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 12153 | . . . . 5 ⊢ 0 ∈ (0[,)+∞) | |
4 | 3 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ∈ (0[,)+∞)) |
5 | eqid 2610 | . . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ 0) = (𝑘 ∈ 𝐴 ↦ 0) | |
6 | 2, 4, 5 | fmptdf 6294 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 0):𝐴⟶(0[,)+∞)) |
7 | 1, 6 | sge0reval 39265 | . 2 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 0)) = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦)), ℝ*, < )) |
8 | eqidd 2611 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (𝑘 ∈ 𝐴 ↦ 0) = (𝑘 ∈ 𝐴 ↦ 0)) | |
9 | eqid 2610 | . . . . . . . . . . 11 ⊢ 0 = 0 | |
10 | 9 | a1i 11 | . . . . . . . . . 10 ⊢ (((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑥) ∧ 𝑘 = 𝑦) → 0 = 0) |
11 | elpwinss 38241 | . . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ⊆ 𝐴) | |
12 | 11 | adantr 480 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑥 ⊆ 𝐴) |
13 | simpr 476 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥) | |
14 | 12, 13 | sseldd 3569 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝐴) |
15 | 0cnd 9912 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 0 ∈ ℂ) | |
16 | 8, 10, 14, 15 | fvmptd 6197 | . . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦) = 0) |
17 | 16 | adantll 746 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦) = 0) |
18 | 17 | sumeq2dv 14281 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦) = Σ𝑦 ∈ 𝑥 0) |
19 | elinel2 3762 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ Fin) | |
20 | olc 398 | . . . . . . . . . 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 14300 | . . . . . . . . 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 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑦 ∈ 𝑥 0 = 0) |
25 | 18, 24 | eqtrd 2644 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦) = 0) |
26 | 25 | mpteq2dva 4672 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦)) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0)) |
27 | 26 | rneqd 5274 | . . . 4 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦)) = ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0)) |
28 | eqid 2610 | . . . . 5 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) | |
29 | 0cnd 9912 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 0 ∈ ℂ) | |
30 | pwfin0 38256 | . . . . . 6 ⊢ (𝒫 𝐴 ∩ Fin) ≠ ∅ | |
31 | 30 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝒫 𝐴 ∩ Fin) ≠ ∅) |
32 | 28, 29, 31 | rnmptc 38348 | . . . 4 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = {0}) |
33 | 27, 32 | eqtrd 2644 | . . 3 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦)) = {0}) |
34 | 33 | supeq1d 8235 | . 2 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 ((𝑘 ∈ 𝐴 ↦ 0)‘𝑦)), ℝ*, < ) = sup({0}, ℝ*, < )) |
35 | xrltso 11850 | . . . 4 ⊢ < Or ℝ* | |
36 | 35 | a1i 11 | . . 3 ⊢ (𝜑 → < Or ℝ*) |
37 | 0xr 9965 | . . . 4 ⊢ 0 ∈ ℝ* | |
38 | 37 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ*) |
39 | supsn 8261 | . . 3 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0) | |
40 | 36, 38, 39 | syl2anc 691 | . 2 ⊢ (𝜑 → sup({0}, ℝ*, < ) = 0) |
41 | 7, 34, 40 | 3eqtrd 2648 | 1 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 0)) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 ≠ wne 2780 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 ↦ cmpt 4643 Or wor 4958 ran crn 5039 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 supcsup 8229 ℂcc 9813 0cc0 9815 +∞cpnf 9950 ℝ*cxr 9952 < clt 9953 ℤ≥cuz 11563 [,)cico 12048 Σcsu 14264 Σ^csumge0 39255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-sumge0 39256 |
This theorem is referenced by: sge0ss 39305 ismeannd 39360 0ome 39419 isomenndlem 39420 ovn0lem 39455 vonct 39584 |
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