fsumsplit - Metamath Proof Explorer

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Theorem fsumsplit 15655

Description: Split a sum into two parts. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 22-Apr-2014.)

**Hypotheses**
Ref	Expression
fsumsplit.1	⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
fsumsplit.2	⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵))
fsumsplit.3	⊢ (𝜑 → 𝑈 ∈ Fin)
fsumsplit.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐶 ∈ ℂ)

**Assertion**
Ref	Expression
fsumsplit	⊢ (𝜑 → Σ𝑘 ∈ 𝑈 𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶))

Distinct variable groups: 𝐴,𝑘 𝐵,𝑘 𝜑,𝑘 𝑈,𝑘 Allowed substitution hint: 𝐶(𝑘)

**Proof of Theorem fsumsplit**
Step	Hyp	Ref	Expression
1		ssun1 4127	. . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
2		fsumsplit.2	. . . . 5 ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵))
3	1, 2	sseqtrrid 3974	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝑈)
4	3	sselda 3930	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝑈)
5		fsumsplit.4	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐶 ∈ ℂ)
6	4, 5	syldan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
7	6	ralrimiva 3125	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ ℂ)
8		fsumsplit.3	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ Fin)
9	8	olcd 874	. . . 4 ⊢ (𝜑 → (𝑈 ⊆ (ℤ_≥‘0) ∨ 𝑈 ∈ Fin))
10		sumss2 15640	. . . 4 ⊢ (((𝐴 ⊆ 𝑈 ∧ ∀𝑘 ∈ 𝐴 𝐶 ∈ ℂ) ∧ (𝑈 ⊆ (ℤ_≥‘0) ∨ 𝑈 ∈ Fin)) → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐴, 𝐶, 0))
11	3, 7, 9, 10	syl21anc 837	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐴, 𝐶, 0))
12		ssun2 4128	. . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
13	12, 2	sseqtrrid 3974	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
14	13	sselda 3930	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝑘 ∈ 𝑈)
15	14, 5	syldan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
16	15	ralrimiva 3125	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
17		sumss2 15640	. . . 4 ⊢ (((𝐵 ⊆ 𝑈 ∧ ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ) ∧ (𝑈 ⊆ (ℤ_≥‘0) ∨ 𝑈 ∈ Fin)) → Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐵, 𝐶, 0))
18	13, 16, 9, 17	syl21anc 837	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐵, 𝐶, 0))
19	11, 18	oveq12d 7373	. 2 ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶) = (Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐴, 𝐶, 0) + Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐵, 𝐶, 0)))
20		0cn 11115	. . . 4 ⊢ 0 ∈ ℂ
21		ifcl 4522	. . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → if(𝑘 ∈ 𝐴, 𝐶, 0) ∈ ℂ)
22	5, 20, 21	sylancl 586	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → if(𝑘 ∈ 𝐴, 𝐶, 0) ∈ ℂ)
23		ifcl 4522	. . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → if(𝑘 ∈ 𝐵, 𝐶, 0) ∈ ℂ)
24	5, 20, 23	sylancl 586	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → if(𝑘 ∈ 𝐵, 𝐶, 0) ∈ ℂ)
25	8, 22, 24	fsumadd 15654	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = (Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐴, 𝐶, 0) + Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐵, 𝐶, 0)))
26	2	eleq2d 2819	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝑈 ↔ 𝑘 ∈ (𝐴 ∪ 𝐵)))
27		elun 4102	. . . . . 6 ⊢ (𝑘 ∈ (𝐴 ∪ 𝐵) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵))
28	26, 27	bitrdi 287	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝑈 ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵)))
29	28	biimpa 476	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵))
30		iftrue 4482	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐶, 0) = 𝐶)
31	30	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐶, 0) = 𝐶)
32		noel 4287	. . . . . . . . . . 11 ⊢ ¬ 𝑘 ∈ ∅
33		fsumsplit.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
34	33	eleq2d 2819	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑘 ∈ (𝐴 ∩ 𝐵) ↔ 𝑘 ∈ ∅))
35		elin 3914	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝐴 ∩ 𝐵) ↔ (𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵))
36	34, 35	bitr3di 286	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ ∅ ↔ (𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)))
37	32, 36	mtbii 326	. . . . . . . . . 10 ⊢ (𝜑 → ¬ (𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵))
38		imnan 399	. . . . . . . . . 10 ⊢ ((𝑘 ∈ 𝐴 → ¬ 𝑘 ∈ 𝐵) ↔ ¬ (𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵))
39	37, 38	sylibr 234	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → ¬ 𝑘 ∈ 𝐵))
40	39	imp 406	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 𝑘 ∈ 𝐵)
41	40	iffalsed 4487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐵, 𝐶, 0) = 0)
42	31, 41	oveq12d 7373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = (𝐶 + 0))
43	6	addridd 11324	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 + 0) = 𝐶)
44	42, 43	eqtrd 2768	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = 𝐶)
45	39	con2d 134	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝐵 → ¬ 𝑘 ∈ 𝐴))
46	45	imp 406	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ¬ 𝑘 ∈ 𝐴)
47	46	iffalsed 4487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝐴, 𝐶, 0) = 0)
48		iftrue 4482	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐵 → if(𝑘 ∈ 𝐵, 𝐶, 0) = 𝐶)
49	48	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝐵, 𝐶, 0) = 𝐶)
50	47, 49	oveq12d 7373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = (0 + 𝐶))
51	15	addlidd 11325	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (0 + 𝐶) = 𝐶)
52	50, 51	eqtrd 2768	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = 𝐶)
53	44, 52	jaodan 959	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵)) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = 𝐶)
54	29, 53	syldan 591	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = 𝐶)
55	54	sumeq2dv 15616	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = Σ𝑘 ∈ 𝑈 𝐶)
56	19, 25, 55	3eqtr2rd 2775	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2113 ∀wral 3048 ∪ cun 3896 ∩ cin 3897 ⊆ wss 3898 ∅c0 4282 ifcif 4476 ‘cfv 6489 (class class class)co 7355 Fincfn 8879 ℂcc 11015 0cc0 11017 + caddc 11020 ℤ_≥cuz 12742 Σcsu 15600

