Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumval | Structured version Visualization version GIF version |
Description: Develop the value of the extended sum. (Contributed by Thierry Arnoux, 4-Jan-2017.) |
Ref | Expression |
---|---|
esumval.p | ⊢ Ⅎ𝑘𝜑 |
esumval.0 | ⊢ Ⅎ𝑘𝐴 |
esumval.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
esumval.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
esumval.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝑥 ↦ 𝐵)) = 𝐶) |
Ref | Expression |
---|---|
esumval | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-esum 31287 | . . 3 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
2 | eqid 2821 | . . . . 5 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s (0[,]+∞)) | |
3 | esumval.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | esumval.p | . . . . . 6 ⊢ Ⅎ𝑘𝜑 | |
5 | esumval.0 | . . . . . 6 ⊢ Ⅎ𝑘𝐴 | |
6 | nfcv 2977 | . . . . . 6 ⊢ Ⅎ𝑘(0[,]+∞) | |
7 | esumval.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
8 | eqid 2821 | . . . . . 6 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
9 | 4, 5, 6, 7, 8 | fmptdF 30401 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
10 | inss1 4205 | . . . . . . . . . . . . . 14 ⊢ (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴 | |
11 | 10 | sseli 3963 | . . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ 𝒫 𝐴) |
12 | 11 | elpwid 4550 | . . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ⊆ 𝐴) |
13 | 12 | adantl 484 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑥 ⊆ 𝐴) |
14 | nfcv 2977 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝑥 | |
15 | 5, 14 | resmptf 5907 | . . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥) = (𝑘 ∈ 𝑥 ↦ 𝐵)) |
16 | 13, 15 | syl 17 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥) = (𝑘 ∈ 𝑥 ↦ 𝐵)) |
17 | 16 | oveq2d 7172 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥)) = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝑥 ↦ 𝐵))) |
18 | esumval.3 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝑥 ↦ 𝐵)) = 𝐶) | |
19 | 17, 18 | eqtr2d 2857 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐶 = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥))) |
20 | 19 | mpteq2dva 5161 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ*𝑠 ↾s (0[,]+∞)) Σg ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥)))) |
21 | 20 | rneqd 5808 | . . . . . 6 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶) = ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ*𝑠 ↾s (0[,]+∞)) Σg ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥)))) |
22 | 21 | supeq1d 8910 | . . . . 5 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ*𝑠 ↾s (0[,]+∞)) Σg ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥))), ℝ*, < )) |
23 | 2, 3, 9, 22 | xrge0tsmsd 30692 | . . . 4 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = {sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ*, < )}) |
24 | 23 | unieqd 4852 | . . 3 ⊢ (𝜑 → ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ∪ {sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ*, < )}) |
25 | 1, 24 | syl5eq 2868 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = ∪ {sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ*, < )}) |
26 | xrltso 12535 | . . . 4 ⊢ < Or ℝ* | |
27 | 26 | supex 8927 | . . 3 ⊢ sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ*, < ) ∈ V |
28 | 27 | unisn 4858 | . 2 ⊢ ∪ {sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ*, < )} = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ*, < ) |
29 | 25, 28 | syl6eq 2872 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2961 ∩ cin 3935 ⊆ wss 3936 𝒫 cpw 4539 {csn 4567 ∪ cuni 4838 ↦ cmpt 5146 ran crn 5556 ↾ cres 5557 (class class class)co 7156 Fincfn 8509 supcsup 8904 0cc0 10537 +∞cpnf 10672 ℝ*cxr 10674 < clt 10675 [,]cicc 12742 ↾s cress 16484 Σg cgsu 16714 ℝ*𝑠cxrs 16773 tsums ctsu 22734 Σ*cesum 31286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-xadd 12509 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-tset 16584 df-ple 16585 df-ds 16587 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-ordt 16774 df-xrs 16775 df-mre 16857 df-mrc 16858 df-acs 16860 df-ps 17810 df-tsr 17811 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-cntz 18447 df-cmn 18908 df-fbas 20542 df-fg 20543 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-ntr 21628 df-nei 21706 df-cn 21835 df-haus 21923 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-tsms 22735 df-esum 31287 |
This theorem is referenced by: esumel 31306 esumnul 31307 esum0 31308 gsumesum 31318 esumlub 31319 esumcst 31322 esumpcvgval 31337 esumcvg 31345 esum2d 31352 |
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