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Theorem regsumsupp 21615

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 21351	. . . 4 ⊢ ℂ = (Base‘ℂ_fld)
2		cnfld0 21385	. . . 4 ⊢ 0 = (0_g‘ℂ_fld)
3		cnring 21383	. . . . 5 ⊢ ℂ_fld ∈ Ring
4		ringcmn 20257	. . . . 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 1139	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → 𝐼 ∈ 𝑉)
7		simp1 1137	. . . . 5 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → 𝐹:𝐼⟶ℝ)
8		ax-resscn 11089	. . . . 5 ⊢ ℝ ⊆ ℂ
9		fss 6679	. . . . 5 ⊢ ((𝐹:𝐼⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝐼⟶ℂ)
10	7, 8, 9	sylancl 587	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → 𝐹:𝐼⟶ℂ)
11		ssidd 3946	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (𝐹 supp 0) ⊆ (𝐹 supp 0))
12		simp2 1138	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → 𝐹 finSupp 0)
13	1, 2, 5, 6, 10, 11, 12	gsumres 19882	. . 3 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (ℂ_fld Σ_g (𝐹 ↾ (𝐹 supp 0))) = (ℂ_fld Σ_g 𝐹))
14		cnfldadd 21353	. . . 4 ⊢ + = (+_g‘ℂ_fld)
15		df-refld 21598	. . . 4 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
16	8	a1i 11	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → ℝ ⊆ ℂ)
17		0red 11141	. . . 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	addlidd 11341	. . . . 5 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ ℂ) → (0 + 𝑥) = 𝑥)
20	18	addridd 11340	. . . . 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 18644	. . 3 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (ℂ_fld Σ_g 𝐹) = (ℝ_fld Σ_g 𝐹))
23	13, 22	eqtr2d 2773	. 2 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (ℝ_fld Σ_g 𝐹) = (ℂ_fld Σ_g (𝐹 ↾ (𝐹 supp 0))))
24		suppssdm 8121	. . . . 5 ⊢ (𝐹 supp 0) ⊆ dom 𝐹
25	24, 7	fssdm 6682	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (𝐹 supp 0) ⊆ 𝐼)
26	7, 25	feqresmpt 6904	. . 3 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (𝐹 ↾ (𝐹 supp 0)) = (𝑥 ∈ (𝐹 supp 0) ↦ (𝐹‘𝑥)))
27	26	oveq2d 7377	. 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 9276	. . 3 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (𝐹 supp 0) ∈ Fin)
29		simpl1 1193	. . . . 5 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ (𝐹 supp 0)) → 𝐹:𝐼⟶ℝ)
30	25	sselda 3922	. . . . 5 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ (𝐹 supp 0)) → 𝑥 ∈ 𝐼)
31	29, 30	ffvelcdmd 7032	. . . 4 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ (𝐹 supp 0)) → (𝐹‘𝑥) ∈ ℝ)
32	8, 31	sselid 3920	. . 3 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ (𝐹 supp 0)) → (𝐹‘𝑥) ∈ ℂ)
33	28, 32	gsumfsum 21427	. 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 2776	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 1087 = wceq 1542 ∈ wcel 2114 ⊆ wss 3890 class class class wbr 5086 ↦ cmpt 5167 ↾ cres 5627 ⟶wf 6489 ‘cfv 6493 (class class class)co 7361 supp csupp 8104 finSupp cfsupp 9268 ℂcc 11030 ℝcr 11031 0cc0 11032 + caddc 11035 Σcsu 15642 Σ_g cgsu 17397 CMndccmn 19749 Ringcrg 20208 ℂ_fldccnfld 21347 ℝ_fldcrefld 21597

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-inf2 9556 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110 ax-addf 11111

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-sup 9349 df-oi 9419 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-z 12519 df-dec 12639 df-uz 12783 df-rp 12937 df-fz 13456 df-fzo 13603 df-seq 13958 df-exp 14018 df-hash 14287 df-cj 15055 df-re 15056 df-im 15057 df-sqrt 15191 df-abs 15192 df-clim 15444 df-sum 15643 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-starv 17229 df-tset 17233 df-ple 17234 df-ds 17236 df-unif 17237 df-0g 17398 df-gsum 17399 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-grp 18906 df-minusg 18907 df-cntz 19286 df-cmn 19751 df-abl 19752 df-mgp 20116 df-ur 20157 df-ring 20210 df-cring 20211 df-cnfld 21348 df-refld 21598

This theorem is referenced by: rrxcph 25372 rrxmval 25385 rrxtopnfi 46736

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