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 1812 | . . . 4 ⊢ Ⅎ𝑥⊤ | |
2 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑥∅ | |
3 | 0ex 5200 | . . . . 5 ⊢ ∅ ∈ V | |
4 | 3 | a1i 11 | . . . 4 ⊢ (⊤ → ∅ ∈ V) |
5 | ral0 4424 | . . . . . 6 ⊢ ∀𝑥 ∈ ∅ 𝐴 ∈ (0[,]+∞) | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (⊤ → ∀𝑥 ∈ ∅ 𝐴 ∈ (0[,]+∞)) |
7 | 6 | r19.21bi 3130 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ∅) → 𝐴 ∈ (0[,]+∞)) |
8 | pw0 4725 | . . . . . . . . . . . . 13 ⊢ 𝒫 ∅ = {∅} | |
9 | 8 | ineq1i 4123 | . . . . . . . . . . . 12 ⊢ (𝒫 ∅ ∩ Fin) = ({∅} ∩ Fin) |
10 | 0fin 8849 | . . . . . . . . . . . . 13 ⊢ ∅ ∈ Fin | |
11 | snssi 4721 | . . . . . . . . . . . . . 14 ⊢ (∅ ∈ Fin → {∅} ⊆ Fin) | |
12 | df-ss 3883 | . . . . . . . . . . . . . 14 ⊢ ({∅} ⊆ Fin ↔ ({∅} ∩ Fin) = {∅}) | |
13 | 11, 12 | sylib 221 | . . . . . . . . . . . . 13 ⊢ (∅ ∈ Fin → ({∅} ∩ Fin) = {∅}) |
14 | 10, 13 | ax-mp 5 | . . . . . . . . . . . 12 ⊢ ({∅} ∩ Fin) = {∅} |
15 | 9, 14 | eqtri 2765 | . . . . . . . . . . 11 ⊢ (𝒫 ∅ ∩ Fin) = {∅} |
16 | 15 | eleq2i 2829 | . . . . . . . . . 10 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↔ 𝑦 ∈ {∅}) |
17 | velsn 4557 | . . . . . . . . . 10 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅) | |
18 | 16, 17 | sylbb 222 | . . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → 𝑦 = ∅) |
19 | 18 | mpteq1d 5144 | . . . . . . . 8 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → (𝑥 ∈ 𝑦 ↦ 𝐴) = (𝑥 ∈ ∅ ↦ 𝐴)) |
20 | mpt0 6520 | . . . . . . . 8 ⊢ (𝑥 ∈ ∅ ↦ 𝐴) = ∅ | |
21 | 19, 20 | eqtrdi 2794 | . . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → (𝑥 ∈ 𝑦 ↦ 𝐴) = ∅) |
22 | 21 | oveq2d 7229 | . . . . . 6 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑥 ∈ 𝑦 ↦ 𝐴)) = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg ∅)) |
23 | xrge00 31014 | . . . . . . 7 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
24 | 23 | gsum0 18156 | . . . . . 6 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) Σg ∅) = 0 |
25 | 22, 24 | eqtrdi 2794 | . . . . 5 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑥 ∈ 𝑦 ↦ 𝐴)) = 0) |
26 | 25 | adantl 485 | . . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ (𝒫 ∅ ∩ Fin)) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑥 ∈ 𝑦 ↦ 𝐴)) = 0) |
27 | 1, 2, 4, 7, 26 | esumval 31726 | . . 3 ⊢ (⊤ → Σ*𝑥 ∈ ∅𝐴 = sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ*, < )) |
28 | 27 | mptru 1550 | . 2 ⊢ Σ*𝑥 ∈ ∅𝐴 = sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ*, < ) |
29 | fconstmpt 5611 | . . . . 5 ⊢ ((𝒫 ∅ ∩ Fin) × {0}) = (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) | |
30 | 29 | eqcomi 2746 | . . . 4 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = ((𝒫 ∅ ∩ Fin) × {0}) |
31 | 0xr 10880 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
32 | 31 | rgenw 3073 | . . . . . 6 ⊢ ∀𝑦 ∈ (𝒫 ∅ ∩ Fin)0 ∈ ℝ* |
33 | eqid 2737 | . . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) | |
34 | 33 | fnmpt 6518 | . . . . . 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 4692 | . . . . . 6 ⊢ {∅} ≠ ∅ |
37 | 15, 36 | eqnetri 3011 | . . . . 5 ⊢ (𝒫 ∅ ∩ Fin) ≠ ∅ |
38 | fconst5 7021 | . . . . 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 692 | . . . 4 ⊢ ((𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = ((𝒫 ∅ ∩ Fin) × {0}) ↔ ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = {0}) |
40 | 30, 39 | mpbi 233 | . . 3 ⊢ ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = {0} |
41 | 40 | supeq1i 9063 | . 2 ⊢ sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ*, < ) = sup({0}, ℝ*, < ) |
42 | xrltso 12731 | . . 3 ⊢ < Or ℝ* | |
43 | supsn 9088 | . . 3 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0) | |
44 | 42, 31, 43 | mp2an 692 | . 2 ⊢ sup({0}, ℝ*, < ) = 0 |
45 | 28, 41, 44 | 3eqtri 2769 | 1 ⊢ Σ*𝑥 ∈ ∅𝐴 = 0 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ⊤wtru 1544 ∈ wcel 2110 ≠ wne 2940 ∀wral 3061 Vcvv 3408 ∩ cin 3865 ⊆ wss 3866 ∅c0 4237 𝒫 cpw 4513 {csn 4541 ↦ cmpt 5135 Or wor 5467 × cxp 5549 ran crn 5552 Fn wfn 6375 (class class class)co 7213 Fincfn 8626 supcsup 9056 0cc0 10729 +∞cpnf 10864 ℝ*cxr 10866 < clt 10867 [,]cicc 12938 ↾s cress 16784 Σg cgsu 16945 ℝ*𝑠cxrs 17005 Σ*cesum 31707 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-xadd 12705 df-ioo 12939 df-ioc 12940 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-seq 13575 df-hash 13897 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-tset 16821 df-ple 16822 df-ds 16824 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-ordt 17006 df-xrs 17007 df-mre 17089 df-mrc 17090 df-acs 17092 df-ps 18072 df-tsr 18073 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-cntz 18711 df-cmn 19172 df-fbas 20360 df-fg 20361 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-ntr 21917 df-nei 21995 df-cn 22124 df-haus 22212 df-fil 22743 df-fm 22835 df-flim 22836 df-flf 22837 df-tsms 23024 df-esum 31708 |
This theorem is referenced by: esumrnmpt2 31748 esum2dlem 31772 ddemeas 31916 carsgclctunlem1 31996 |
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