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Theorem regsumsupp 21663

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 21391	. . . 4 ⊢ ℂ = (Base‘ℂ_fld)
2		cnfld0 21428	. . . 4 ⊢ 0 = (0_g‘ℂ_fld)
3		cnring 21426	. . . . 5 ⊢ ℂ_fld ∈ Ring
4		ringcmn 20305	. . . . 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 1138	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → 𝐼 ∈ 𝑉)
7		simp1 1136	. . . . 5 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → 𝐹:𝐼⟶ℝ)
8		ax-resscn 11241	. . . . 5 ⊢ ℝ ⊆ ℂ
9		fss 6763	. . . . 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 4032	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (𝐹 supp 0) ⊆ (𝐹 supp 0))
12		simp2 1137	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → 𝐹 finSupp 0)
13	1, 2, 5, 6, 10, 11, 12	gsumres 19955	. . 3 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (ℂ_fld Σ_g (𝐹 ↾ (𝐹 supp 0))) = (ℂ_fld Σ_g 𝐹))
14		cnfldadd 21393	. . . 4 ⊢ + = (+_g‘ℂ_fld)
15		df-refld 21646	. . . 4 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
16	8	a1i 11	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → ℝ ⊆ ℂ)
17		0red 11293	. . . 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 11491	. . . . 5 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ ℂ) → (0 + 𝑥) = 𝑥)
20	18	addridd 11490	. . . . 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 18720	. . 3 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (ℂ_fld Σ_g 𝐹) = (ℝ_fld Σ_g 𝐹))
23	13, 22	eqtr2d 2781	. 2 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (ℝ_fld Σ_g 𝐹) = (ℂ_fld Σ_g (𝐹 ↾ (𝐹 supp 0))))
24		suppssdm 8218	. . . . 5 ⊢ (𝐹 supp 0) ⊆ dom 𝐹
25	24, 7	fssdm 6766	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (𝐹 supp 0) ⊆ 𝐼)
26	7, 25	feqresmpt 6991	. . 3 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (𝐹 ↾ (𝐹 supp 0)) = (𝑥 ∈ (𝐹 supp 0) ↦ (𝐹‘𝑥)))
27	26	oveq2d 7464	. 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 9439	. . 3 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (𝐹 supp 0) ∈ Fin)
29		simpl1 1191	. . . . 5 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ (𝐹 supp 0)) → 𝐹:𝐼⟶ℝ)
30	25	sselda 4008	. . . . 5 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ (𝐹 supp 0)) → 𝑥 ∈ 𝐼)
31	29, 30	ffvelcdmd 7119	. . . 4 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ (𝐹 supp 0)) → (𝐹‘𝑥) ∈ ℝ)
32	8, 31	sselid 4006	. . 3 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ (𝐹 supp 0)) → (𝐹‘𝑥) ∈ ℂ)
33	28, 32	gsumfsum 21475	. 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 2784	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 1537 ∈ wcel 2108 ⊆ wss 3976 class class class wbr 5166 ↦ cmpt 5249 ↾ cres 5702 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 supp csupp 8201 finSupp cfsupp 9431 ℂcc 11182 ℝcr 11183 0cc0 11184 + caddc 11187 Σcsu 15734 Σ_g cgsu 17500 CMndccmn 19822 Ringcrg 20260 ℂ_fldccnfld 21387 ℝ_fldcrefld 21645

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-rp 13058 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-sum 15735 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-0g 17501 df-gsum 17502 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-grp 18976 df-minusg 18977 df-cntz 19357 df-cmn 19824 df-abl 19825 df-mgp 20162 df-ur 20209 df-ring 20262 df-cring 20263 df-cnfld 21388 df-refld 21646

This theorem is referenced by: rrxcph 25445 rrxmval 25458 rrxtopnfi 46208

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