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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 31350 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = sup(ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴), ℝ*, < )) |
4 | nfv 1915 | . . . . . 6 ⊢ Ⅎ𝑛𝜑 | |
5 | nfcv 2979 | . . . . . . 7 ⊢ Ⅎ𝑛𝑧 | |
6 | nfmpt1 5166 | . . . . . . . 8 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴) | |
7 | 6 | nfrn 5826 | . . . . . . 7 ⊢ Ⅎ𝑛ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴) |
8 | 5, 7 | nfel 2994 | . . . . . 6 ⊢ Ⅎ𝑛 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴) |
9 | 4, 8 | nfan 1900 | . . . . 5 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴)) |
10 | simpr 487 | . . . . . 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 773 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = Σ*𝑘 ∈ (1...𝑛)𝐴) → 𝜑) | |
12 | simplr 767 | . . . . . . 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 586 | . . . . . 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 5090 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = Σ*𝑘 ∈ (1...𝑛)𝐴) → 𝑧 ≤ 𝐵) |
16 | eqid 2823 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴) = (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴) | |
17 | esumex 31290 | . . . . . . . 8 ⊢ Σ*𝑘 ∈ (1...𝑛)𝐴 ∈ V | |
18 | 16, 17 | elrnmpti 5834 | . . . . . . 7 ⊢ (𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴) ↔ ∃𝑛 ∈ ℕ 𝑧 = Σ*𝑘 ∈ (1...𝑛)𝐴) |
19 | 18 | biimpi 218 | . . . . . 6 ⊢ (𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴) → ∃𝑛 ∈ ℕ 𝑧 = Σ*𝑘 ∈ (1...𝑛)𝐴) |
20 | 19 | adantl 484 | . . . . 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 3333 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴)) → 𝑧 ≤ 𝐵) |
22 | 21 | ralrimiva 3184 | . . 3 ⊢ (𝜑 → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴)𝑧 ≤ 𝐵) |
23 | ovexd 7193 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ V) | |
24 | simpll 765 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑) | |
25 | fz1ssnn 12941 | . . . . . . . . . . . 12 ⊢ (1...𝑛) ⊆ ℕ | |
26 | 25 | a1i 11 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ) |
27 | 26 | sselda 3969 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ) |
28 | 24, 27, 2 | syl2anc 586 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (0[,]+∞)) |
29 | 28 | ralrimiva 3184 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞)) |
30 | nfcv 2979 | . . . . . . . . 9 ⊢ Ⅎ𝑘(1...𝑛) | |
31 | 30 | esumcl 31291 | . . . . . . . 8 ⊢ (((1...𝑛) ∈ V ∧ ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞)) → Σ*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞)) |
32 | 23, 29, 31 | syl2anc 586 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞)) |
33 | 32 | ralrimiva 3184 | . . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ Σ*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞)) |
34 | 16 | rnmptss 6888 | . . . . . 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 12822 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
37 | 35, 36 | sstrdi 3981 | . . . 4 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴) ⊆ ℝ*) |
38 | 36, 1 | sseldi 3967 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
39 | supxrleub 12722 | . . . 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 586 | . . 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 259 | . 2 ⊢ (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴), ℝ*, < ) ≤ 𝐵) |
42 | 3, 41 | eqbrtrd 5090 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ ℕ𝐴 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 Vcvv 3496 ⊆ wss 3938 class class class wbr 5068 ↦ cmpt 5148 ran crn 5558 (class class class)co 7158 supcsup 8906 0cc0 10539 1c1 10540 +∞cpnf 10674 ℝ*cxr 10676 < clt 10677 ≤ cle 10678 ℕcn 11640 [,]cicc 12744 ...cfz 12895 Σ*cesum 31288 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14428 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-ef 15423 df-sin 15425 df-cos 15426 df-pi 15428 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-ordt 16776 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-ps 17812 df-tsr 17813 df-plusf 17853 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-subrg 19535 df-abv 19590 df-lmod 19638 df-scaf 19639 df-sra 19946 df-rgmod 19947 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-tmd 22682 df-tgp 22683 df-tsms 22737 df-trg 22770 df-xms 22932 df-ms 22933 df-tms 22934 df-nm 23194 df-ngp 23195 df-nrg 23197 df-nlm 23198 df-ii 23487 df-cncf 23488 df-limc 24466 df-dv 24467 df-log 25142 df-esum 31289 |
This theorem is referenced by: carsggect 31578 carsgclctunlem2 31579 |
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