esumgect - Mathbox for Thierry Arnoux

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Theorem esumgect 33076

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 33075	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ ℕ𝐴 = sup(ran (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴), ℝ^*, < ))
4		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑛𝜑
5		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑛𝑧
6		nfmpt1 5255	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴)
7	6	nfrn 5949	. . . . . . 7 ⊢ Ⅎ𝑛ran (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴)
8	5, 7	nfel 2917	. . . . . 6 ⊢ Ⅎ𝑛 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴)
9	4, 8	nfan 1902	. . . . 5 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴))
10		simpr 485	. . . . . 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 584	. . . . . 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 5169	. . . . 5 ⊢ ((((𝜑 ∧ 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = Σ^𝑘 ∈ (1...𝑛)𝐴) → 𝑧 ≤ 𝐵)
16		eqid 2732	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) = (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴)
17		esumex 33015	. . . . . . . 8 ⊢ Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ V
18	16, 17	elrnmpti 5957	. . . . . . 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 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 3265	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴)) → 𝑧 ≤ 𝐵)
22	21	ralrimiva 3146	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴)𝑧 ≤ 𝐵)
23		ovexd 7440	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ V)
24		simpll 765	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
25		fz1ssnn 13528	. . . . . . . . . . . 12 ⊢ (1...𝑛) ⊆ ℕ
26	25	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
27	26	sselda 3981	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
28	24, 27, 2	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (0[,]+∞))
29	28	ralrimiva 3146	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
30		nfcv 2903	. . . . . . . . 9 ⊢ Ⅎ𝑘(1...𝑛)
31	30	esumcl 33016	. . . . . . . 8 ⊢ (((1...𝑛) ∈ V ∧ ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
32	23, 29, 31	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
33	32	ralrimiva 3146	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
34	16	rnmptss 7118	. . . . . 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 13403	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ^*
37	35, 36	sstrdi 3993	. . . 4 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) ⊆ ℝ^)
38	36, 1	sselid 3979	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
39		supxrleub 13301	. . . 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 584	. . 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 256	. 2 ⊢ (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴), ℝ^, < ) ≤ 𝐵)
42	3, 41	eqbrtrd 5169	1 ⊢ (𝜑 → Σ^*𝑘 ∈ ℕ𝐴 ≤ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 Vcvv 3474 ⊆ wss 3947 class class class wbr 5147 ↦ cmpt 5230 ran crn 5676 (class class class)co 7405 supcsup 9431 0cc0 11106 1c1 11107 +∞cpnf 11241 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 ℕcn 12208 [,]cicc 13323 ...cfz 13480 Σ^*cesum 33013

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 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 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 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 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-ordt 17443 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-ps 18515 df-tsr 18516 df-plusf 18556 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-cntz 19175 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-cring 20052 df-subrg 20353 df-abv 20417 df-lmod 20465 df-scaf 20466 df-sra 20777 df-rgmod 20778 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-tmd 23567 df-tgp 23568 df-tsms 23622 df-trg 23655 df-xms 23817 df-ms 23818 df-tms 23819 df-nm 24082 df-ngp 24083 df-nrg 24085 df-nlm 24086 df-ii 24384 df-cncf 24385 df-limc 25374 df-dv 25375 df-log 26056 df-esum 33014

This theorem is referenced by: carsggect 33305 carsgclctunlem2 33306

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