esumpfinvalf - Mathbox for Thierry Arnoux

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Theorem esumpfinvalf 30478

Description: Same as esumpfinval 30477, 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 30430	. . . 4 ⊢ Σ^𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
2		xrge0base 30025	. . . . . 6 ⊢ (0[,]+∞) = (Base‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
3		xrge00 30026	. . . . . 6 ⊢ 0 = (0_g‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
4		xrge0cmn 20003	. . . . . . 7 ⊢ (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd
5	4	a1i 11	. . . . . 6 ⊢ (𝜑 → (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd)
6		xrge0tps 30328	. . . . . . 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 2913	. . . . . . 7 ⊢ Ⅎ𝑘(0[,]+∞)
12		icossicc 12466	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
13		esumpfinvalf.b	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))
14	12, 13	sseldi 3750	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
15		eqid 2771	. . . . . . 7 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
16	9, 10, 11, 14, 15	fmptdF 29796	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞))
17		c0ex 10240	. . . . . . . 8 ⊢ 0 ∈ V
18	17	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ∈ V)
19	16, 8, 18	fdmfifsupp 8445	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) finSupp 0)
20		xrge0topn 30329	. . . . . . 7 ⊢ (TopOpen‘(ℝ^*_𝑠 ↾_s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))
21	20	eqcomi 2780	. . . . . 6 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) = (TopOpen‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
22		xrhaus 29875	. . . . . . . 8 ⊢ (ordTop‘ ≤ ) ∈ Haus
23		ovex 6827	. . . . . . . 8 ⊢ (0[,]+∞) ∈ V
24		resthaus 21393	. . . . . . . 8 ⊢ (((ordTop‘ ≤ ) ∈ Haus ∧ (0[,]+∞) ∈ V) → ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ Haus)
25	22, 23, 24	mp2an 672	. . . . . . 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 22164	. . . . 5 ⊢ (𝜑 → ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = {((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))})
28	27	unieqd 4585	. . . 4 ⊢ (𝜑 → ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ∪ {((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))})
29	1, 28	syl5eq 2817	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 = ∪ {((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))})
30		ovex 6827	. . . 4 ⊢ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V
31	30	unisn 4590	. . 3 ⊢ ∪ {((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))} = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))
32	29, 31	syl6eq 2821	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)))
33		nfcv 2913	. . . 4 ⊢ Ⅎ𝑘(0[,)+∞)
34	9, 10, 33, 13, 15	fmptdF 29796	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,)+∞))
35		esumpfinvallem 30476	. . 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 573	. 2 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)) = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)))
37		rge0ssre 12487	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
38		ax-resscn 10199	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
39	37, 38	sstri 3761	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℂ
40	39, 13	sseldi 3750	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
41	40	sbt 2566	. . . . 5 ⊢ [𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
42		sbim 2542	. . . . . 6 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) → [𝑙 / 𝑘]𝐵 ∈ ℂ))
43		sban 2546	. . . . . . . 8 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) ↔ ([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ 𝐴))
44	9	sbf 2527	. . . . . . . . 9 ⊢ ([𝑙 / 𝑘]𝜑 ↔ 𝜑)
45	10	clelsb3f 2917	. . . . . . . . 9 ⊢ ([𝑙 / 𝑘]𝑘 ∈ 𝐴 ↔ 𝑙 ∈ 𝐴)
46	44, 45	anbi12i 612	. . . . . . . 8 ⊢ (([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑙 ∈ 𝐴))
47	43, 46	bitri 264	. . . . . . 7 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑙 ∈ 𝐴))
48		sbsbc 3591	. . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ [𝑙 / 𝑘]𝐵 ∈ ℂ)
49		vex 3354	. . . . . . . . 9 ⊢ 𝑙 ∈ V
50		sbcel1g 4132	. . . . . . . . 9 ⊢ (𝑙 ∈ V → ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ))
51	49, 50	ax-mp 5	. . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ)
52	48, 51	bitri 264	. . . . . . 7 ⊢ ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ)
53	47, 52	imbi12i 339	. . . . . 6 ⊢ (([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) → [𝑙 / 𝑘]𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ))
54	42, 53	bitri 264	. . . . 5 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ))
55	41, 54	mpbi 220	. . . 4 ⊢ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ)
56	8, 55	gsumfsum 20028	. . 3 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵)) = Σ𝑙 ∈ 𝐴 ⦋𝑙 / 𝑘⦌𝐵)
57		nfcv 2913	. . . . 5 ⊢ Ⅎ𝑙𝐴
58		nfcv 2913	. . . . 5 ⊢ Ⅎ𝑙𝐵
59		nfcsb1v 3698	. . . . 5 ⊢ Ⅎ𝑘⦋𝑙 / 𝑘⦌𝐵
60		csbeq1a 3691	. . . . 5 ⊢ (𝑘 = 𝑙 → 𝐵 = ⦋𝑙 / 𝑘⦌𝐵)
61	10, 57, 58, 59, 60	cbvmptf 4883	. . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵)
62	61	oveq2i 6807	. . 3 ⊢ (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)) = (ℂ_fld Σ_g (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵))
63	60, 57, 10, 58, 59	cbvsum 14633	. . 3 ⊢ Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑙 ∈ 𝐴 ⦋𝑙 / 𝑘⦌𝐵
64	56, 62, 63	3eqtr4g 2830	. 2 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵)
65	32, 36, 64	3eqtr2d 2811	1 ⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴𝐵 = Σ𝑘 ∈ 𝐴 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 Ⅎwnf 1856 [wsb 2049 ∈ wcel 2145 Ⅎwnfc 2900 Vcvv 3351 [wsbc 3587 ⦋csb 3682 {csn 4317 ∪ cuni 4575 ↦ cmpt 4864 ⟶wf 6026 ‘cfv 6030 (class class class)co 6796 Fincfn 8113 ℂcc 10140 ℝcr 10141 0cc0 10142 +∞cpnf 10277 ≤ cle 10281 [,)cico 12382 [,]cicc 12383 Σcsu 14624 ↾_s cress 16065 ↾_t crest 16289 TopOpenctopn 16290 Σ_g cgsu 16309 ordTopcordt 16367 ℝ^*_𝑠cxrs 16368 CMndccmn 18400 ℂ_fldccnfld 19961 TopSpctps 20957 Hauscha 21333 tsums ctsu 22149 Σ^*cesum 30429

