esumsup - Mathbox for Thierry Arnoux

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Theorem esumsup 33076

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 7112	. . 3 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶(0[,]+∞))
3		nfmpt1 5256	. . . 4 ⊢ Ⅎ𝑘(𝑘 ∈ ℕ ↦ 𝐴)
4	3	esumfsup 33057	. . 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 486	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
7		eqid 2733	. . . . 5 ⊢ (𝑘 ∈ ℕ ↦ 𝐴) = (𝑘 ∈ ℕ ↦ 𝐴)
8	7	fvmpt2 7007	. . . 4 ⊢ ((𝑘 ∈ ℕ ∧ 𝐴 ∈ (0[,]+∞)) → ((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = 𝐴)
9	6, 1, 8	syl2anc 585	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = 𝐴)
10	9	esumeq2dv 33025	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ ℕ((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = Σ^𝑘 ∈ ℕ𝐴)
11		1z 12589	. . . . . . . . 9 ⊢ 1 ∈ ℤ
12		seqfn 13975	. . . . . . . . 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 12862	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
15	14	fneq2i 6645	. . . . . . . 8 ⊢ (seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) Fn ℕ ↔ seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) Fn (ℤ_≥‘1))
16	13, 15	mpbir 230	. . . . . . 7 ⊢ seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) Fn ℕ
17		nfcv 2904	. . . . . . . 8 ⊢ Ⅎ𝑛seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴))
18	17	dffn5f 6961	. . . . . . 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 229	. . . . . 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 13529	. . . . . . . . . . 11 ⊢ (1...𝑛) ⊆ ℕ
22	21	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
23	22	sselda 3982	. . . . . . . . 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 585	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (0[,]+∞))
26	23, 25, 8	syl2anc 585	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = 𝐴)
27	26	esumeq2dv 33025	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ^𝑘 ∈ (1...𝑛)((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = Σ^𝑘 ∈ (1...𝑛)𝐴)
28	3	esumfzf 33056	. . . . . . . 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 581	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = (seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴))‘𝑛))
30	27, 29	eqtr3d 2775	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)𝐴 = (seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴))‘𝑛))
31	30	mpteq2dva 5248	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) = (𝑛 ∈ ℕ ↦ (seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴))‘𝑛)))
32	20, 31	eqtr4d 2776	. . . 4 ⊢ (𝜑 → seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) = (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴))
33	32	rneqd 5936	. . 3 ⊢ (𝜑 → ran seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) = ran (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴))
34	33	supeq1d 9438	. 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 2781	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 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3948 ↦ cmpt 5231 ran crn 5677 Fn wfn 6536 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 supcsup 9432 0cc0 11107 1c1 11108 +∞cpnf 11242 ℝ^*cxr 11244 < clt 11245 ℕcn 12209 ℤcz 12555 ℤ_≥cuz 12819 +_𝑒 cxad 13087 [,]cicc 13324 ...cfz 13481 seqcseq 13963 Σ^*cesum 33014

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-supp 8144 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-2o 8464 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ioc 13326 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15011 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-limsup 15412 df-clim 15429 df-rlim 15430 df-sum 15630 df-ef 16008 df-sin 16010 df-cos 16011 df-pi 16013 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-starv 17209 df-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ds 17216 df-unif 17217 df-hom 17218 df-cco 17219 df-rest 17365 df-topn 17366 df-0g 17384 df-gsum 17385 df-topgen 17386 df-pt 17387 df-prds 17390 df-ordt 17444 df-xrs 17445 df-qtop 17450 df-imas 17451 df-xps 17453 df-mre 17527 df-mrc 17528 df-acs 17530 df-ps 18516 df-tsr 18517 df-plusf 18557 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-mhm 18668 df-submnd 18669 df-grp 18819 df-minusg 18820 df-sbg 18821 df-mulg 18946 df-subg 18998 df-cntz 19176 df-cmn 19645 df-abl 19646 df-mgp 19983 df-ur 20000 df-ring 20052 df-cring 20053 df-subrg 20354 df-abv 20418 df-lmod 20466 df-scaf 20467 df-sra 20778 df-rgmod 20779 df-psmet 20929 df-xmet 20930 df-met 20931 df-bl 20932 df-mopn 20933 df-fbas 20934 df-fg 20935 df-cnfld 20938 df-top 22388 df-topon 22405 df-topsp 22427 df-bases 22441 df-cld 22515 df-ntr 22516 df-cls 22517 df-nei 22594 df-lp 22632 df-perf 22633 df-cn 22723 df-cnp 22724 df-haus 22811 df-tx 23058 df-hmeo 23251 df-fil 23342 df-fm 23434 df-flim 23435 df-flf 23436 df-tmd 23568 df-tgp 23569 df-tsms 23623 df-trg 23656 df-xms 23818 df-ms 23819 df-tms 23820 df-nm 24083 df-ngp 24084 df-nrg 24086 df-nlm 24087 df-ii 24385 df-cncf 24386 df-limc 25375 df-dv 25376 df-log 26057 df-esum 33015

This theorem is referenced by: esumgect 33077

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