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Theorem esumadd 33676

Description: Addition of infinite sums. (Contributed by Thierry Arnoux, 24-Mar-2017.)

**Hypotheses**
Ref	Expression
esumadd.0	⊢ (𝜑 → 𝐴 ∈ 𝑉)
esumadd.1	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
esumadd.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞))

**Assertion**
Ref	Expression
esumadd	⊢ (𝜑 → Σ^𝑘 ∈ 𝐴(𝐵 +_𝑒 𝐶) = (Σ^𝑘 ∈ 𝐴𝐵 +_𝑒 Σ^*𝑘 ∈ 𝐴𝐶))

Distinct variable groups: 𝐴,𝑘 𝑘,𝑉 𝜑,𝑘 Allowed substitution hints: 𝐵(𝑘) 𝐶(𝑘)

**Proof of Theorem esumadd**
Step	Hyp	Ref	Expression
1		nfv 1910	. 2 ⊢ Ⅎ𝑘𝜑
2		nfcv 2899	. 2 ⊢ Ⅎ𝑘𝐴
3		esumadd.0	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		esumadd.1	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
5		esumadd.2	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞))
6		ge0xaddcl 13471	. . 3 ⊢ ((𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → (𝐵 +_𝑒 𝐶) ∈ (0[,]+∞))
7	4, 5, 6	syl2anc 583	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 +_𝑒 𝐶) ∈ (0[,]+∞))
8		xrge0base 32741	. . . 4 ⊢ (0[,]+∞) = (Base‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
9		xrge0plusg 32743	. . . 4 ⊢ +_𝑒 = (+_g‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
10		xrge0cmn 21340	. . . . 5 ⊢ (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd
11	10	a1i 11	. . . 4 ⊢ (𝜑 → (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd)
12		xrge0tmd 33546	. . . . 5 ⊢ (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ TopMnd
13	12	a1i 11	. . . 4 ⊢ (𝜑 → (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ TopMnd)
14	4	fmpttd 7125	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞))
15	5	fmpttd 7125	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶(0[,]+∞))
16	1, 2, 3, 4	esumel 33666	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ∈ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)))
17	1, 2, 3, 5	esumel 33666	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐶 ∈ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)))
18	8, 9, 11, 13, 3, 14, 15, 16, 17	tsmsadd 24050	. . 3 ⊢ (𝜑 → (Σ^𝑘 ∈ 𝐴𝐵 +_𝑒 Σ^𝑘 ∈ 𝐴𝐶) ∈ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) tsums ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘_f +_𝑒 (𝑘 ∈ 𝐴 ↦ 𝐶))))
19		eqidd 2729	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵))
20		eqidd 2729	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐴 ↦ 𝐶))
21	3, 4, 5, 19, 20	offval2 7705	. . . 4 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘_f +_𝑒 (𝑘 ∈ 𝐴 ↦ 𝐶)) = (𝑘 ∈ 𝐴 ↦ (𝐵 +_𝑒 𝐶)))
22	21	oveq2d 7436	. . 3 ⊢ (𝜑 → ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘_f +_𝑒 (𝑘 ∈ 𝐴 ↦ 𝐶))) = ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ (𝐵 +_𝑒 𝐶))))
23	18, 22	eleqtrd 2831	. 2 ⊢ (𝜑 → (Σ^𝑘 ∈ 𝐴𝐵 +_𝑒 Σ^𝑘 ∈ 𝐴𝐶) ∈ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ (𝐵 +_𝑒 𝐶))))
24	1, 2, 3, 7, 23	esumid 33663	1 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴(𝐵 +_𝑒 𝐶) = (Σ^𝑘 ∈ 𝐴𝐵 +_𝑒 Σ^*𝑘 ∈ 𝐴𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ↦ cmpt 5231 (class class class)co 7420 ∘_f cof 7683 0cc0 11138 +∞cpnf 11275 +_𝑒 cxad 13122 [,]cicc 13359 ↾_s cress 17208 ℝ^*_𝑠cxrs 17481 CMndccmn 19734 TopMndctmd 23973 tsums ctsu 24029 Σ^*cesum 33646

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 ax-mulf 11218

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-er 8724 df-map 8846 df-pm 8847 df-ixp 8916 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fsupp 9386 df-fi 9434 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-ioc 13361 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-mod 13867 df-seq 13999 df-exp 14059 df-fac 14265 df-bc 14294 df-hash 14322 df-shft 15046 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-limsup 15447 df-clim 15464 df-rlim 15465 df-sum 15665 df-ef 16043 df-sin 16045 df-cos 16046 df-pi 16048 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-rest 17403 df-topn 17404 df-0g 17422 df-gsum 17423 df-topgen 17424 df-pt 17425 df-prds 17428 df-ordt 17482 df-xrs 17483 df-qtop 17488 df-imas 17489 df-xps 17491 df-mre 17565 df-mrc 17566 df-acs 17568 df-ps 18557 df-tsr 18558 df-plusf 18598 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-grp 18892 df-minusg 18893 df-sbg 18894 df-mulg 19023 df-subg 19077 df-cntz 19267 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-cring 20175 df-subrng 20482 df-subrg 20507 df-abv 20696 df-lmod 20744 df-scaf 20745 df-sra 21057 df-rgmod 21058 df-psmet 21270 df-xmet 21271 df-met 21272 df-bl 21273 df-mopn 21274 df-fbas 21275 df-fg 21276 df-cnfld 21279 df-top 22795 df-topon 22812 df-topsp 22834 df-bases 22848 df-cld 22922 df-ntr 22923 df-cls 22924 df-nei 23001 df-lp 23039 df-perf 23040 df-cn 23130 df-cnp 23131 df-haus 23218 df-tx 23465 df-hmeo 23658 df-fil 23749 df-fm 23841 df-flim 23842 df-flf 23843 df-tmd 23975 df-tgp 23976 df-tsms 24030 df-trg 24063 df-xms 24225 df-ms 24226 df-tms 24227 df-nm 24490 df-ngp 24491 df-nrg 24493 df-nlm 24494 df-ii 24796 df-cncf 24797 df-limc 25794 df-dv 25795 df-log 26489 df-esum 33647

This theorem is referenced by: esumle 33677

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