esumpfinvalf - Mathbox for Thierry Arnoux

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Theorem esumpfinvalf 34375

Description: Same as esumpfinval 34374, 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 34327	. . . 4 ⊢ Σ^𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
2		xrge0base 17639	. . . . . 6 ⊢ (0[,]+∞) = (Base‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
3		xrge00 33194	. . . . . 6 ⊢ 0 = (0_g‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
4		xrge0cmn 21498	. . . . . . 7 ⊢ (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd
5	4	a1i 11	. . . . . 6 ⊢ (𝜑 → (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd)
6		xrge0tps 34241	. . . . . . 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 2926	. . . . . . 7 ⊢ Ⅎ𝑘(0[,]+∞)
12		icossicc 13442	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
13		esumpfinvalf.b	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))
14	12, 13	sselid 3936	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
15		eqid 2764	. . . . . . 7 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
16	9, 10, 11, 14, 15	fmptdF 32860	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞))
17		c0ex 11175	. . . . . . . 8 ⊢ 0 ∈ V
18	17	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ∈ V)
19	16, 8, 18	fdmfifsupp 9323	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) finSupp 0)
20		xrge0topn 34242	. . . . . . 7 ⊢ (TopOpen‘(ℝ^*_𝑠 ↾_s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))
21	20	eqcomi 2773	. . . . . 6 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) = (TopOpen‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
22		xrhaus 23447	. . . . . . . 8 ⊢ (ordTop‘ ≤ ) ∈ Haus
23		ovex 7431	. . . . . . . 8 ⊢ (0[,]+∞) ∈ V
24		resthaus 23430	. . . . . . . 8 ⊢ (((ordTop‘ ≤ ) ∈ Haus ∧ (0[,]+∞) ∈ V) → ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ Haus)
25	22, 23, 24	mp2an 702	. . . . . . 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 24203	. . . . 5 ⊢ (𝜑 → ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = {((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))})
28	27	unieqd 4880	. . . 4 ⊢ (𝜑 → ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ∪ {((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))})
29	1, 28	eqtrid 2811	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 = ∪ {((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))})
30		ovex 7431	. . . 4 ⊢ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V
31	30	unisn 4886	. . 3 ⊢ ∪ {((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))} = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))
32	29, 31	eqtrdi 2815	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)))
33		nfcv 2926	. . . 4 ⊢ Ⅎ𝑘(0[,)+∞)
34	9, 10, 33, 13, 15	fmptdF 32860	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,)+∞))
35		esumpfinvallem 34373	. . 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 593	. 2 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)) = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)))
37		rge0ssre 13462	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
38		ax-resscn 11132	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
39	37, 38	sstri 3947	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℂ
40	39, 13	sselid 3936	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
41	40	sbt 2097	. . . . 5 ⊢ [𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
42		sbim 2339	. . . . . 6 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) → [𝑙 / 𝑘]𝐵 ∈ ℂ))
43		sban 2115	. . . . . . . 8 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) ↔ ([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ 𝐴))
44	9	sbf 2307	. . . . . . . . 9 ⊢ ([𝑙 / 𝑘]𝜑 ↔ 𝜑)
45	10	clelsb1fw 2930	. . . . . . . . 9 ⊢ ([𝑙 / 𝑘]𝑘 ∈ 𝐴 ↔ 𝑙 ∈ 𝐴)
46	44, 45	anbi12i 637	. . . . . . . 8 ⊢ (([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑙 ∈ 𝐴))
47	43, 46	bitri 277	. . . . . . 7 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑙 ∈ 𝐴))
48		sbsbc 3750	. . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ [𝑙 / 𝑘]𝐵 ∈ ℂ)
49		sbcel1g 4372	. . . . . . . . 9 ⊢ (𝑙 ∈ V → ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ))
50	49	elv 3461	. . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ)
51	48, 50	bitri 277	. . . . . . 7 ⊢ ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ)
52	47, 51	imbi12i 352	. . . . . 6 ⊢ (([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) → [𝑙 / 𝑘]𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ))
53	42, 52	bitri 277	. . . . 5 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ))
54	41, 53	mpbi 232	. . . 4 ⊢ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ)
55	8, 54	gsumfsum 21488	. . 3 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵)) = Σ𝑙 ∈ 𝐴 ⦋𝑙 / 𝑘⦌𝐵)
56		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑙𝐴
57		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑙𝐵
58		nfcsb1v 3878	. . . . 5 ⊢ Ⅎ𝑘⦋𝑙 / 𝑘⦌𝐵
59		csbeq1a 3868	. . . . 5 ⊢ (𝑘 = 𝑙 → 𝐵 = ⦋𝑙 / 𝑘⦌𝐵)
60	10, 56, 57, 58, 59	cbvmptf 5202	. . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵)
61	60	oveq2i 7409	. . 3 ⊢ (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)) = (ℂ_fld Σ_g (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵))
62	59, 57, 58	cbvsum 15724	. . 3 ⊢ Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑙 ∈ 𝐴 ⦋𝑙 / 𝑘⦌𝐵
63	55, 61, 62	3eqtr4g 2824	. 2 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵)
64	32, 36, 63	3eqtr2d 2805	1 ⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴𝐵 = Σ𝑘 ∈ 𝐴 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1562 Ⅎwnf 1805 [wsb 2092 ∈ wcel 2144 Ⅎwnfc 2911 Vcvv 3456 [wsbc 3746 ⦋csb 3854 {csn 4584 ∪ cuni 4867 ↦ cmpt 5183 ⟶wf 6519 ‘cfv 6523 (class class class)co 7398 Fincfn 8929 ℂcc 11073 ℝcr 11074 0cc0 11075 +∞cpnf 11215 ≤ cle 11219 [,)cico 13353 [,]cicc 13354 Σcsu 15715 ↾_s cress 17268 ↾_t crest 17451 TopOpenctopn 17452 Σ_g cgsu 17471 ordTopcordt 17531 ℝ^*_𝑠cxrs 17532 CMndccmn 19822 ℂ_fldccnfld 21426 TopSpctps 22994 Hauscha 23370 tsums ctsu 24188 Σ^*cesum 34326

