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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 1770 | . . . 4 ⊢ Ⅎ𝑥⊤ | |
2 | nfcv 2793 | . . . 4 ⊢ Ⅎ𝑥∅ | |
3 | 0ex 4823 | . . . . 5 ⊢ ∅ ∈ V | |
4 | 3 | a1i 11 | . . . 4 ⊢ (⊤ → ∅ ∈ V) |
5 | ral0 4109 | . . . . . 6 ⊢ ∀𝑥 ∈ ∅ 𝐴 ∈ (0[,]+∞) | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (⊤ → ∀𝑥 ∈ ∅ 𝐴 ∈ (0[,]+∞)) |
7 | 6 | r19.21bi 2961 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ∅) → 𝐴 ∈ (0[,]+∞)) |
8 | pw0 4375 | . . . . . . . . . . . . 13 ⊢ 𝒫 ∅ = {∅} | |
9 | 8 | ineq1i 3843 | . . . . . . . . . . . 12 ⊢ (𝒫 ∅ ∩ Fin) = ({∅} ∩ Fin) |
10 | 0fin 8229 | . . . . . . . . . . . . 13 ⊢ ∅ ∈ Fin | |
11 | snssi 4371 | . . . . . . . . . . . . . 14 ⊢ (∅ ∈ Fin → {∅} ⊆ Fin) | |
12 | df-ss 3621 | . . . . . . . . . . . . . 14 ⊢ ({∅} ⊆ Fin ↔ ({∅} ∩ Fin) = {∅}) | |
13 | 11, 12 | sylib 208 | . . . . . . . . . . . . 13 ⊢ (∅ ∈ Fin → ({∅} ∩ Fin) = {∅}) |
14 | 10, 13 | ax-mp 5 | . . . . . . . . . . . 12 ⊢ ({∅} ∩ Fin) = {∅} |
15 | 9, 14 | eqtri 2673 | . . . . . . . . . . 11 ⊢ (𝒫 ∅ ∩ Fin) = {∅} |
16 | 15 | eleq2i 2722 | . . . . . . . . . 10 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↔ 𝑦 ∈ {∅}) |
17 | velsn 4226 | . . . . . . . . . 10 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅) | |
18 | 16, 17 | sylbb 209 | . . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → 𝑦 = ∅) |
19 | 18 | mpteq1d 4771 | . . . . . . . 8 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → (𝑥 ∈ 𝑦 ↦ 𝐴) = (𝑥 ∈ ∅ ↦ 𝐴)) |
20 | mpt0 6059 | . . . . . . . 8 ⊢ (𝑥 ∈ ∅ ↦ 𝐴) = ∅ | |
21 | 19, 20 | syl6eq 2701 | . . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → (𝑥 ∈ 𝑦 ↦ 𝐴) = ∅) |
22 | 21 | oveq2d 6706 | . . . . . 6 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑥 ∈ 𝑦 ↦ 𝐴)) = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg ∅)) |
23 | xrge00 29814 | . . . . . . 7 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
24 | 23 | gsum0 17325 | . . . . . 6 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) Σg ∅) = 0 |
25 | 22, 24 | syl6eq 2701 | . . . . 5 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑥 ∈ 𝑦 ↦ 𝐴)) = 0) |
26 | 25 | adantl 481 | . . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ (𝒫 ∅ ∩ Fin)) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑥 ∈ 𝑦 ↦ 𝐴)) = 0) |
27 | 1, 2, 4, 7, 26 | esumval 30236 | . . 3 ⊢ (⊤ → Σ*𝑥 ∈ ∅𝐴 = sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ*, < )) |
28 | 27 | trud 1533 | . 2 ⊢ Σ*𝑥 ∈ ∅𝐴 = sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ*, < ) |
29 | fconstmpt 5197 | . . . . 5 ⊢ ((𝒫 ∅ ∩ Fin) × {0}) = (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) | |
30 | 29 | eqcomi 2660 | . . . 4 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = ((𝒫 ∅ ∩ Fin) × {0}) |
31 | 0xr 10124 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
32 | 31 | rgenw 2953 | . . . . . 6 ⊢ ∀𝑦 ∈ (𝒫 ∅ ∩ Fin)0 ∈ ℝ* |
33 | eqid 2651 | . . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) | |
34 | 33 | fnmpt 6058 | . . . . . 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 4340 | . . . . . 6 ⊢ {∅} ≠ ∅ |
37 | 15, 36 | eqnetri 2893 | . . . . 5 ⊢ (𝒫 ∅ ∩ Fin) ≠ ∅ |
38 | fconst5 6512 | . . . . 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 708 | . . . 4 ⊢ ((𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = ((𝒫 ∅ ∩ Fin) × {0}) ↔ ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = {0}) |
40 | 30, 39 | mpbi 220 | . . 3 ⊢ ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = {0} |
41 | 40 | supeq1i 8394 | . 2 ⊢ sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ*, < ) = sup({0}, ℝ*, < ) |
42 | xrltso 12012 | . . 3 ⊢ < Or ℝ* | |
43 | supsn 8419 | . . 3 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0) | |
44 | 42, 31, 43 | mp2an 708 | . 2 ⊢ sup({0}, ℝ*, < ) = 0 |
45 | 28, 41, 44 | 3eqtri 2677 | 1 ⊢ Σ*𝑥 ∈ ∅𝐴 = 0 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1523 ⊤wtru 1524 ∈ wcel 2030 ≠ wne 2823 ∀wral 2941 Vcvv 3231 ∩ cin 3606 ⊆ wss 3607 ∅c0 3948 𝒫 cpw 4191 {csn 4210 ↦ cmpt 4762 Or wor 5063 × cxp 5141 ran crn 5144 Fn wfn 5921 (class class class)co 6690 Fincfn 7997 supcsup 8387 0cc0 9974 +∞cpnf 10109 ℝ*cxr 10111 < clt 10112 [,]cicc 12216 ↾s cress 15905 Σg cgsu 16148 ℝ*𝑠cxrs 16207 Σ*cesum 30217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-fi 8358 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-q 11827 df-xadd 11985 df-ioo 12217 df-ioc 12218 df-ico 12219 df-icc 12220 df-fz 12365 df-fzo 12505 df-seq 12842 df-hash 13158 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-tset 16007 df-ple 16008 df-ds 16011 df-rest 16130 df-topn 16131 df-0g 16149 df-gsum 16150 df-topgen 16151 df-ordt 16208 df-xrs 16209 df-mre 16293 df-mrc 16294 df-acs 16296 df-ps 17247 df-tsr 17248 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-cntz 17796 df-cmn 18241 df-fbas 19791 df-fg 19792 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-ntr 20872 df-nei 20950 df-cn 21079 df-haus 21167 df-fil 21697 df-fm 21789 df-flim 21790 df-flf 21791 df-tsms 21977 df-esum 30218 |
This theorem is referenced by: esumrnmpt2 30258 esum2dlem 30282 ddemeas 30427 carsgclctunlem1 30507 |
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