Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0revalmpt | Structured version Visualization version GIF version |
Description: Value of the sum of nonnegative extended reals, when all terms in the sum are reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
sge0revalmpt.1 | ⊢ Ⅎ𝑥𝜑 |
sge0revalmpt.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
sge0revalmpt.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) |
Ref | Expression |
---|---|
sge0revalmpt | ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = sup(ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑥 ∈ 𝑦 𝐵), ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sge0revalmpt.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | sge0revalmpt.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
3 | sge0revalmpt.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) | |
4 | eqid 2818 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
5 | 2, 3, 4 | fmptdf 6873 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,)+∞)) |
6 | 1, 5 | sge0reval 42531 | . 2 ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = sup(ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑧 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)), ℝ*, < )) |
7 | fveq2 6663 | . . . . . . . 8 ⊢ (𝑧 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) | |
8 | nfcv 2974 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑦 | |
9 | nfcv 2974 | . . . . . . . 8 ⊢ Ⅎ𝑧𝑦 | |
10 | nfmpt1 5155 | . . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | |
11 | nfcv 2974 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝑧 | |
12 | 10, 11 | nffv 6673 | . . . . . . . 8 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) |
13 | nfcv 2974 | . . . . . . . 8 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) | |
14 | 7, 8, 9, 12, 13 | cbvsum 15040 | . . . . . . 7 ⊢ Σ𝑧 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) = Σ𝑥 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) |
15 | 14 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑧 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) = Σ𝑥 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) |
16 | nfv 1906 | . . . . . . . . 9 ⊢ Ⅎ𝑥 𝑦 ∈ (𝒫 𝐴 ∩ Fin) | |
17 | 2, 16 | nfan 1891 | . . . . . . . 8 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) |
18 | elpwinss 41188 | . . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ⊆ 𝐴) | |
19 | 18 | adantr 481 | . . . . . . . . . . . 12 ⊢ ((𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑥 ∈ 𝑦) → 𝑦 ⊆ 𝐴) |
20 | simpr 485 | . . . . . . . . . . . 12 ⊢ ((𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ 𝑦) | |
21 | 19, 20 | sseldd 3965 | . . . . . . . . . . 11 ⊢ ((𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ 𝐴) |
22 | 21 | adantll 710 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ 𝐴) |
23 | simpll 763 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → 𝜑) | |
24 | 23, 22, 3 | syl2anc 584 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → 𝐵 ∈ (0[,)+∞)) |
25 | 4 | fvmpt2 6771 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ (0[,)+∞)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
26 | 22, 24, 25 | syl2anc 584 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
27 | 26 | ex 413 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑥 ∈ 𝑦 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)) |
28 | 17, 27 | ralrimi 3213 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ∀𝑥 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
29 | sumeq2 15039 | . . . . . . 7 ⊢ (∀𝑥 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵 → Σ𝑥 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = Σ𝑥 ∈ 𝑦 𝐵) | |
30 | 28, 29 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑥 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = Σ𝑥 ∈ 𝑦 𝐵) |
31 | 15, 30 | eqtrd 2853 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → Σ𝑧 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) = Σ𝑥 ∈ 𝑦 𝐵) |
32 | 31 | mpteq2dva 5152 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑧 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)) = (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑥 ∈ 𝑦 𝐵)) |
33 | 32 | rneqd 5801 | . . 3 ⊢ (𝜑 → ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑧 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)) = ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑥 ∈ 𝑦 𝐵)) |
34 | 33 | supeq1d 8898 | . 2 ⊢ (𝜑 → sup(ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑧 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)), ℝ*, < ) = sup(ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑥 ∈ 𝑦 𝐵), ℝ*, < )) |
35 | 6, 34 | eqtrd 2853 | 1 ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = sup(ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑥 ∈ 𝑦 𝐵), ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 ∀wral 3135 ∩ cin 3932 ⊆ wss 3933 𝒫 cpw 4535 ↦ cmpt 5137 ran crn 5549 ‘cfv 6348 (class class class)co 7145 Fincfn 8497 supcsup 8892 0cc0 10525 +∞cpnf 10660 ℝ*cxr 10662 < clt 10663 [,)cico 12728 Σcsu 15030 Σ^csumge0 42521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-ico 12732 df-icc 12733 df-fz 12881 df-seq 13358 df-sum 15031 df-sumge0 42522 |
This theorem is referenced by: sge0f1o 42541 sge0xaddlem1 42592 sge0xaddlem2 42593 sge0reuz 42606 |
Copyright terms: Public domain | W3C validator |