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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 13498 | . . . . 5 ⊢ 0 ∈ (0[,)+∞) | |
| 4 | 3 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ∈ (0[,)+∞)) |
| 5 | 2, 4 | fmptd2f 45240 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 0):𝐴⟶(0[,)+∞)) |
| 6 | 1, 5 | sge0reval 46387 | . 2 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 0)) = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦)), ℝ*, < )) |
| 7 | eqidd 2738 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (𝑘 ∈ 𝐴 ↦ 0) = (𝑘 ∈ 𝐴 ↦ 0)) | |
| 8 | eqidd 2738 | . . . . . . . . . 10 ⊢ (((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑥) ∧ 𝑘 = 𝑦) → 0 = 0) | |
| 9 | elpwinss 45054 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ⊆ 𝐴) | |
| 10 | 9 | sselda 3983 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝐴) |
| 11 | 0cnd 11254 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 0 ∈ ℂ) | |
| 12 | 7, 8, 10, 11 | fvmptd 7023 | . . . . . . . . 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 15738 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦) = Σ𝑦 ∈ 𝑥 0) |
| 15 | elinel2 4202 | . . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ Fin) | |
| 16 | olc 869 | . . . . . . . . 9 ⊢ (𝑥 ∈ Fin → (𝑥 ⊆ (ℤ≥‘𝐵) ∨ 𝑥 ∈ Fin)) | |
| 17 | sumz 15758 | . . . . . . . . 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 2777 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦) = 0) |
| 21 | 20 | mpteq2dva 5242 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦)) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0)) |
| 22 | 21 | rneqd 5949 | . . . 4 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦)) = ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0)) |
| 23 | eqid 2737 | . . . . 5 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) | |
| 24 | pwfin0 45067 | . . . . . 6 ⊢ (𝒫 𝐴 ∩ Fin) ≠ ∅ | |
| 25 | 24 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝒫 𝐴 ∩ Fin) ≠ ∅) |
| 26 | 23, 25 | rnmptc 7227 | . . . 4 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = {0}) |
| 27 | 22, 26 | eqtrd 2777 | . . 3 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦)) = {0}) |
| 28 | 27 | supeq1d 9486 | . 2 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦)), ℝ*, < ) = sup({0}, ℝ*, < )) |
| 29 | xrltso 13183 | . . . 4 ⊢ < Or ℝ* | |
| 30 | 29 | a1i 11 | . . 3 ⊢ (𝜑 → < Or ℝ*) |
| 31 | 0xr 11308 | . . 3 ⊢ 0 ∈ ℝ* | |
| 32 | supsn 9512 | . . 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 2781 | 1 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 0)) = 0) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 ≠ wne 2940 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 {csn 4626 ↦ cmpt 5225 Or wor 5591 ran crn 5686 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 supcsup 9480 ℂcc 11153 0cc0 11155 +∞cpnf 11292 ℝ*cxr 11294 < clt 11295 ℤ≥cuz 12878 [,)cico 13389 Σcsu 15722 Σ^csumge0 46377 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-sumge0 46378 |
| This theorem is referenced by: sge0ss 46427 ismeannd 46482 0ome 46544 isomenndlem 46545 ovn0lem 46580 vonct 46708 |
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