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Theorem regsumsupp 20827

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 20601	. . . 4 ⊢ ℂ = (Base‘ℂ_fld)
2		cnfld0 20622	. . . 4 ⊢ 0 = (0_g‘ℂ_fld)
3		cnring 20620	. . . . 5 ⊢ ℂ_fld ∈ Ring
4		ringcmn 19820	. . . . 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 1137	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → 𝐼 ∈ 𝑉)
7		simp1 1135	. . . . 5 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → 𝐹:𝐼⟶ℝ)
8		ax-resscn 10928	. . . . 5 ⊢ ℝ ⊆ ℂ
9		fss 6617	. . . . 5 ⊢ ((𝐹:𝐼⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝐼⟶ℂ)
10	7, 8, 9	sylancl 586	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → 𝐹:𝐼⟶ℂ)
11		ssidd 3944	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (𝐹 supp 0) ⊆ (𝐹 supp 0))
12		simp2 1136	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → 𝐹 finSupp 0)
13	1, 2, 5, 6, 10, 11, 12	gsumres 19514	. . 3 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (ℂ_fld Σ_g (𝐹 ↾ (𝐹 supp 0))) = (ℂ_fld Σ_g 𝐹))
14		cnfldadd 20602	. . . 4 ⊢ + = (+_g‘ℂ_fld)
15		df-refld 20810	. . . 4 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
16	8	a1i 11	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → ℝ ⊆ ℂ)
17		0red 10978	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → 0 ∈ ℝ)
18		simpr 485	. . . . . 6 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
19	18	addid2d 11176	. . . . 5 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ ℂ) → (0 + 𝑥) = 𝑥)
20	18	addid1d 11175	. . . . 5 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ ℂ) → (𝑥 + 0) = 𝑥)
21	19, 20	jca 512	. . . 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 18366	. . 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 7993	. . . . 5 ⊢ (𝐹 supp 0) ⊆ dom 𝐹
25	24, 7	fssdm 6620	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (𝐹 supp 0) ⊆ 𝐼)
26	7, 25	feqresmpt 6838	. . 3 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (𝐹 ↾ (𝐹 supp 0)) = (𝑥 ∈ (𝐹 supp 0) ↦ (𝐹‘𝑥)))
27	26	oveq2d 7291	. 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 9135	. . 3 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (𝐹 supp 0) ∈ Fin)
29		simpl1 1190	. . . . 5 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ (𝐹 supp 0)) → 𝐹:𝐼⟶ℝ)
30	25	sselda 3921	. . . . 5 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ (𝐹 supp 0)) → 𝑥 ∈ 𝐼)
31	29, 30	ffvelrnd 6962	. . . 4 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ (𝐹 supp 0)) → (𝐹‘𝑥) ∈ ℝ)
32	8, 31	sselid 3919	. . 3 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ (𝐹 supp 0)) → (𝐹‘𝑥) ∈ ℂ)
33	28, 32	gsumfsum 20665	. 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 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ⊆ wss 3887 class class class wbr 5074 ↦ cmpt 5157 ↾ cres 5591 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 supp csupp 7977 finSupp cfsupp 9128 ℂcc 10869 ℝcr 10870 0cc0 10871 + caddc 10874 Σcsu 15397 Σ_g cgsu 17151 CMndccmn 19386 Ringcrg 19783 ℂ_fldccnfld 20597 ℝ_fldcrefld 20809

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-sup 9201 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-sum 15398 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-0g 17152 df-gsum 17153 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-grp 18580 df-minusg 18581 df-cntz 18923 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-cnfld 20598 df-refld 20810

This theorem is referenced by: rrxcph 24556 rrxmval 24569 rrxtopnfi 43828

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