esumval - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > esumval		Structured version Visualization version GIF version

Theorem esumval 34436

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 34418	. . 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 2925	. . . . . 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	fmptdF 33001	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞))
10		inss1 4189	. . . . . . . . . . . . . 14 ⊢ (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
11	10	sseli 3933	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ 𝒫 𝐴)
12	11	elpwid 4571	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ⊆ 𝐴)
13	12	adantl 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑥 ⊆ 𝐴)
14		nfcv 2925	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝑥
15	5, 14	resmptf 6041	. . . . . . . . . . 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 7426	. . . . . . . . 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 5928	. . . . . 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 9402	. . . . 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 33393	. . . 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 13161	. . . 4 ⊢ < Or ℝ^*
27	26	supex 9420	. . 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

Syntax hints: → wi 4 ∧ wa 400 = wceq 1570 Ⅎwnf 1813 ∈ wcel 2143 Ⅎwnfc 2910 ∩ cin 3904 ⊆ wss 3905 𝒫 cpw 4562 {csn 4589 ∪ cuni 4872 ↦ cmpt 5192 ran crn 5662 ↾ cres 5663 (class class class)co 7410 Fincfn 8939 supcsup 9396 0cc0 11095 +∞cpnf 11235 ℝ^*cxr 11237 < clt 11238 [,]cicc 13370 ↾_s cress 17285 Σ_g cgsu 17488 ℝ^*_𝑠cxrs 17549 tsums ctsu 24283 Σ^*cesum 34417

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-xadd 13133 df-ioo 13371 df-ioc 13372 df-ico 13373 df-icc 13374 df-fz 13531 df-fzo 13679 df-seq 14034 df-hash 14363 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-tset 17324 df-ple 17325 df-ds 17327 df-rest 17470 df-topn 17471 df-0g 17489 df-gsum 17490 df-topgen 17491 df-ordt 17550 df-xrs 17551 df-mre 17633 df-mrc 17634 df-acs 17636 df-ps 18617 df-tsr 18618 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-submnd 18837 df-cntz 19382 df-cmn 19847 df-fbas 21519 df-fg 21520 df-top 23051 df-topon 23068 df-topsp 23090 df-bases 23103 df-ntr 23177 df-nei 23255 df-cn 23384 df-haus 23472 df-fil 24003 df-fm 24095 df-flim 24096 df-flf 24097 df-tsms 24284 df-esum 34418

This theorem is referenced by: esumel 34437 esumnul 34438 esum0 34439 gsumesum 34449 esumlub 34450 esumcst 34453 esumpcvgval 34468 esumcvg 34476 esum2d 34483

W3C validator