esumpfinvalf - Mathbox for Thierry Arnoux

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Theorem esumpfinvalf 34233

Description: Same as esumpfinval 34232, minus distinct variable restrictions. (Contributed by Thierry Arnoux, 28-Aug-2017.) (Proof shortened by AV, 25-Jul-2019.)

**Hypotheses**
Ref	Expression
esumpfinvalf.1	⊢ Ⅎ𝑘𝐴
esumpfinvalf.2	⊢ Ⅎ𝑘𝜑
esumpfinvalf.a	⊢ (𝜑 → 𝐴 ∈ Fin)
esumpfinvalf.b	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))

**Assertion**
Ref	Expression
esumpfinvalf	⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴𝐵 = Σ𝑘 ∈ 𝐴 𝐵)

Proof of Theorem esumpfinvalf Dummy variable 𝑙 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-esum 34185	. . . 4 ⊢ Σ^𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
2		xrge0base 17528	. . . . . 6 ⊢ (0[,]+∞) = (Base‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
3		xrge00 33096	. . . . . 6 ⊢ 0 = (0_g‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
4		xrge0cmn 21399	. . . . . . 7 ⊢ (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd
5	4	a1i 11	. . . . . 6 ⊢ (𝜑 → (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd)
6		xrge0tps 34099	. . . . . . 7 ⊢ (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ TopSp
7	6	a1i 11	. . . . . 6 ⊢ (𝜑 → (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ TopSp)
8		esumpfinvalf.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Fin)
9		esumpfinvalf.2	. . . . . . 7 ⊢ Ⅎ𝑘𝜑
10		esumpfinvalf.1	. . . . . . 7 ⊢ Ⅎ𝑘𝐴
11		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑘(0[,]+∞)
12		icossicc 13352	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
13		esumpfinvalf.b	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))
14	12, 13	sselid 3931	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
15		eqid 2736	. . . . . . 7 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
16	9, 10, 11, 14, 15	fmptdF 32734	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞))
17		c0ex 11126	. . . . . . . 8 ⊢ 0 ∈ V
18	17	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ∈ V)
19	16, 8, 18	fdmfifsupp 9278	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) finSupp 0)
20		xrge0topn 34100	. . . . . . 7 ⊢ (TopOpen‘(ℝ^*_𝑠 ↾_s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))
21	20	eqcomi 2745	. . . . . 6 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) = (TopOpen‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
22		xrhaus 23329	. . . . . . . 8 ⊢ (ordTop‘ ≤ ) ∈ Haus
23		ovex 7391	. . . . . . . 8 ⊢ (0[,]+∞) ∈ V
24		resthaus 23312	. . . . . . . 8 ⊢ (((ordTop‘ ≤ ) ∈ Haus ∧ (0[,]+∞) ∈ V) → ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ Haus)
25	22, 23, 24	mp2an 692	. . . . . . 7 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ Haus
26	25	a1i 11	. . . . . 6 ⊢ (𝜑 → ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ Haus)
27	2, 3, 5, 7, 8, 16, 19, 21, 26	haustsmsid 24085	. . . . 5 ⊢ (𝜑 → ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = {((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))})
28	27	unieqd 4876	. . . 4 ⊢ (𝜑 → ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ∪ {((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))})
29	1, 28	eqtrid 2783	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 = ∪ {((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))})
30		ovex 7391	. . . 4 ⊢ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V
31	30	unisn 4882	. . 3 ⊢ ∪ {((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))} = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))
32	29, 31	eqtrdi 2787	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)))
33		nfcv 2898	. . . 4 ⊢ Ⅎ𝑘(0[,)+∞)
34	9, 10, 33, 13, 15	fmptdF 32734	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,)+∞))
35		esumpfinvallem 34231	. . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,)+∞)) → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)) = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)))
