esumgect - Mathbox for Thierry Arnoux

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Theorem esumgect 30152

Description: "Send 𝑛 to +∞ " in an inequality with an extended sum. (Contributed by Thierry Arnoux, 24-May-2020.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
esumgect	⊢ (𝜑 → Σ^*𝑘 ∈ ℕ𝐴 ≤ 𝐵)

Distinct variable groups: 𝐴,𝑛 𝐵,𝑛 𝑘,𝑛,𝜑 Allowed substitution hints: 𝐴(𝑘) 𝐵(𝑘)

Proof of Theorem esumgect Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		esumsup.1	. . 3 ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞))
2		esumsup.2	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
3	1, 2	esumsup 30151	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ ℕ𝐴 = sup(ran (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴), ℝ^*, < ))
4		nfv 1843	. . . . . 6 ⊢ Ⅎ𝑛𝜑
5		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑛𝑧
6		nfmpt1 4747	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴)
7	6	nfrn 5368	. . . . . . 7 ⊢ Ⅎ𝑛ran (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴)
8	5, 7	nfel 2777	. . . . . 6 ⊢ Ⅎ𝑛 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴)
9	4, 8	nfan 1828	. . . . 5 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴))
10		simpr 477	. . . . . 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 798	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = Σ^𝑘 ∈ (1...𝑛)𝐴) → 𝜑)
12		simplr 792	. . . . . . 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 693	. . . . . 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 4675	. . . . 5 ⊢ ((((𝜑 ∧ 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = Σ^𝑘 ∈ (1...𝑛)𝐴) → 𝑧 ≤ 𝐵)
16		eqid 2622	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) = (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴)
17		esumex 30091	. . . . . . . 8 ⊢ Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ V
18	16, 17	elrnmpti 5376	. . . . . . 7 ⊢ (𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) ↔ ∃𝑛 ∈ ℕ 𝑧 = Σ^𝑘 ∈ (1...𝑛)𝐴)
19	18	biimpi 206	. . . . . 6 ⊢ (𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) → ∃𝑛 ∈ ℕ 𝑧 = Σ^𝑘 ∈ (1...𝑛)𝐴)
20	19	adantl 482	. . . . 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 3076	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴)) → 𝑧 ≤ 𝐵)
22	21	ralrimiva 2966	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴)𝑧 ≤ 𝐵)
23		ovexd 6680	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ V)
24		simpll 790	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
25		fz1ssnn 12372	. . . . . . . . . . . 12 ⊢ (1...𝑛) ⊆ ℕ
26	25	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
27	26	sselda 3603	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
28	24, 27, 2	syl2anc 693	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (0[,]+∞))
29	28	ralrimiva 2966	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
30		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑘(1...𝑛)
31	30	esumcl 30092	. . . . . . . 8 ⊢ (((1...𝑛) ∈ V ∧ ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
32	23, 29, 31	syl2anc 693	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
33	32	ralrimiva 2966	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
34	16	rnmptss 6392	. . . . . 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 12256	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ^*
37	35, 36	syl6ss 3615	. . . 4 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) ⊆ ℝ^)
38	36, 1	sseldi 3601	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
39		supxrleub 12156	. . . 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 693	. . 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 247	. 2 ⊢ (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴), ℝ^, < ) ≤ 𝐵)
42	3, 41	eqbrtrd 4675	1 ⊢ (𝜑 → Σ^*𝑘 ∈ ℕ𝐴 ≤ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 Vcvv 3200 ⊆ wss 3574 class class class wbr 4653 ↦ cmpt 4729 ran crn 5115 (class class class)co 6650 supcsup 8346 0cc0 9936 1c1 9937 +∞cpnf 10071 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 ℕcn 11020 [,]cicc 12178 ...cfz 12326 Σ^*cesum 30089

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-ordt 16161 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-ps 17200 df-tsr 17201 df-plusf 17241 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-subrg 18778 df-abv 18817 df-lmod 18865 df-scaf 18866 df-sra 19172 df-rgmod 19173 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-tmd 21876 df-tgp 21877 df-tsms 21930 df-trg 21963 df-xms 22125 df-ms 22126 df-tms 22127 df-nm 22387 df-ngp 22388 df-nrg 22390 df-nlm 22391 df-ii 22680 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303 df-esum 30090

This theorem is referenced by: carsggect 30380 carsgclctunlem2 30381

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