fsumsplit - Metamath Proof Explorer

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Theorem fsumsplit 15725

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 4172	. . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
2		fsumsplit.2	. . . . 5 ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵))
3	1, 2	sseqtrrid 4033	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝑈)
4	3	sselda 3980	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝑈)
5		fsumsplit.4	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐶 ∈ ℂ)
6	4, 5	syldan 589	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
7	6	ralrimiva 3142	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ ℂ)
8		fsumsplit.3	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ Fin)
9	8	olcd 872	. . . 4 ⊢ (𝜑 → (𝑈 ⊆ (ℤ_≥‘0) ∨ 𝑈 ∈ Fin))
10		sumss2 15710	. . . 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 836	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐴, 𝐶, 0))
12		ssun2 4173	. . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
13	12, 2	sseqtrrid 4033	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
14	13	sselda 3980	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝑘 ∈ 𝑈)
15	14, 5	syldan 589	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
16	15	ralrimiva 3142	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
17		sumss2 15710	. . . 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 836	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐵, 𝐶, 0))
19	11, 18	oveq12d 7442	. 2 ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶) = (Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐴, 𝐶, 0) + Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐵, 𝐶, 0)))
20		0cn 11242	. . . 4 ⊢ 0 ∈ ℂ
21		ifcl 4575	. . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → if(𝑘 ∈ 𝐴, 𝐶, 0) ∈ ℂ)
22	5, 20, 21	sylancl 584	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → if(𝑘 ∈ 𝐴, 𝐶, 0) ∈ ℂ)
23		ifcl 4575	. . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → if(𝑘 ∈ 𝐵, 𝐶, 0) ∈ ℂ)
24	5, 20, 23	sylancl 584	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → if(𝑘 ∈ 𝐵, 𝐶, 0) ∈ ℂ)
25	8, 22, 24	fsumadd 15724	. 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 2814	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝑈 ↔ 𝑘 ∈ (𝐴 ∪ 𝐵)))
27		elun 4147	. . . . . 6 ⊢ (𝑘 ∈ (𝐴 ∪ 𝐵) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵))
28	26, 27	bitrdi 286	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝑈 ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵)))
29	28	biimpa 475	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵))
30		iftrue 4536	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐶, 0) = 𝐶)
31	30	adantl 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐶, 0) = 𝐶)
32		noel 4332	. . . . . . . . . . 11 ⊢ ¬ 𝑘 ∈ ∅
33		fsumsplit.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
34	33	eleq2d 2814	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑘 ∈ (𝐴 ∩ 𝐵) ↔ 𝑘 ∈ ∅))
35		elin 3963	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝐴 ∩ 𝐵) ↔ (𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵))
36	34, 35	bitr3di 285	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ ∅ ↔ (𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)))
37	32, 36	mtbii 325	. . . . . . . . . 10 ⊢ (𝜑 → ¬ (𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵))
38		imnan 398	. . . . . . . . . 10 ⊢ ((𝑘 ∈ 𝐴 → ¬ 𝑘 ∈ 𝐵) ↔ ¬ (𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵))
39	37, 38	sylibr 233	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → ¬ 𝑘 ∈ 𝐵))
40	39	imp 405	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 𝑘 ∈ 𝐵)
41	40	iffalsed 4541	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐵, 𝐶, 0) = 0)
42	31, 41	oveq12d 7442	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = (𝐶 + 0))
43	6	addridd 11450	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 + 0) = 𝐶)
44	42, 43	eqtrd 2767	. . . . 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 405	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ¬ 𝑘 ∈ 𝐴)
47	46	iffalsed 4541	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝐴, 𝐶, 0) = 0)
48		iftrue 4536	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐵 → if(𝑘 ∈ 𝐵, 𝐶, 0) = 𝐶)
49	48	adantl 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝐵, 𝐶, 0) = 𝐶)
50	47, 49	oveq12d 7442	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = (0 + 𝐶))
51	15	addlidd 11451	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (0 + 𝐶) = 𝐶)
52	50, 51	eqtrd 2767	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = 𝐶)
53	44, 52	jaodan 955	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵)) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = 𝐶)
54	29, 53	syldan 589	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = 𝐶)
55	54	sumeq2dv 15687	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = Σ𝑘 ∈ 𝑈 𝐶)
56	19, 25, 55	3eqtr2rd 2774	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ∀wral 3057 ∪ cun 3945 ∩ cin 3946 ⊆ wss 3947 ∅c0 4324 ifcif 4530 ‘cfv 6551 (class class class)co 7424 Fincfn 8968 ℂcc 11142 0cc0 11144 + caddc 11147 ℤ_≥cuz 12858 Σcsu 15670

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-inf2 9670 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 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 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-sup 9471 df-oi 9539 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-n0 12509 df-z 12595 df-uz 12859 df-rp 13013 df-fz 13523 df-fzo 13666 df-seq 14005 df-exp 14065 df-hash 14328 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-clim 15470 df-sum 15671

This theorem is referenced by: fsumsplitf 15726 sumpr 15732 sumtp 15733 fsumm1 15735 fsum1p 15737 fsumsplitsnun 15739 fsum2dlem 15754 fsumless 15780 fsumabs 15785 fsumrlim 15795 fsumo1 15796 o1fsum 15797 cvgcmpce 15802 fsumiun 15805 incexclem 15820 incexc 15821 isumltss 15832 climcndslem1 15833 climcndslem2 15834 mertenslem1 15868 bitsinv1 16422 bitsinvp1 16429 sylow2a 19579 fsumcn 24806 ovolfiniun 25448 volfiniun 25494 uniioombllem3 25532 itgfsum 25774 dvmptfsum 25925 vieta1lem2 26264 mtest 26358 birthdaylem2 26902 fsumharmonic 26962 ftalem5 27027 chtprm 27103 chtdif 27108 perfectlem2 27181 lgsquadlem2 27332 dchrisumlem1 27440 dchrisumlem2 27441 rpvmasum2 27463 dchrisum0lem1b 27466 dchrisum0lem3 27470 pntrsumbnd2 27518 pntrlog2bndlem6 27534 pntpbnd2 27538 pntlemf 27556 axlowdimlem16 28786 axlowdimlem17 28787 vtxdgoddnumeven 29385 indsumin 33646 signsplypnf 34187 fsum2dsub 34244 hgt750lemd 34285 tgoldbachgtde 34297 sticksstones6 41627 sticksstones7 41628 sumcubes 41876 jm2.22 42419 jm2.23 42420 sumpair 44400 sumnnodd 45020 stoweidlem11 45401 stoweidlem26 45416 stoweidlem44 45434 sge0resplit 45796 sge0split 45799 fsumsplitsndif 46715 perfectALTVlem2 47064

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