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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0z | Structured version Visualization version GIF version |
Description: Any nonnegative extended sum of zero is zero. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
sge0z.1 | ⊢ Ⅎ𝑘𝜑 |
sge0z.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
sge0z | ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 0)) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sge0z.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | sge0z.1 | . . . 4 ⊢ Ⅎ𝑘𝜑 | |
3 | 0e0icopnf 12572 | . . . . 5 ⊢ 0 ∈ (0[,)+∞) | |
4 | 3 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ∈ (0[,)+∞)) |
5 | eqid 2825 | . . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ 0) = (𝑘 ∈ 𝐴 ↦ 0) | |
6 | 2, 4, 5 | fmptdf 6636 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 0):𝐴⟶(0[,)+∞)) |
7 | 1, 6 | sge0reval 41373 | . 2 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 0)) = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦)), ℝ*, < )) |
8 | eqidd 2826 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (𝑘 ∈ 𝐴 ↦ 0) = (𝑘 ∈ 𝐴 ↦ 0)) | |
9 | eqid 2825 | . . . . . . . . . . 11 ⊢ 0 = 0 | |
10 | 9 | a1i 11 | . . . . . . . . . 10 ⊢ (((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑥) ∧ 𝑘 = 𝑦) → 0 = 0) |
11 | elpwinss 40026 | . . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ⊆ 𝐴) | |
12 | 11 | adantr 474 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑥 ⊆ 𝐴) |
13 | simpr 479 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥) | |
14 | 12, 13 | sseldd 3828 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝐴) |
15 | 0cnd 10349 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 0 ∈ ℂ) | |
16 | 8, 10, 14, 15 | fvmptd 6535 | . . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦) = 0) |
17 | 16 | adantll 705 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦) = 0) |
18 | 17 | sumeq2dv 14810 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦) = Σ𝑦 ∈ 𝑥 0) |
19 | elinel2 4027 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ Fin) | |
20 | olc 899 | . . . . . . . . . 10 ⊢ (𝑥 ∈ Fin → (𝑥 ⊆ (ℤ≥‘𝐵) ∨ 𝑥 ∈ Fin)) | |
21 | 19, 20 | syl 17 | . . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → (𝑥 ⊆ (ℤ≥‘𝐵) ∨ 𝑥 ∈ Fin)) |
22 | sumz 14830 | . . . . . . . . 9 ⊢ ((𝑥 ⊆ (ℤ≥‘𝐵) ∨ 𝑥 ∈ Fin) → Σ𝑦 ∈ 𝑥 0 = 0) | |
23 | 21, 22 | syl 17 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → Σ𝑦 ∈ 𝑥 0 = 0) |
24 | 23 | adantl 475 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑦 ∈ 𝑥 0 = 0) |
25 | 18, 24 | eqtrd 2861 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦) = 0) |
26 | 25 | mpteq2dva 4967 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦)) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0)) |
27 | 26 | rneqd 5585 | . . . 4 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦)) = ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0)) |
28 | eqid 2825 | . . . . 5 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) | |
29 | 0cnd 10349 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 0 ∈ ℂ) | |
30 | pwfin0 40041 | . . . . . 6 ⊢ (𝒫 𝐴 ∩ Fin) ≠ ∅ | |
31 | 30 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝒫 𝐴 ∩ Fin) ≠ ∅) |
32 | 28, 29, 31 | rnmptc 40155 | . . . 4 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = {0}) |
33 | 27, 32 | eqtrd 2861 | . . 3 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦)) = {0}) |
34 | 33 | supeq1d 8621 | . 2 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦)), ℝ*, < ) = sup({0}, ℝ*, < )) |
35 | xrltso 12260 | . . . 4 ⊢ < Or ℝ* | |
36 | 35 | a1i 11 | . . 3 ⊢ (𝜑 → < Or ℝ*) |
37 | 0xr 10403 | . . . 4 ⊢ 0 ∈ ℝ* | |
38 | 37 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ*) |
39 | supsn 8647 | . . 3 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0) | |
40 | 36, 38, 39 | syl2anc 579 | . 2 ⊢ (𝜑 → sup({0}, ℝ*, < ) = 0) |
41 | 7, 34, 40 | 3eqtrd 2865 | 1 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 0)) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∨ wo 878 = wceq 1656 Ⅎwnf 1882 ∈ wcel 2164 ≠ wne 2999 ∩ cin 3797 ⊆ wss 3798 ∅c0 4144 𝒫 cpw 4378 {csn 4397 ↦ cmpt 4952 Or wor 5262 ran crn 5343 ‘cfv 6123 (class class class)co 6905 Fincfn 8222 supcsup 8615 ℂcc 10250 0cc0 10252 +∞cpnf 10388 ℝ*cxr 10390 < clt 10391 ℤ≥cuz 11968 [,)cico 12465 Σcsu 14793 Σ^csumge0 41363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-sup 8617 df-oi 8684 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-n0 11619 df-z 11705 df-uz 11969 df-rp 12113 df-ico 12469 df-icc 12470 df-fz 12620 df-fzo 12761 df-seq 13096 df-exp 13155 df-hash 13411 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-clim 14596 df-sum 14794 df-sumge0 41364 |
This theorem is referenced by: sge0ss 41413 ismeannd 41468 0ome 41530 isomenndlem 41531 ovn0lem 41566 vonct 41694 |
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