esumgect - Mathbox for Thierry Arnoux

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Theorem esumgect 34234

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 34233	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ ℕ𝐴 = sup(ran (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴), ℝ^*, < ))
4		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑛𝜑
5		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑛𝑧
6		nfmpt1 5184	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴)
7	6	nfrn 5907	. . . . . . 7 ⊢ Ⅎ𝑛ran (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴)
8	5, 7	nfel 2913	. . . . . 6 ⊢ Ⅎ𝑛 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴)
9	4, 8	nfan 1901	. . . . 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 775	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = Σ^𝑘 ∈ (1...𝑛)𝐴) → 𝜑)
12		simplr 769	. . . . . . 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 585	. . . . . 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 5107	. . . . 5 ⊢ ((((𝜑 ∧ 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = Σ^𝑘 ∈ (1...𝑛)𝐴) → 𝑧 ≤ 𝐵)
16		eqid 2736	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) = (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴)
17		esumex 34173	. . . . . . . 8 ⊢ Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ V
18	16, 17	elrnmpti 5917	. . . . . . 7 ⊢ (𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) ↔ ∃𝑛 ∈ ℕ 𝑧 = Σ^𝑘 ∈ (1...𝑛)𝐴)
19	18	biimpi 216	. . . . . 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 3246	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴)) → 𝑧 ≤ 𝐵)
22	21	ralrimiva 3129	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴)𝑧 ≤ 𝐵)
23		ovexd 7402	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ V)
24		simpll 767	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
25		fz1ssnn 13509	. . . . . . . . . . . 12 ⊢ (1...𝑛) ⊆ ℕ
26	25	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
27	26	sselda 3921	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
28	24, 27, 2	syl2anc 585	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (0[,]+∞))
29	28	ralrimiva 3129	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
30		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑘(1...𝑛)
31	30	esumcl 34174	. . . . . . . 8 ⊢ (((1...𝑛) ∈ V ∧ ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
32	23, 29, 31	syl2anc 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
33	32	ralrimiva 3129	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
34	16	rnmptss 7075	. . . . . 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 13383	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ^*
37	35, 36	sstrdi 3934	. . . 4 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) ⊆ ℝ^)
38	36, 1	sselid 3919	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
39		supxrleub 13278	. . . 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 585	. . 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 5107	1 ⊢ (𝜑 → Σ^*𝑘 ∈ ℕ𝐴 ≤ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3051 ∃wrex 3061 Vcvv 3429 ⊆ wss 3889 class class class wbr 5085 ↦ cmpt 5166 ran crn 5632 (class class class)co 7367 supcsup 9353 0cc0 11038 1c1 11039 +∞cpnf 11176 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 ℕcn 12174 [,]cicc 13301 ...cfz 13461 Σ^*cesum 34171

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-ef 16032 df-sin 16034 df-cos 16035 df-pi 16037 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-ordt 17465 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-ps 18532 df-tsr 18533 df-plusf 18607 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-mulg 19044 df-subg 19099 df-cntz 19292 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-cring 20217 df-subrng 20523 df-subrg 20547 df-abv 20786 df-lmod 20857 df-scaf 20858 df-sra 21168 df-rgmod 21169 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-tmd 24037 df-tgp 24038 df-tsms 24092 df-trg 24125 df-xms 24285 df-ms 24286 df-tms 24287 df-nm 24547 df-ngp 24548 df-nrg 24550 df-nlm 24551 df-ii 24844 df-cncf 24845 df-limc 25833 df-dv 25834 df-log 26520 df-esum 34172

This theorem is referenced by: carsggect 34462 carsgclctunlem2 34463

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