esumpfinvalf - Mathbox for Thierry Arnoux

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Theorem esumpfinvalf 34475

Description: Same as esumpfinval 34474, 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 34427	. . . 4 ⊢ Σ^𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
2		xrge0base 17667	. . . . . 6 ⊢ (0[,]+∞) = (Base‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
3		xrge00 33343	. . . . . 6 ⊢ 0 = (0_g‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
4		xrge0cmn 21605	. . . . . . 7 ⊢ (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd
5	4	a1i 11	. . . . . 6 ⊢ (𝜑 → (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd)
6		xrge0tps 34341	. . . . . . 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 2924	. . . . . . 7 ⊢ Ⅎ𝑘(0[,]+∞)
12		icossicc 13469	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
13		esumpfinvalf.b	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))
14	12, 13	sselid 3934	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
15		eqid 2762	. . . . . . 7 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
16	9, 10, 11, 14, 15	fmptdF 33012	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞))
17		c0ex 11206	. . . . . . . 8 ⊢ 0 ∈ V
18	17	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ∈ V)
19	16, 8, 18	fdmfifsupp 9333	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) finSupp 0)
20		xrge0topn 34342	. . . . . . 7 ⊢ (TopOpen‘(ℝ^*_𝑠 ↾_s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))
21	20	eqcomi 2771	. . . . . 6 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) = (TopOpen‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
22		xrhaus 23553	. . . . . . . 8 ⊢ (ordTop‘ ≤ ) ∈ Haus
23		ovex 7445	. . . . . . . 8 ⊢ (0[,]+∞) ∈ V
24		resthaus 23536	. . . . . . . 8 ⊢ (((ordTop‘ ≤ ) ∈ Haus ∧ (0[,]+∞) ∈ V) → ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ Haus)
25	22, 23, 24	mp2an 704	. . . . . . 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 24309	. . . . 5 ⊢ (𝜑 → ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = {((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))})
28	27	unieqd 4884	. . . 4 ⊢ (𝜑 → ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ∪ {((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))})
29	1, 28	eqtrid 2809	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 = ∪ {((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))})
30		ovex 7445	. . . 4 ⊢ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V
31	30	unisn 4890	. . 3 ⊢ ∪ {((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))} = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))
32	29, 31	eqtrdi 2813	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)))
33		nfcv 2924	. . . 4 ⊢ Ⅎ𝑘(0[,)+∞)
34	9, 10, 33, 13, 15	fmptdF 33012	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,)+∞))
35		esumpfinvallem 34473	. . 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 595	. 2 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)) = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)))
37		rge0ssre 13489	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
38		ax-resscn 11163	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
39	37, 38	sstri 3945	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℂ
40	39, 13	sselid 3934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
41	40	sbt 2099	. . . . 5 ⊢ [𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
42		sbim 2337	. . . . . 6 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) → [𝑙 / 𝑘]𝐵 ∈ ℂ))
43		sban 2113	. . . . . . . 8 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) ↔ ([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ 𝐴))
44	9	sbf 2305	. . . . . . . . 9 ⊢ ([𝑙 / 𝑘]𝜑 ↔ 𝜑)
45	10	clelsb1fw 2928	. . . . . . . . 9 ⊢ ([𝑙 / 𝑘]𝑘 ∈ 𝐴 ↔ 𝑙 ∈ 𝐴)
46	44, 45	anbi12i 639	. . . . . . . 8 ⊢ (([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑙 ∈ 𝐴))
47	43, 46	bitri 278	. . . . . . 7 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑙 ∈ 𝐴))
48		sbsbc 3747	. . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ [𝑙 / 𝑘]𝐵 ∈ ℂ)
49		sbcel1g 4380	. . . . . . . . 9 ⊢ (𝑙 ∈ V → ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ))
50	49	elv 3459	. . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ)
51	48, 50	bitri 278	. . . . . . 7 ⊢ ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ)
52	47, 51	imbi12i 353	. . . . . 6 ⊢ (([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) → [𝑙 / 𝑘]𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ))
53	42, 52	bitri 278	. . . . 5 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ))
54	41, 53	mpbi 233	. . . 4 ⊢ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ)
55	8, 54	gsumfsum 21595	. . 3 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵)) = Σ𝑙 ∈ 𝐴 ⦋𝑙 / 𝑘⦌𝐵)
56		nfcv 2924	. . . . 5 ⊢ Ⅎ𝑙𝐴
57		nfcv 2924	. . . . 5 ⊢ Ⅎ𝑙𝐵
58		nfcsb1v 3876	. . . . 5 ⊢ Ⅎ𝑘⦋𝑙 / 𝑘⦌𝐵
59		csbeq1a 3866	. . . . 5 ⊢ (𝑘 = 𝑙 → 𝐵 = ⦋𝑙 / 𝑘⦌𝐵)
60	10, 56, 57, 58, 59	cbvmptf 5210	. . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵)
61	60	oveq2i 7423	. . 3 ⊢ (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)) = (ℂ_fld Σ_g (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵))
62	59, 57, 58	cbvsum 15753	. . 3 ⊢ Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑙 ∈ 𝐴 ⦋𝑙 / 𝑘⦌𝐵
63	55, 61, 62	3eqtr4g 2822	. 2 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵)
64	32, 36, 63	3eqtr2d 2803	1 ⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴𝐵 = Σ𝑘 ∈ 𝐴 𝐵)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1569 Ⅎwnf 1812 [wsb 2095 ∈ wcel 2142 Ⅎwnfc 2909 Vcvv 3454 [wsbc 3743 ⦋csb 3852 {csn 4588 ∪ cuni 4871 ↦ cmpt 5191 ⟶wf 6532 ‘cfv 6536 (class class class)co 7412 Fincfn 8941 ℂcc 11104 ℝcr 11105 0cc0 11106 +∞cpnf 11246 ≤ cle 11250 [,)cico 13380 [,]cicc 13381 Σcsu 15744 ↾_s cress 17296 ↾_t crest 17479 TopOpenctopn 17480 Σ_g cgsu 17499 ordTopcordt 17559 ℝ^*_𝑠cxrs 17560 CMndccmn 19856 ℂ_fldccnfld 21533 TopSpctps 23100 Hauscha 23476 tsums ctsu 24294 Σ^*cesum 34426

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-inf2 9608 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 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 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-er 8692 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-fi 9369 df-sup 9400 df-oi 9470 df-card 9932 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12869 df-rp 13023 df-xadd 13144 df-ico 13384 df-icc 13385 df-fz 13542 df-fzo 13690 df-seq 14045 df-exp 14105 df-hash 14374 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-clim 15546 df-sum 15745 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-starv 17331 df-tset 17335 df-ple 17336 df-ds 17338 df-unif 17339 df-rest 17481 df-topn 17482 df-0g 17500 df-gsum 17501 df-topgen 17502 df-ordt 17561 df-xrs 17562 df-ps 18628 df-tsr 18629 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-submnd 18848 df-grp 19009 df-minusg 19010 df-cntz 19393 df-cmn 19858 df-abl 19859 df-mgp 20223 df-ur 20270 df-ring 20323 df-cring 20324 df-fbas 21530 df-fg 21531 df-cnfld 21534 df-top 23062 df-topon 23079 df-topsp 23101 df-bases 23114 df-cld 23187 df-ntr 23188 df-cls 23189 df-nei 23266 df-cn 23395 df-haus 23483 df-fil 24014 df-fm 24106 df-flim 24107 df-flf 24108 df-tsms 24295 df-esum 34427

This theorem is used by: volfiniune 34629

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