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Theorem regsumsupp 20739

Description: The group sum over the real numbers, expressed as a finite sum. (Contributed by Thierry Arnoux, 22-Jun-2019.) (Proof shortened by AV, 19-Jul-2019.)

**Assertion**
Ref	Expression
regsumsupp	⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (ℝ_fld Σ_g 𝐹) = Σ𝑥 ∈ (𝐹 supp 0)(𝐹‘𝑥))

Distinct variable groups: 𝑥,𝐹 𝑥,𝐼 𝑥,𝑉

**Proof of Theorem regsumsupp**
Step	Hyp	Ref	Expression
1		cnfldbas 20514	. . . 4 ⊢ ℂ = (Base‘ℂ_fld)
2		cnfld0 20534	. . . 4 ⊢ 0 = (0_g‘ℂ_fld)
3		cnring 20532	. . . . 5 ⊢ ℂ_fld ∈ Ring
4		ringcmn 19735	. . . . 5 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ CMnd)
5	3, 4	mp1i 13	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → ℂ_fld ∈ CMnd)
6		simp3 1136	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → 𝐼 ∈ 𝑉)
7		simp1 1134	. . . . 5 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → 𝐹:𝐼⟶ℝ)
8		ax-resscn 10859	. . . . 5 ⊢ ℝ ⊆ ℂ
9		fss 6601	. . . . 5 ⊢ ((𝐹:𝐼⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝐼⟶ℂ)
10	7, 8, 9	sylancl 585	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → 𝐹:𝐼⟶ℂ)
11		ssidd 3940	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (𝐹 supp 0) ⊆ (𝐹 supp 0))
12		simp2 1135	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → 𝐹 finSupp 0)
13	1, 2, 5, 6, 10, 11, 12	gsumres 19429	. . 3 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (ℂ_fld Σ_g (𝐹 ↾ (𝐹 supp 0))) = (ℂ_fld Σ_g 𝐹))
14		cnfldadd 20515	. . . 4 ⊢ + = (+_g‘ℂ_fld)
15		df-refld 20722	. . . 4 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
16	8	a1i 11	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → ℝ ⊆ ℂ)
17		0red 10909	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → 0 ∈ ℝ)
18		simpr 484	. . . . . 6 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
19	18	addid2d 11106	. . . . 5 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ ℂ) → (0 + 𝑥) = 𝑥)
20	18	addid1d 11105	. . . . 5 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ ℂ) → (𝑥 + 0) = 𝑥)
21	19, 20	jca 511	. . . 4 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ ℂ) → ((0 + 𝑥) = 𝑥 ∧ (𝑥 + 0) = 𝑥))
22	1, 14, 15, 5, 6, 16, 7, 17, 21	gsumress 18281	. . 3 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (ℂ_fld Σ_g 𝐹) = (ℝ_fld Σ_g 𝐹))
23	13, 22	eqtr2d 2779	. 2 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (ℝ_fld Σ_g 𝐹) = (ℂ_fld Σ_g (𝐹 ↾ (𝐹 supp 0))))
24		suppssdm 7964	. . . . 5 ⊢ (𝐹 supp 0) ⊆ dom 𝐹
25	24, 7	fssdm 6604	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (𝐹 supp 0) ⊆ 𝐼)
26	7, 25	feqresmpt 6820	. . 3 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (𝐹 ↾ (𝐹 supp 0)) = (𝑥 ∈ (𝐹 supp 0) ↦ (𝐹‘𝑥)))
27	26	oveq2d 7271	. 2 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (ℂ_fld Σ_g (𝐹 ↾ (𝐹 supp 0))) = (ℂ_fld Σ_g (𝑥 ∈ (𝐹 supp 0) ↦ (𝐹‘𝑥))))
28	12	fsuppimpd 9065	. . 3 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (𝐹 supp 0) ∈ Fin)
29		simpl1 1189	. . . . 5 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ (𝐹 supp 0)) → 𝐹:𝐼⟶ℝ)
30	25	sselda 3917	. . . . 5 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ (𝐹 supp 0)) → 𝑥 ∈ 𝐼)
31	29, 30	ffvelrnd 6944	. . . 4 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ (𝐹 supp 0)) → (𝐹‘𝑥) ∈ ℝ)
32	8, 31	sselid 3915	. . 3 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ (𝐹 supp 0)) → (𝐹‘𝑥) ∈ ℂ)
33	28, 32	gsumfsum 20577	. 2 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (ℂ_fld Σ_g (𝑥 ∈ (𝐹 supp 0) ↦ (𝐹‘𝑥))) = Σ𝑥 ∈ (𝐹 supp 0)(𝐹‘𝑥))
34	23, 27, 33	3eqtrd 2782	1 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (ℝ_fld Σ_g 𝐹) = Σ𝑥 ∈ (𝐹 supp 0)(𝐹‘𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ⊆ wss 3883 class class class wbr 5070 ↦ cmpt 5153 ↾ cres 5582 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 supp csupp 7948 finSupp cfsupp 9058 ℂcc 10800 ℝcr 10801 0cc0 10802 + caddc 10805 Σcsu 15325 Σ_g cgsu 17068 CMndccmn 19301 Ringcrg 19698 ℂ_fldccnfld 20510 ℝ_fldcrefld 20721

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-sup 9131 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-sum 15326 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-0g 17069 df-gsum 17070 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-grp 18495 df-minusg 18496 df-cntz 18838 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-cnfld 20511 df-refld 20722

This theorem is referenced by: rrxcph 24461 rrxmval 24474 rrxtopnfi 43718

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