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-inf2 9542 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-pre-sup 11095

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-om 7806 df-1st 7930 df-2nd 7931 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-er 8631 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9337 df-oi 9407 df-card 9843 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-div 11786 df-nn 12137 df-2 12199 df-3 12200 df-n0 12393 df-z 12480 df-uz 12743 df-rp 12897 df-fz 13415 df-fzo 13562 df-seq 13916 df-exp 13976 df-hash 14245 df-cj 15013 df-re 15014 df-im 15015 df-sqrt 15149 df-abs 15150 df-clim 15402 df-sum 15601

This theorem is referenced by: fsumsplitf 15656 sumpr 15662 sumtp 15663 fsumm1 15665 fsum1p 15667 fsumsplitsnun 15669 fsum2dlem 15684 fsumless 15710 fsumabs 15715 fsumrlim 15725 fsumo1 15726 o1fsum 15727 cvgcmpce 15732 fsumiun 15735 incexclem 15750 incexc 15751 isumltss 15762 climcndslem1 15763 climcndslem2 15764 mertenslem1 15798 bitsinv1 16360 bitsinvp1 16367 sylow2a 19539 fsumcn 24808 ovolfiniun 25449 volfiniun 25495 uniioombllem3 25533 itgfsum 25775 dvmptfsum 25926 vieta1lem2 26266 mtest 26360 birthdaylem2 26909 fsumharmonic 26969 ftalem5 27034 chtprm 27110 chtdif 27115 perfectlem2 27188 lgsquadlem2 27339 dchrisumlem1 27447 dchrisumlem2 27448 rpvmasum2 27470 dchrisum0lem1b 27473 dchrisum0lem3 27477 pntrsumbnd2 27525 pntrlog2bndlem6 27541 pntpbnd2 27545 pntlemf 27563 axlowdimlem16 28956 axlowdimlem17 28957 vtxdgoddnumeven 29553 indsumin 32871 signsplypnf 34635 fsum2dsub 34692 hgt750lemd 34733 tgoldbachgtde 34745 sticksstones6 42317 sticksstones7 42318 sumcubes 42483 jm2.22 43152 jm2.23 43153 sumpair 45196 sumnnodd 45792 stoweidlem11 46171 stoweidlem26 46186 stoweidlem44 46204 sge0resplit 46566 sge0split 46569 fsumsplitsndif 47535 perfectALTVlem2 47884

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