36	8, 34, 35	syl2anc 584	. 2 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)) = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)))
37		rge0ssre 13372	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
38		ax-resscn 11083	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
39	37, 38	sstri 3943	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℂ
40	39, 13	sselid 3931	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
41	40	sbt 2071	. . . . 5 ⊢ [𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
42		sbim 2309	. . . . . 6 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) → [𝑙 / 𝑘]𝐵 ∈ ℂ))
43		sban 2085	. . . . . . . 8 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) ↔ ([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ 𝐴))
44	9	sbf 2277	. . . . . . . . 9 ⊢ ([𝑙 / 𝑘]𝜑 ↔ 𝜑)
45	10	clelsb1fw 2902	. . . . . . . . 9 ⊢ ([𝑙 / 𝑘]𝑘 ∈ 𝐴 ↔ 𝑙 ∈ 𝐴)
46	44, 45	anbi12i 628	. . . . . . . 8 ⊢ (([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑙 ∈ 𝐴))
47	43, 46	bitri 275	. . . . . . 7 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑙 ∈ 𝐴))
48		sbsbc 3744	. . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ [𝑙 / 𝑘]𝐵 ∈ ℂ)
49		sbcel1g 4368	. . . . . . . . 9 ⊢ (𝑙 ∈ V → ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ))
50	49	elv 3445	. . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ)
51	48, 50	bitri 275	. . . . . . 7 ⊢ ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ)
52	47, 51	imbi12i 350	. . . . . 6 ⊢ (([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) → [𝑙 / 𝑘]𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ))
53	42, 52	bitri 275	. . . . 5 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ))
54	41, 53	mpbi 230	. . . 4 ⊢ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ)
55	8, 54	gsumfsum 21389	. . 3 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵)) = Σ𝑙 ∈ 𝐴 ⦋𝑙 / 𝑘⦌𝐵)
56		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑙𝐴
57		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑙𝐵
58		nfcsb1v 3873	. . . . 5 ⊢ Ⅎ𝑘⦋𝑙 / 𝑘⦌𝐵
59		csbeq1a 3863	. . . . 5 ⊢ (𝑘 = 𝑙 → 𝐵 = ⦋𝑙 / 𝑘⦌𝐵)
60	10, 56, 57, 58, 59	cbvmptf 5198	. . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵)
61	60	oveq2i 7369	. . 3 ⊢ (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)) = (ℂ_fld Σ_g (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵))
62	59, 57, 58	cbvsum 15618	. . 3 ⊢ Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑙 ∈ 𝐴 ⦋𝑙 / 𝑘⦌𝐵
63	55, 61, 62	3eqtr4g 2796	. 2 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵)
64	32, 36, 63	3eqtr2d 2777	1 ⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴𝐵 = Σ𝑘 ∈ 𝐴 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 Ⅎwnf 1784 [wsb 2067 ∈ wcel 2113 Ⅎwnfc 2883 Vcvv 3440 [wsbc 3740 ⦋csb 3849 {csn 4580 ∪ cuni 4863 ↦ cmpt 5179 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 Fincfn 8883 ℂcc 11024 ℝcr 11025 0cc0 11026 +∞cpnf 11163 ≤ cle 11167 [,)cico 13263 [,]cicc 13264 Σcsu 15609 ↾_s cress 17157 ↾_t crest 17340 TopOpenctopn 17341 Σ_g cgsu 17360 ordTopcordt 17420 ℝ^*_𝑠cxrs 17421 CMndccmn 19709 ℂ_fldccnfld 21309 TopSpctps 22876 Hauscha 23252 tsums ctsu 24070 Σ^*cesum 34184

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 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-rp 12906 df-xadd 13027 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-clim 15411 df-sum 15610 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-ordt 17422 df-xrs 17423 df-ps 18489 df-tsr 18490 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-grp 18866 df-minusg 18867 df-cntz 19246 df-cmn 19711 df-abl 19712 df-mgp 20076 df-ur 20117 df-ring 20170 df-cring 20171 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-cn 23171 df-haus 23259 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-tsms 24071 df-esum 34185

This theorem is referenced by: volfiniune 34387

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