esumsup - Mathbox for Thierry Arnoux

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Theorem esumsup 34072

Description: Express an extended sum as a supremum of extended sums. (Contributed by Thierry Arnoux, 24-May-2020.)

**Hypotheses**
Ref	Expression
esumsup.1	⊢ (𝜑 → 𝐵 ∈ (0[,]+∞))
esumsup.2	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))

**Assertion**
Ref	Expression
esumsup	⊢ (𝜑 → Σ^𝑘 ∈ ℕ𝐴 = sup(ran (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴), ℝ^*, < ))

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

**Proof of Theorem esumsup**
Step	Hyp	Ref	Expression
1		esumsup.2	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
2	1	fmpttd 7069	. . 3 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶(0[,]+∞))
3		nfmpt1 5201	. . . 4 ⊢ Ⅎ𝑘(𝑘 ∈ ℕ ↦ 𝐴)
4	3	esumfsup 34053	. . 3 ⊢ ((𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶(0[,]+∞) → Σ^𝑘 ∈ ℕ((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = sup(ran seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)), ℝ^, < ))
5	2, 4	syl 17	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ ℕ((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = sup(ran seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)), ℝ^, < ))
6		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
7		eqid 2729	. . . . 5 ⊢ (𝑘 ∈ ℕ ↦ 𝐴) = (𝑘 ∈ ℕ ↦ 𝐴)
8	7	fvmpt2 6961	. . . 4 ⊢ ((𝑘 ∈ ℕ ∧ 𝐴 ∈ (0[,]+∞)) → ((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = 𝐴)
9	6, 1, 8	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = 𝐴)
10	9	esumeq2dv 34021	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ ℕ((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = Σ^𝑘 ∈ ℕ𝐴)
11		1z 12539	. . . . . . . . 9 ⊢ 1 ∈ ℤ
12		seqfn 13954	. . . . . . . . 9 ⊢ (1 ∈ ℤ → seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) Fn (ℤ_≥‘1))
13	11, 12	ax-mp 5	. . . . . . . 8 ⊢ seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) Fn (ℤ_≥‘1)
14		nnuz 12812	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
15	14	fneq2i 6598	. . . . . . . 8 ⊢ (seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) Fn ℕ ↔ seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) Fn (ℤ_≥‘1))
16	13, 15	mpbir 231	. . . . . . 7 ⊢ seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) Fn ℕ
17		nfcv 2891	. . . . . . . 8 ⊢ Ⅎ𝑛seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴))
18	17	dffn5f 6914	. . . . . . 7 ⊢ (seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) Fn ℕ ↔ seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) = (𝑛 ∈ ℕ ↦ (seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴))‘𝑛)))
19	16, 18	mpbi 230	. . . . . 6 ⊢ seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) = (𝑛 ∈ ℕ ↦ (seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴))‘𝑛))
20	19	a1i 11	. . . . 5 ⊢ (𝜑 → seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) = (𝑛 ∈ ℕ ↦ (seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴))‘𝑛)))
21		fz1ssnn 13492	. . . . . . . . . . 11 ⊢ (1...𝑛) ⊆ ℕ
22	21	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
23	22	sselda 3943	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
24		simpll 766	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
25	24, 23, 1	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (0[,]+∞))
26	23, 25, 8	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = 𝐴)
27	26	esumeq2dv 34021	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ^𝑘 ∈ (1...𝑛)((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = Σ^𝑘 ∈ (1...𝑛)𝐴)
28	3	esumfzf 34052	. . . . . . . 8 ⊢ (((𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = (seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴))‘𝑛))
29	2, 28	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = (seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴))‘𝑛))
30	27, 29	eqtr3d 2766	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)𝐴 = (seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴))‘𝑛))
31	30	mpteq2dva 5195	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) = (𝑛 ∈ ℕ ↦ (seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴))‘𝑛)))
32	20, 31	eqtr4d 2767	. . . 4 ⊢ (𝜑 → seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) = (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴))
33	32	rneqd 5891	. . 3 ⊢ (𝜑 → ran seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) = ran (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴))
34	33	supeq1d 9373	. 2 ⊢ (𝜑 → sup(ran seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)), ℝ^, < ) = sup(ran (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴), ℝ^*, < ))
35	5, 10, 34	3eqtr3d 2772	1 ⊢ (𝜑 → Σ^𝑘 ∈ ℕ𝐴 = sup(ran (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴), ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ⊆ wss 3911 ↦ cmpt 5183 ran crn 5632 Fn wfn 6494 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 supcsup 9367 0cc0 11044 1c1 11045 +∞cpnf 11181 ℝ^*cxr 11183 < clt 11184 ℕcn 12162 ℤcz 12505 ℤ_≥cuz 12769 +_𝑒 cxad 13046 [,]cicc 13285 ...cfz 13444 seqcseq 13942 Σ^*cesum 34010

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123 ax-mulf 11124

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 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 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ioc 13287 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-mod 13808 df-seq 13943 df-exp 14003 df-fac 14215 df-bc 14244 df-hash 14272 df-shft 15009 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-limsup 15413 df-clim 15430 df-rlim 15431 df-sum 15629 df-ef 16009 df-sin 16011 df-cos 16012 df-pi 16014 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-ordt 17440 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-ps 18507 df-tsr 18508 df-plusf 18548 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-mhm 18692 df-submnd 18693 df-grp 18850 df-minusg 18851 df-sbg 18852 df-mulg 18982 df-subg 19037 df-cntz 19231 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-cring 20156 df-subrng 20466 df-subrg 20490 df-abv 20729 df-lmod 20800 df-scaf 20801 df-sra 21112 df-rgmod 21113 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22866 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-lp 23056 df-perf 23057 df-cn 23147 df-cnp 23148 df-haus 23235 df-tx 23482 df-hmeo 23675 df-fil 23766 df-fm 23858 df-flim 23859 df-flf 23860 df-tmd 23992 df-tgp 23993 df-tsms 24047 df-trg 24080 df-xms 24241 df-ms 24242 df-tms 24243 df-nm 24503 df-ngp 24504 df-nrg 24506 df-nlm 24507 df-ii 24803 df-cncf 24804 df-limc 25800 df-dv 25801 df-log 26498 df-esum 34011

This theorem is referenced by: esumgect 34073

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