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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumgect | Structured version Visualization version GIF version |
Description: "Send 𝑛 to +∞ " in an inequality with an extended sum. (Contributed by Thierry Arnoux, 24-May-2020.) |
Ref | Expression |
---|---|
esumsup.1 | ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) |
esumsup.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞)) |
esumgect.1 | ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ*𝑘 ∈ (1...𝑛)𝐴 ≤ 𝐵) |
Ref | Expression |
---|---|
esumgect | ⊢ (𝜑 → Σ*𝑘 ∈ ℕ𝐴 ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | esumsup.1 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) | |
2 | esumsup.2 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞)) | |
3 | 1, 2 | esumsup 33576 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = sup(ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴), ℝ*, < )) |
4 | nfv 1909 | . . . . . 6 ⊢ Ⅎ𝑛𝜑 | |
5 | nfcv 2895 | . . . . . . 7 ⊢ Ⅎ𝑛𝑧 | |
6 | nfmpt1 5246 | . . . . . . . 8 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴) | |
7 | 6 | nfrn 5941 | . . . . . . 7 ⊢ Ⅎ𝑛ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴) |
8 | 5, 7 | nfel 2909 | . . . . . 6 ⊢ Ⅎ𝑛 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴) |
9 | 4, 8 | nfan 1894 | . . . . 5 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴)) |
10 | simpr 484 | . . . . . 6 ⊢ ((((𝜑 ∧ 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = Σ*𝑘 ∈ (1...𝑛)𝐴) → 𝑧 = Σ*𝑘 ∈ (1...𝑛)𝐴) | |
11 | simplll 772 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = Σ*𝑘 ∈ (1...𝑛)𝐴) → 𝜑) | |
12 | simplr 766 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = Σ*𝑘 ∈ (1...𝑛)𝐴) → 𝑛 ∈ ℕ) | |
13 | esumgect.1 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ*𝑘 ∈ (1...𝑛)𝐴 ≤ 𝐵) | |
14 | 11, 12, 13 | syl2anc 583 | . . . . . 6 ⊢ ((((𝜑 ∧ 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = Σ*𝑘 ∈ (1...𝑛)𝐴) → Σ*𝑘 ∈ (1...𝑛)𝐴 ≤ 𝐵) |
15 | 10, 14 | eqbrtrd 5160 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = Σ*𝑘 ∈ (1...𝑛)𝐴) → 𝑧 ≤ 𝐵) |
16 | eqid 2724 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴) = (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴) | |
17 | esumex 33516 | . . . . . . . 8 ⊢ Σ*𝑘 ∈ (1...𝑛)𝐴 ∈ V | |
18 | 16, 17 | elrnmpti 5949 | . . . . . . 7 ⊢ (𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴) ↔ ∃𝑛 ∈ ℕ 𝑧 = Σ*𝑘 ∈ (1...𝑛)𝐴) |
19 | 18 | biimpi 215 | . . . . . 6 ⊢ (𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴) → ∃𝑛 ∈ ℕ 𝑧 = Σ*𝑘 ∈ (1...𝑛)𝐴) |
20 | 19 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴)) → ∃𝑛 ∈ ℕ 𝑧 = Σ*𝑘 ∈ (1...𝑛)𝐴) |
21 | 9, 15, 20 | r19.29af 3257 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴)) → 𝑧 ≤ 𝐵) |
22 | 21 | ralrimiva 3138 | . . 3 ⊢ (𝜑 → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴)𝑧 ≤ 𝐵) |
23 | ovexd 7436 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ V) | |
24 | simpll 764 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑) | |
25 | fz1ssnn 13529 | . . . . . . . . . . . 12 ⊢ (1...𝑛) ⊆ ℕ | |
26 | 25 | a1i 11 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ) |
27 | 26 | sselda 3974 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ) |
28 | 24, 27, 2 | syl2anc 583 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (0[,]+∞)) |
29 | 28 | ralrimiva 3138 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞)) |
30 | nfcv 2895 | . . . . . . . . 9 ⊢ Ⅎ𝑘(1...𝑛) | |
31 | 30 | esumcl 33517 | . . . . . . . 8 ⊢ (((1...𝑛) ∈ V ∧ ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞)) → Σ*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞)) |
32 | 23, 29, 31 | syl2anc 583 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞)) |
33 | 32 | ralrimiva 3138 | . . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ Σ*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞)) |
34 | 16 | rnmptss 7114 | . . . . . 6 ⊢ (∀𝑛 ∈ ℕ Σ*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞) → ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴) ⊆ (0[,]+∞)) |
35 | 33, 34 | syl 17 | . . . . 5 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴) ⊆ (0[,]+∞)) |
36 | iccssxr 13404 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
37 | 35, 36 | sstrdi 3986 | . . . 4 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴) ⊆ ℝ*) |
38 | 36, 1 | sselid 3972 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
39 | supxrleub 13302 | . . . 4 ⊢ ((ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴) ⊆ ℝ* ∧ 𝐵 ∈ ℝ*) → (sup(ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴), ℝ*, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴)𝑧 ≤ 𝐵)) | |
40 | 37, 38, 39 | syl2anc 583 | . . 3 ⊢ (𝜑 → (sup(ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴), ℝ*, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴)𝑧 ≤ 𝐵)) |
41 | 22, 40 | mpbird 257 | . 2 ⊢ (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴), ℝ*, < ) ≤ 𝐵) |
42 | 3, 41 | eqbrtrd 5160 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ ℕ𝐴 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ∃wrex 3062 Vcvv 3466 ⊆ wss 3940 class class class wbr 5138 ↦ cmpt 5221 ran crn 5667 (class class class)co 7401 supcsup 9431 0cc0 11106 1c1 11107 +∞cpnf 11242 ℝ*cxr 11244 < clt 11245 ≤ cle 11246 ℕcn 12209 [,]cicc 13324 ...cfz 13481 Σ*cesum 33514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ioc 13326 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15011 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-limsup 15412 df-clim 15429 df-rlim 15430 df-sum 15630 df-ef 16008 df-sin 16010 df-cos 16011 df-pi 16013 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-ordt 17446 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-ps 18521 df-tsr 18522 df-plusf 18562 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18703 df-submnd 18704 df-grp 18856 df-minusg 18857 df-sbg 18858 df-mulg 18986 df-subg 19040 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-subrng 20436 df-subrg 20461 df-abv 20650 df-lmod 20698 df-scaf 20699 df-sra 21011 df-rgmod 21012 df-psmet 21220 df-xmet 21221 df-met 21222 df-bl 21223 df-mopn 21224 df-fbas 21225 df-fg 21226 df-cnfld 21229 df-top 22718 df-topon 22735 df-topsp 22757 df-bases 22771 df-cld 22845 df-ntr 22846 df-cls 22847 df-nei 22924 df-lp 22962 df-perf 22963 df-cn 23053 df-cnp 23054 df-haus 23141 df-tx 23388 df-hmeo 23581 df-fil 23672 df-fm 23764 df-flim 23765 df-flf 23766 df-tmd 23898 df-tgp 23899 df-tsms 23953 df-trg 23986 df-xms 24148 df-ms 24149 df-tms 24150 df-nm 24413 df-ngp 24414 df-nrg 24416 df-nlm 24417 df-ii 24719 df-cncf 24720 df-limc 25717 df-dv 25718 df-log 26407 df-esum 33515 |
This theorem is referenced by: carsggect 33806 carsgclctunlem2 33807 |
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