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-inf2 9598 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-se 5603 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-isom 6532 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-om 7849 df-1st 7972 df-2nd 7973 df-supp 8143 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-2o 8440 df-er 8680 df-map 8812 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-fsupp 9310 df-fi 9359 df-sup 9390 df-oi 9460 df-card 9899 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-div 11847 df-nn 12213 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12484 df-z 12571 df-dec 12691 df-uz 12842 df-rp 12996 df-xadd 13117 df-ico 13357 df-icc 13358 df-fz 13515 df-fzo 13662 df-seq 14017 df-exp 14077 df-hash 14346 df-cj 15128 df-re 15129 df-im 15130 df-sqrt 15264 df-abs 15265 df-clim 15517 df-sum 15716 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17248 df-ress 17269 df-plusg 17301 df-mulr 17302 df-starv 17303 df-tset 17307 df-ple 17308 df-ds 17310 df-unif 17311 df-rest 17453 df-topn 17454 df-0g 17472 df-gsum 17473 df-topgen 17474 df-ordt 17533 df-xrs 17534 df-ps 18600 df-tsr 18601 df-mgm 18676 df-sgrp 18755 df-mnd 18771 df-submnd 18820 df-grp 18980 df-minusg 18981 df-cntz 19359 df-cmn 19824 df-abl 19825 df-mgp 20189 df-ur 20234 df-ring 20287 df-cring 20288 df-fbas 21423 df-fg 21424 df-cnfld 21427 df-top 22956 df-topon 22973 df-topsp 22995 df-bases 23008 df-cld 23081 df-ntr 23082 df-cls 23083 df-nei 23160 df-cn 23289 df-haus 23377 df-fil 23908 df-fm 24000 df-flim 24001 df-flf 24002 df-tsms 24189 df-esum 34327

This theorem is referenced by: volfiniune 34529

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