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-inf2 8706 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-pre-sup 10220 ax-addf 10221 ax-mulf 10222

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-iin 4658 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-isom 6039 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-om 7217 df-1st 7319 df-2nd 7320 df-supp 7451 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-1o 7717 df-oadd 7721 df-er 7900 df-map 8015 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-fsupp 8436 df-fi 8477 df-sup 8508 df-oi 8575 df-card 8969 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-div 10891 df-nn 11227 df-2 11285 df-3 11286 df-4 11287 df-5 11288 df-6 11289 df-7 11290 df-8 11291 df-9 11292 df-n0 11500 df-z 11585 df-dec 11701 df-uz 11894 df-rp 12036 df-xadd 12152 df-ico 12386 df-icc 12387 df-fz 12534 df-fzo 12674 df-seq 13009 df-exp 13068 df-hash 13322 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-clim 14427 df-sum 14625 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-starv 16164 df-tset 16168 df-ple 16169 df-ds 16172 df-unif 16173 df-rest 16291 df-topn 16292 df-0g 16310 df-gsum 16311 df-topgen 16312 df-ordt 16369 df-xrs 16370 df-ps 17408 df-tsr 17409 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-submnd 17544 df-grp 17633 df-minusg 17634 df-cntz 17957 df-cmn 18402 df-abl 18403 df-mgp 18698 df-ur 18710 df-ring 18757 df-cring 18758 df-fbas 19958 df-fg 19959 df-cnfld 19962 df-top 20919 df-topon 20936 df-topsp 20958 df-bases 20971 df-cld 21044 df-ntr 21045 df-cls 21046 df-nei 21123 df-cn 21252 df-haus 21340 df-fil 21870 df-fm 21962 df-flim 21963 df-flf 21964 df-tsms 22150 df-esum 30430

This theorem is referenced by: volfiniune 30633

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