esumval - Mathbox for Thierry Arnoux

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Theorem esumval 34789

Description: Develop the value of the extended sum. (Contributed by Thierry Arnoux, 4-Jan-2017.)

**Hypotheses**
Ref	Expression
esumval.p	⊢ Ⅎ𝑘𝜑
esumval.0	⊢ Ⅎ𝑘𝐴
esumval.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
esumval.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
esumval.3	⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝑥 ↦ 𝐵)) = 𝐶)

**Assertion**
Ref	Expression
esumval	⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ^, < ))

Distinct variable groups: 𝑥,𝑘 𝑥,𝐴 𝜑,𝑥 𝑥,𝐵 Allowed substitution hints: 𝜑(𝑘) 𝐴(𝑘) 𝐵(𝑘) 𝐶(𝑥, 𝑘) 𝑉(𝑥, 𝑘)

**Proof of Theorem esumval**
Step	Hyp	Ref	Expression
1		df-esum 34771	. . 3 ⊢ Σ^𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
2		eqid 2763	. . . . 5 ⊢ (ℝ^_𝑠 ↾_s (0[,]+∞)) = (ℝ^_𝑠 ↾_s (0[,]+∞))
3		esumval.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		esumval.p	. . . . . 6 ⊢ Ⅎ𝑘𝜑
5		esumval.0	. . . . . 6 ⊢ Ⅎ𝑘𝐴
6		nfcv 2926	. . . . . 6 ⊢ Ⅎ𝑘(0[,]+∞)
7		esumval.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
8		eqid 2763	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
9	4, 5, 6, 7, 8	fmptdf2 33348	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞))
10		inss1 4136	. . . . . . . . . . . . . 14 ⊢ (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
11	10	sseli 3909	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ 𝒫 𝐴)
12	11	elpwid 4569	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ⊆ 𝐴)
13	12	adantl 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑥 ⊆ 𝐴)
14		nfcv 2926	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝑥
15	5, 14	resmptf 6049	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥) = (𝑘 ∈ 𝑥 ↦ 𝐵))
16	13, 15	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥) = (𝑘 ∈ 𝑥 ↦ 𝐵))
17	16	oveq2d 7467	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥)) = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝑥 ↦ 𝐵)))
18		esumval.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝑥 ↦ 𝐵)) = 𝐶)
19	17, 18	eqtr2d 2799	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐶 = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥)))
20	19	mpteq2dva 5204	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥))))
21	20	rneqd 5932	. . . . . 6 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶) = ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥))))
22	21	supeq1d 9475	. . . . 5 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ^, < ) = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥))), ℝ^*, < ))
23	2, 3, 9, 22	xrge0tsmsd 33733	. . . 4 ⊢ (𝜑 → ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = {sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ^, < )})
24	23	unieqd 4885	. . 3 ⊢ (𝜑 → ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ∪ {sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ^, < )})
25	1, 24	eqtrid 2810	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 = ∪ {sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ^, < )})
26		xrltso 13310	. . . 4 ⊢ < Or ℝ^*
27	26	supex 9493	. . 3 ⊢ sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ^*, < ) ∈ V
28	27	unisn 4891	. 2 ⊢ ∪ {sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ^, < )} = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ^, < )
29	25, 28	eqtrdi 2814	1 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ^, < ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 Ⅎwnf 1816 ∈ wcel 2147 Ⅎwnfc 2911 ⊆ wss 3900 ∩ cin 4092 𝒫 cpw 4560 {csn 4588 ∪ cuni 4872 ↦ cmpt 5192 ran crn 5659 ↾ cres 5660 (class class class)co 7451 Fincfn 9011 supcsup 9469 0cc0 11239 +∞cpnf 11379 ℝ^*cxr 11381 < clt 11382 [,]cicc 13519 ↾_s cress 17452 Σ_g cgsu 17661 ℝ^*_𝑠cxrs 17722 tsums ctsu 24500 Σ^*cesum 34770

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-er 8755 df-map 8887 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9541 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-xadd 13282 df-ioo 13520 df-ioc 13521 df-ico 13522 df-icc 13523 df-fz 13680 df-fzo 13829 df-seq 14185 df-hash 14516 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-tset 17492 df-ple 17493 df-ds 17495 df-rest 17643 df-topn 17644 df-0g 17662 df-gsum 17663 df-topgen 17664 df-ordt 17723 df-xrs 17724 df-mre 17806 df-mrc 17807 df-acs 17809 df-ps 18790 df-tsr 18791 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-submnd 19029 df-cntz 19582 df-cmn 20047 df-fbas 21726 df-fg 21727 df-top 23267 df-topon 23284 df-topsp 23306 df-bases 23319 df-ntr 23393 df-nei 23471 df-cn 23600 df-haus 23688 df-fil 24220 df-fm 24312 df-flim 24313 df-flf 24314 df-tsms 24501 df-esum 34771

This theorem is used by: esumel 34790 esumnul 34791 esum0 34792 gsumesum 34802 esumlub 34803 esumcst 34806 esumpcvgval 34821 esumcvg 34829 esum2d 34836

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