tsr 17810 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-cntz 18446 df-cmn 18907 df-fbas 20541 df-fg 20542 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-ntr 21627 df-nei 21705 df-cn 21834 df-haus 21922 df-fil 22453 df-fm 22545 df-flim 22546 df-flf 22547 df-tsms 22734 df-esum 31287 |
This theorem is referenced by: esumrnmpt2 31327 esum2dlem 31351 ddemeas 31495 carsgclctunlem1 31575 |
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