esumval - Mathbox for Thierry Arnoux

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Theorem esumval 33573

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 33555	. . 3 ⊢ Σ^𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
2		eqid 2726	. . . . 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 2897	. . . . . 6 ⊢ Ⅎ𝑘(0[,]+∞)
7		esumval.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
8		eqid 2726	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
9	4, 5, 6, 7, 8	fmptdF 32385	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞))
10		inss1 4223	. . . . . . . . . . . . . 14 ⊢ (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
11	10	sseli 3973	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ 𝒫 𝐴)
12	11	elpwid 4606	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ⊆ 𝐴)
13	12	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑥 ⊆ 𝐴)
14		nfcv 2897	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝑥
15	5, 14	resmptf 6032	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥) = (𝑘 ∈ 𝑥 ↦ 𝐵))
16	13, 15	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥) = (𝑘 ∈ 𝑥 ↦ 𝐵))
17	16	oveq2d 7420	. . . . . . . . 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 2767	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐶 = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥)))
20	19	mpteq2dva 5241	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥))))
21	20	rneqd 5930	. . . . . 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 9440	. . . . 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 32712	. . . 4 ⊢ (𝜑 → ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = {sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ^, < )})
24	23	unieqd 4915	. . 3 ⊢ (𝜑 → ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ∪ {sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ^, < )})
25	1, 24	eqtrid 2778	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 = ∪ {sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ^, < )})
26		xrltso 13123	. . . 4 ⊢ < Or ℝ^*
27	26	supex 9457	. . 3 ⊢ sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ^*, < ) ∈ V
28	27	unisn 4923	. 2 ⊢ ∪ {sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ^, < )} = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ^, < )
29	25, 28	eqtrdi 2782	1 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 Ⅎwnfc 2877 ∩ cin 3942 ⊆ wss 3943 𝒫 cpw 4597 {csn 4623 ∪ cuni 4902 ↦ cmpt 5224 ran crn 5670 ↾ cres 5671 (class class class)co 7404 Fincfn 8938 supcsup 9434 0cc0 11109 +∞cpnf 11246 ℝ^*cxr 11248 < clt 11249 [,]cicc 13330 ↾_s cress 17179 Σ_g cgsu 17392 ℝ^*_𝑠cxrs 17452 tsums ctsu 23980 Σ^*cesum 33554

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 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 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-xadd 13096 df-ioo 13331 df-ioc 13332 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-seq 13970 df-hash 14293 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-tset 17222 df-ple 17223 df-ds 17225 df-rest 17374 df-topn 17375 df-0g 17393 df-gsum 17394 df-topgen 17395 df-ordt 17453 df-xrs 17454 df-mre 17536 df-mrc 17537 df-acs 17539 df-ps 18528 df-tsr 18529 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-submnd 18711 df-cntz 19230 df-cmn 19699 df-fbas 21232 df-fg 21233 df-top 22746 df-topon 22763 df-topsp 22785 df-bases 22799 df-ntr 22874 df-nei 22952 df-cn 23081 df-haus 23169 df-fil 23700 df-fm 23792 df-flim 23793 df-flf 23794 df-tsms 23981 df-esum 33555

This theorem is referenced by: esumel 33574 esumnul 33575 esum0 33576 gsumesum 33586 esumlub 33587 esumcst 33590 esumpcvgval 33605 esumcvg 33613 esum2d 33620

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