esumval - Mathbox for Thierry Arnoux

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Theorem esumval 34131

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 34113	. . 3 ⊢ Σ^𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
2		eqid 2733	. . . . 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 2895	. . . . . 6 ⊢ Ⅎ𝑘(0[,]+∞)
7		esumval.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
8		eqid 2733	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
9	4, 5, 6, 7, 8	fmptdF 32660	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞))
10		inss1 4186	. . . . . . . . . . . . . 14 ⊢ (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
11	10	sseli 3926	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ 𝒫 𝐴)
12	11	elpwid 4560	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ⊆ 𝐴)
13	12	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑥 ⊆ 𝐴)
14		nfcv 2895	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝑥
15	5, 14	resmptf 5995	. . . . . . . . . . 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 7371	. . . . . . . . 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 2769	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐶 = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥)))
20	19	mpteq2dva 5188	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥))))
21	20	rneqd 5884	. . . . . 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 9341	. . . . 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 33083	. . . 4 ⊢ (𝜑 → ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = {sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ^, < )})
24	23	unieqd 4873	. . 3 ⊢ (𝜑 → ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ∪ {sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ^, < )})
25	1, 24	eqtrid 2780	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 = ∪ {sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ^, < )})
26		xrltso 13046	. . . 4 ⊢ < Or ℝ^*
27	26	supex 9359	. . 3 ⊢ sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ^*, < ) ∈ V
28	27	unisn 4879	. 2 ⊢ ∪ {sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ^, < )} = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ^, < )
29	25, 28	eqtrdi 2784	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 1541 Ⅎwnf 1784 ∈ wcel 2113 Ⅎwnfc 2880 ∩ cin 3897 ⊆ wss 3898 𝒫 cpw 4551 {csn 4577 ∪ cuni 4860 ↦ cmpt 5176 ran crn 5622 ↾ cres 5623 (class class class)co 7355 Fincfn 8879 supcsup 9335 0cc0 11017 +∞cpnf 11154 ℝ^*cxr 11156 < clt 11157 [,]cicc 13255 ↾_s cress 17148 Σ_g cgsu 17351 ℝ^*_𝑠cxrs 17412 tsums ctsu 24061 Σ^*cesum 34112

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-pre-sup 11095

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-om 7806 df-1st 7930 df-2nd 7931 df-supp 8100 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-er 8631 df-map 8761 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9257 df-fi 9306 df-sup 9337 df-inf 9338 df-oi 9407 df-card 9843 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-div 11786 df-nn 12137 df-2 12199 df-3 12200 df-4 12201 df-5 12202 df-6 12203 df-7 12204 df-8 12205 df-9 12206 df-n0 12393 df-z 12480 df-dec 12599 df-uz 12743 df-q 12853 df-xadd 13018 df-ioo 13256 df-ioc 13257 df-ico 13258 df-icc 13259 df-fz 13415 df-fzo 13562 df-seq 13916 df-hash 14245 df-struct 17065 df-sets 17082 df-slot 17100 df-ndx 17112 df-base 17128 df-ress 17149 df-plusg 17181 df-mulr 17182 df-tset 17187 df-ple 17188 df-ds 17190 df-rest 17333 df-topn 17334 df-0g 17352 df-gsum 17353 df-topgen 17354 df-ordt 17413 df-xrs 17414 df-mre 17496 df-mrc 17497 df-acs 17499 df-ps 18480 df-tsr 18481 df-mgm 18556 df-sgrp 18635 df-mnd 18651 df-submnd 18700 df-cntz 19237 df-cmn 19702 df-fbas 21297 df-fg 21298 df-top 22829 df-topon 22846 df-topsp 22868 df-bases 22881 df-ntr 22955 df-nei 23033 df-cn 23162 df-haus 23250 df-fil 23781 df-fm 23873 df-flim 23874 df-flf 23875 df-tsms 24062 df-esum 34113

This theorem is referenced by: esumel 34132 esumnul 34133 esum0 34134 gsumesum 34144 esumlub 34145 esumcst 34148 esumpcvgval 34163 esumcvg 34171 esum2d 34178

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