esumsup - Mathbox for Thierry Arnoux

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Theorem esumsup 32057

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 6989	. . 3 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶(0[,]+∞))
3		nfmpt1 5182	. . . 4 ⊢ Ⅎ𝑘(𝑘 ∈ ℕ ↦ 𝐴)
4	3	esumfsup 32038	. . 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 485	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
7		eqid 2738	. . . . 5 ⊢ (𝑘 ∈ ℕ ↦ 𝐴) = (𝑘 ∈ ℕ ↦ 𝐴)
8	7	fvmpt2 6886	. . . 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 32006	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ ℕ((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = Σ^𝑘 ∈ ℕ𝐴)
11		1z 12350	. . . . . . . . 9 ⊢ 1 ∈ ℤ
12		seqfn 13733	. . . . . . . . 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 12621	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
15	14	fneq2i 6531	. . . . . . . 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 2907	. . . . . . . 8 ⊢ Ⅎ𝑛seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴))
18	17	dffn5f 6840	. . . . . . 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 13287	. . . . . . . . . . 11 ⊢ (1...𝑛) ⊆ ℕ
22	21	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
23	22	sselda 3921	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
24		simpll 764	. . . . . . . . . 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 32006	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ^𝑘 ∈ (1...𝑛)((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = Σ^𝑘 ∈ (1...𝑛)𝐴)
28	3	esumfzf 32037	. . . . . . . 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 2780	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)𝐴 = (seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴))‘𝑛))
31	30	mpteq2dva 5174	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) = (𝑛 ∈ ℕ ↦ (seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴))‘𝑛)))
32	20, 31	eqtr4d 2781	. . . 4 ⊢ (𝜑 → seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) = (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴))
33	32	rneqd 5847	. . 3 ⊢ (𝜑 → ran seq1( +_𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) = ran (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴))
34	33	supeq1d 9205	. 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 2786	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 396 = wceq 1539 ∈ wcel 2106 ⊆ wss 3887 ↦ cmpt 5157 ran crn 5590 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 supcsup 9199 0cc0 10871 1c1 10872 +∞cpnf 11006 ℝ^*cxr 11008 < clt 11009 ℕcn 11973 ℤcz 12319 ℤ_≥cuz 12582 +_𝑒 cxad 12846 [,]cicc 13082 ...cfz 13239 seqcseq 13721 Σ^*cesum 31995

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-ef 15777 df-sin 15779 df-cos 15780 df-pi 15782 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-ordt 17212 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-ps 18284 df-tsr 18285 df-plusf 18325 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-cntz 18923 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-subrg 20022 df-abv 20077 df-lmod 20125 df-scaf 20126 df-sra 20434 df-rgmod 20435 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-tmd 23223 df-tgp 23224 df-tsms 23278 df-trg 23311 df-xms 23473 df-ms 23474 df-tms 23475 df-nm 23738 df-ngp 23739 df-nrg 23741 df-nlm 23742 df-ii 24040 df-cncf 24041 df-limc 25030 df-dv 25031 df-log 25712 df-esum 31996

This theorem is referenced by: esumgect 32058

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