esumgect - Mathbox for Thierry Arnoux

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Theorem esumgect 33750

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 33749	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ ℕ𝐴 = sup(ran (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴), ℝ^*, < ))
4		nfv 1909	. . . . . 6 ⊢ Ⅎ𝑛𝜑
5		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑛𝑧
6		nfmpt1 5260	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴)
7	6	nfrn 5958	. . . . . . 7 ⊢ Ⅎ𝑛ran (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴)
8	5, 7	nfel 2914	. . . . . 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 483	. . . . . 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 582	. . . . . 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 5174	. . . . 5 ⊢ ((((𝜑 ∧ 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = Σ^𝑘 ∈ (1...𝑛)𝐴) → 𝑧 ≤ 𝐵)
16		eqid 2728	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) = (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴)
17		esumex 33689	. . . . . . . 8 ⊢ Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ V
18	16, 17	elrnmpti 5966	. . . . . . 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 480	. . . . 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 3263	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴)) → 𝑧 ≤ 𝐵)
22	21	ralrimiva 3143	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴)𝑧 ≤ 𝐵)
23		ovexd 7461	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ V)
24		simpll 765	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
25		fz1ssnn 13574	. . . . . . . . . . . 12 ⊢ (1...𝑛) ⊆ ℕ
26	25	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
27	26	sselda 3982	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
28	24, 27, 2	syl2anc 582	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (0[,]+∞))
29	28	ralrimiva 3143	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
30		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑘(1...𝑛)
31	30	esumcl 33690	. . . . . . . 8 ⊢ (((1...𝑛) ∈ V ∧ ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
32	23, 29, 31	syl2anc 582	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
33	32	ralrimiva 3143	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
34	16	rnmptss 7138	. . . . . 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 13449	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ^*
37	35, 36	sstrdi 3994	. . . 4 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) ⊆ ℝ^)
38	36, 1	sselid 3980	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
39		supxrleub 13347	. . . 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 582	. . 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 5174	1 ⊢ (𝜑 → Σ^*𝑘 ∈ ℕ𝐴 ≤ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3058 ∃wrex 3067 Vcvv 3473 ⊆ wss 3949 class class class wbr 5152 ↦ cmpt 5235 ran crn 5683 (class class class)co 7426 supcsup 9473 0cc0 11148 1c1 11149 +∞cpnf 11285 ℝ^*cxr 11287 < clt 11288 ≤ cle 11289 ℕcn 12252 [,]cicc 13369 ...cfz 13526 Σ^*cesum 33687

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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-inf2 9674 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 ax-addf 11227 ax-mulf 11228

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-er 8733 df-map 8855 df-pm 8856 df-ixp 8925 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-fsupp 9396 df-fi 9444 df-sup 9475 df-inf 9476 df-oi 9543 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13134 df-xadd 13135 df-xmul 13136 df-ioo 13370 df-ioc 13371 df-ico 13372 df-icc 13373 df-fz 13527 df-fzo 13670 df-fl 13799 df-mod 13877 df-seq 14009 df-exp 14069 df-fac 14275 df-bc 14304 df-hash 14332 df-shft 15056 df-cj 15088 df-re 15089 df-im 15090 df-sqrt 15224 df-abs 15225 df-limsup 15457 df-clim 15474 df-rlim 15475 df-sum 15675 df-ef 16053 df-sin 16055 df-cos 16056 df-pi 16058 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-starv 17257 df-sca 17258 df-vsca 17259 df-ip 17260 df-tset 17261 df-ple 17262 df-ds 17264 df-unif 17265 df-hom 17266 df-cco 17267 df-rest 17413 df-topn 17414 df-0g 17432 df-gsum 17433 df-topgen 17434 df-pt 17435 df-prds 17438 df-ordt 17492 df-xrs 17493 df-qtop 17498 df-imas 17499 df-xps 17501 df-mre 17575 df-mrc 17576 df-acs 17578 df-ps 18567 df-tsr 18568 df-plusf 18608 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-mhm 18749 df-submnd 18750 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19038 df-subg 19092 df-cntz 19282 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-ring 20189 df-cring 20190 df-subrng 20497 df-subrg 20522 df-abv 20711 df-lmod 20759 df-scaf 20760 df-sra 21072 df-rgmod 21073 df-psmet 21285 df-xmet 21286 df-met 21287 df-bl 21288 df-mopn 21289 df-fbas 21290 df-fg 21291 df-cnfld 21294 df-top 22824 df-topon 22841 df-topsp 22863 df-bases 22877 df-cld 22951 df-ntr 22952 df-cls 22953 df-nei 23030 df-lp 23068 df-perf 23069 df-cn 23159 df-cnp 23160 df-haus 23247 df-tx 23494 df-hmeo 23687 df-fil 23778 df-fm 23870 df-flim 23871 df-flf 23872 df-tmd 24004 df-tgp 24005 df-tsms 24059 df-trg 24092 df-xms 24254 df-ms 24255 df-tms 24256 df-nm 24519 df-ngp 24520 df-nrg 24522 df-nlm 24523 df-ii 24825 df-cncf 24826 df-limc 25823 df-dv 25824 df-log 26518 df-esum 33688

This theorem is referenced by: carsggect 33979 carsgclctunlem2 33980

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