fsumsplit - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > fsumsplit		Structured version Visualization version GIF version

Theorem fsumsplit 15827

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 4124	. . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
2		fsumsplit.2	. . . . 5 ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵))
3	1, 2	sseqtrrid 3974	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝑈)
4	3	sselda 3931	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝑈)
5		fsumsplit.4	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐶 ∈ ℂ)
6	4, 5	syldan 603	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
7	6	ralrimiva 3154	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ ℂ)
8		fsumsplit.3	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ Fin)
9	8	olcd 888	. . . 4 ⊢ (𝜑 → (𝑈 ⊆ (ℤ_≥‘0) ∨ 𝑈 ∈ Fin))
10		sumss2 15812	. . . 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 851	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐴, 𝐶, 0))
12		ssun2 4125	. . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
13	12, 2	sseqtrrid 3974	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
14	13	sselda 3931	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝑘 ∈ 𝑈)
15	14, 5	syldan 603	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
16	15	ralrimiva 3154	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
17		sumss2 15812	. . . 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 851	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐵, 𝐶, 0))
19	11, 18	oveq12d 7431	. 2 ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶) = (Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐴, 𝐶, 0) + Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐵, 𝐶, 0)))
20		0cn 11222	. . . 4 ⊢ 0 ∈ ℂ
21		ifcl 4528	. . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → if(𝑘 ∈ 𝐴, 𝐶, 0) ∈ ℂ)
22	5, 20, 21	sylancl 598	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → if(𝑘 ∈ 𝐴, 𝐶, 0) ∈ ℂ)
23		ifcl 4528	. . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → if(𝑘 ∈ 𝐵, 𝐶, 0) ∈ ℂ)
24	5, 20, 23	sylancl 598	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → if(𝑘 ∈ 𝐵, 𝐶, 0) ∈ ℂ)
25	8, 22, 24	fsumadd 15826	. 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 2846	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝑈 ↔ 𝑘 ∈ (𝐴 ∪ 𝐵)))
27		elun 4100	. . . . . 6 ⊢ (𝑘 ∈ (𝐴 ∪ 𝐵) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵))
28	26, 27	bitrdi 290	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝑈 ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵)))
29	28	biimpa 482	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵))
30		iftrue 4488	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐶, 0) = 𝐶)
31	30	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐶, 0) = 𝐶)
32		noel 4284	. . . . . . . . . . 11 ⊢ ¬ 𝑘 ∈ ∅
33		fsumsplit.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
34	33	eleq2d 2846	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑘 ∈ (𝐴 ∩ 𝐵) ↔ 𝑘 ∈ ∅))
35		elin 3915	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝐴 ∩ 𝐵) ↔ (𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵))
36	34, 35	bitr3di 289	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ ∅ ↔ (𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)))
37	32, 36	mtbii 329	. . . . . . . . . 10 ⊢ (𝜑 → ¬ (𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵))
38		imnan 405	. . . . . . . . . 10 ⊢ ((𝑘 ∈ 𝐴 → ¬ 𝑘 ∈ 𝐵) ↔ ¬ (𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵))
39	37, 38	sylibr 237	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → ¬ 𝑘 ∈ 𝐵))
40	39	imp 412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 𝑘 ∈ 𝐵)
41	40	iffalsed 4493	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐵, 𝐶, 0) = 0)
42	31, 41	oveq12d 7431	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = (𝐶 + 0))
43	6	addridd 11434	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 + 0) = 𝐶)
44	42, 43	eqtrd 2795	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = 𝐶)
45	39	con2d 135	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝐵 → ¬ 𝑘 ∈ 𝐴))
46	45	imp 412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ¬ 𝑘 ∈ 𝐴)
47	46	iffalsed 4493	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝐴, 𝐶, 0) = 0)
48		iftrue 4488	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐵 → if(𝑘 ∈ 𝐵, 𝐶, 0) = 𝐶)
49	48	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝐵, 𝐶, 0) = 𝐶)
50	47, 49	oveq12d 7431	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = (0 + 𝐶))
51	15	addlidd 11435	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (0 + 𝐶) = 𝐶)
52	50, 51	eqtrd 2795	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = 𝐶)
53	44, 52	jaodan 972	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵)) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = 𝐶)
54	29, 53	syldan 603	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = 𝐶)
55	54	sumeq2dv 15789	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = Σ𝑘 ∈ 𝑈 𝐶)
56	19, 25, 55	3eqtr2rd 2802	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2145 ∀wral 3076 ∪ cun 3897 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 ifcif 4482 ‘cfv 6533 (class class class)co 7413 Fincfn 8952 ℂcc 11122 0cc0 11124 + caddc 11127 ℤ_≥cuz 12887 Σcsu 15773

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-inf2 9620 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-oi 9482 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-n0 12529 df-z 12616 df-uz 12888 df-rp 13043 df-fz 13562 df-fzo 13710 df-seq 14066 df-exp 14126 df-hash 14395 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-clim 15575 df-sum 15774

This theorem is used by: fsumsplitf 15828 sumpr 15834 sumtp 15835 fsumm1 15837 fsum1p 15839 fsumsplitsnun 15841 fsum2dlem 15856 fsumless 15883 fsumabs 15888 fsumrlim 15898 fsumo1 15899 o1fsum 15900 cvgcmpce 15905 fsumiun 15908 incexclem 15925 incexc 15926 isumltss 15937 climcndslem1 15938 climcndslem2 15939 mertenslem1 15973 bitsinv1 16532 bitsinvp1 16539 sylow2a 19746 fsumcn 25098 ovolfiniun 25729 volfiniun 25775 uniioombllem3 25813 itgfsum 26054 dvmptfsum 26202 vieta1lem2 26543 mtest 26640 birthdaylem2 27189 fsumharmonic 27248 ftalem5 27313 chtprm 27389 chtdif 27394 perfectlem2 27466 lgsquadlem2 27617 dchrisumlem1 27725 dchrisumlem2 27726 rpvmasum2 27748 dchrisum0lem1b 27751 dchrisum0lem3 27755 pntrsumbnd2 27803 pntrlog2bndlem6 27819 pntpbnd2 27823 pntlemf 27841 axlowdimlem16 29414 axlowdimlem17 29415 vtxdgoddnumeven 30013 indsumin 33307 signsplypnf 35058 fsum2dsub 35115 hgt750lemd 35156 tgoldbachgtde 35168 sticksstones6 43017 sticksstones7 43018 sumcubes 43188 jm2.22 43836 jm2.23 43837 sumpair 45869 sumnnodd 46460 stoweidlem11 46839 stoweidlem26 46854 stoweidlem44 46872 sge0resplit 47234 sge0split 47237 fsumsplitsndif 48269 perfectALTVlem2 48638

W3C validator