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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 13494 | . . . . 5 ⊢ 0 ∈ (0[,)+∞) | |
4 | 3 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ∈ (0[,)+∞)) |
5 | 2, 4 | fmptd2f 45177 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 0):𝐴⟶(0[,)+∞)) |
6 | 1, 5 | sge0reval 46327 | . 2 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 0)) = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦)), ℝ*, < )) |
7 | eqidd 2735 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (𝑘 ∈ 𝐴 ↦ 0) = (𝑘 ∈ 𝐴 ↦ 0)) | |
8 | eqidd 2735 | . . . . . . . . . 10 ⊢ (((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑥) ∧ 𝑘 = 𝑦) → 0 = 0) | |
9 | elpwinss 44988 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ⊆ 𝐴) | |
10 | 9 | sselda 3994 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝐴) |
11 | 0cnd 11251 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 0 ∈ ℂ) | |
12 | 7, 8, 10, 11 | fvmptd 7022 | . . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦) = 0) |
13 | 12 | adantll 714 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦) = 0) |
14 | 13 | sumeq2dv 15734 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦) = Σ𝑦 ∈ 𝑥 0) |
15 | elinel2 4211 | . . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ Fin) | |
16 | olc 868 | . . . . . . . . 9 ⊢ (𝑥 ∈ Fin → (𝑥 ⊆ (ℤ≥‘𝐵) ∨ 𝑥 ∈ Fin)) | |
17 | sumz 15754 | . . . . . . . . 9 ⊢ ((𝑥 ⊆ (ℤ≥‘𝐵) ∨ 𝑥 ∈ Fin) → Σ𝑦 ∈ 𝑥 0 = 0) | |
18 | 15, 16, 17 | 3syl 18 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → Σ𝑦 ∈ 𝑥 0 = 0) |
19 | 18 | adantl 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑦 ∈ 𝑥 0 = 0) |
20 | 14, 19 | eqtrd 2774 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦) = 0) |
21 | 20 | mpteq2dva 5247 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦)) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0)) |
22 | 21 | rneqd 5951 | . . . 4 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦)) = ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0)) |
23 | eqid 2734 | . . . . 5 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) | |
24 | pwfin0 45001 | . . . . . 6 ⊢ (𝒫 𝐴 ∩ Fin) ≠ ∅ | |
25 | 24 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝒫 𝐴 ∩ Fin) ≠ ∅) |
26 | 23, 25 | rnmptc 7226 | . . . 4 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = {0}) |
27 | 22, 26 | eqtrd 2774 | . . 3 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦)) = {0}) |
28 | 27 | supeq1d 9483 | . 2 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦)), ℝ*, < ) = sup({0}, ℝ*, < )) |
29 | xrltso 13179 | . . . 4 ⊢ < Or ℝ* | |
30 | 29 | a1i 11 | . . 3 ⊢ (𝜑 → < Or ℝ*) |
31 | 0xr 11305 | . . 3 ⊢ 0 ∈ ℝ* | |
32 | supsn 9509 | . . 3 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0) | |
33 | 30, 31, 32 | sylancl 586 | . 2 ⊢ (𝜑 → sup({0}, ℝ*, < ) = 0) |
34 | 6, 28, 33 | 3eqtrd 2778 | 1 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 0)) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1536 Ⅎwnf 1779 ∈ wcel 2105 ≠ wne 2937 ∩ cin 3961 ⊆ wss 3962 ∅c0 4338 𝒫 cpw 4604 {csn 4630 ↦ cmpt 5230 Or wor 5595 ran crn 5689 ‘cfv 6562 (class class class)co 7430 Fincfn 8983 supcsup 9477 ℂcc 11150 0cc0 11152 +∞cpnf 11289 ℝ*cxr 11291 < clt 11292 ℤ≥cuz 12875 [,)cico 13385 Σcsu 15718 Σ^csumge0 46317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-sum 15719 df-sumge0 46318 |
This theorem is referenced by: sge0ss 46367 ismeannd 46422 0ome 46484 isomenndlem 46485 ovn0lem 46520 vonct 46648 |
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