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Theorem regsumsupp 21587

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 21324	. . . 4 ⊢ ℂ = (Base‘ℂ_fld)
2		cnfld0 21360	. . . 4 ⊢ 0 = (0_g‘ℂ_fld)
3		cnring 21358	. . . . 5 ⊢ ℂ_fld ∈ Ring
4		ringcmn 20247	. . . . 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 11191	. . . . 5 ⊢ ℝ ⊆ ℂ
9		fss 6727	. . . . 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 3987	. . . 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 19899	. . 3 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (ℂ_fld Σ_g (𝐹 ↾ (𝐹 supp 0))) = (ℂ_fld Σ_g 𝐹))
14		cnfldadd 21326	. . . 4 ⊢ + = (+_g‘ℂ_fld)
15		df-refld 21570	. . . 4 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
16	8	a1i 11	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → ℝ ⊆ ℂ)
17		0red 11243	. . . 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 11441	. . . . 5 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ ℂ) → (0 + 𝑥) = 𝑥)
20	18	addridd 11440	. . . . 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 18665	. . 3 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (ℂ_fld Σ_g 𝐹) = (ℝ_fld Σ_g 𝐹))
23	13, 22	eqtr2d 2772	. 2 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (ℝ_fld Σ_g 𝐹) = (ℂ_fld Σ_g (𝐹 ↾ (𝐹 supp 0))))
24		suppssdm 8181	. . . . 5 ⊢ (𝐹 supp 0) ⊆ dom 𝐹
25	24, 7	fssdm 6730	. . . 4 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (𝐹 supp 0) ⊆ 𝐼)
26	7, 25	feqresmpt 6953	. . 3 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (𝐹 ↾ (𝐹 supp 0)) = (𝑥 ∈ (𝐹 supp 0) ↦ (𝐹‘𝑥)))
27	26	oveq2d 7426	. 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 9386	. . 3 ⊢ ((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) → (𝐹 supp 0) ∈ Fin)
29		simpl1 1192	. . . . 5 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ (𝐹 supp 0)) → 𝐹:𝐼⟶ℝ)
30	25	sselda 3963	. . . . 5 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ (𝐹 supp 0)) → 𝑥 ∈ 𝐼)
31	29, 30	ffvelcdmd 7080	. . . 4 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ (𝐹 supp 0)) → (𝐹‘𝑥) ∈ ℝ)
32	8, 31	sselid 3961	. . 3 ⊢ (((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ (𝐹 supp 0)) → (𝐹‘𝑥) ∈ ℂ)
33	28, 32	gsumfsum 21407	. 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 2775	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 1086 = wceq 1540 ∈ wcel 2109 ⊆ wss 3931 class class class wbr 5124 ↦ cmpt 5206 ↾ cres 5661 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 supp csupp 8164 finSupp cfsupp 9378 ℂcc 11132 ℝcr 11133 0cc0 11134 + caddc 11137 Σcsu 15707 Σ_g cgsu 17459 CMndccmn 19766 Ringcrg 20198 ℂ_fldccnfld 21320 ℝ_fldcrefld 21569

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 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 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-sup 9459 df-oi 9529 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-rp 13014 df-fz 13530 df-fzo 13677 df-seq 14025 df-exp 14085 df-hash 14354 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-clim 15509 df-sum 15708 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-0g 17460 df-gsum 17461 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-grp 18924 df-minusg 18925 df-cntz 19305 df-cmn 19768 df-abl 19769 df-mgp 20106 df-ur 20147 df-ring 20200 df-cring 20201 df-cnfld 21321 df-refld 21570

This theorem is referenced by: rrxcph 25349 rrxmval 25362 rrxtopnfi 46283

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