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Mirrors > Home > MPE Home > Th. List > fsumsplit | Structured version Visualization version GIF version |
Description: Split a sum into two parts. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 22-Apr-2014.) |
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) |
Ref | Expression |
---|---|
fsumsplit | ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun1 4145 | . . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
2 | fsumsplit.2 | . . . . 5 ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵)) | |
3 | 1, 2 | sseqtrrid 4017 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝑈) |
4 | 3 | sselda 3964 | . . . . . 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 3179 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ ℂ) |
8 | fsumsplit.3 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ Fin) | |
9 | 8 | olcd 870 | . . . 4 ⊢ (𝜑 → (𝑈 ⊆ (ℤ≥‘0) ∨ 𝑈 ∈ Fin)) |
10 | sumss2 15071 | . . . 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 833 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐴, 𝐶, 0)) |
12 | ssun2 4146 | . . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
13 | 12, 2 | sseqtrrid 4017 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝑈) |
14 | 13 | sselda 3964 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝑘 ∈ 𝑈) |
15 | 14, 5 | syldan 591 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ) |
16 | 15 | ralrimiva 3179 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ) |
17 | sumss2 15071 | . . . 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 833 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐵, 𝐶, 0)) |
19 | 11, 18 | oveq12d 7163 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶) = (Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐴, 𝐶, 0) + Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐵, 𝐶, 0))) |
20 | 0cn 10621 | . . . 4 ⊢ 0 ∈ ℂ | |
21 | ifcl 4507 | . . . 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 4507 | . . . 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 15084 | . 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 2895 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝑈 ↔ 𝑘 ∈ (𝐴 ∪ 𝐵))) |
27 | elun 4122 | . . . . . 6 ⊢ (𝑘 ∈ (𝐴 ∪ 𝐵) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵)) | |
28 | 26, 27 | syl6bb 288 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝑈 ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵))) |
29 | 28 | biimpa 477 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵)) |
30 | iftrue 4469 | . . . . . . . 8 ⊢ (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐶, 0) = 𝐶) | |
31 | 30 | adantl 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐶, 0) = 𝐶) |
32 | noel 4293 | . . . . . . . . . . 11 ⊢ ¬ 𝑘 ∈ ∅ | |
33 | elin 4166 | . . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝐴 ∩ 𝐵) ↔ (𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) | |
34 | fsumsplit.1 | . . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) | |
35 | 34 | eleq2d 2895 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝑘 ∈ (𝐴 ∩ 𝐵) ↔ 𝑘 ∈ ∅)) |
36 | 33, 35 | syl5rbbr 287 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ ∅ ↔ (𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵))) |
37 | 32, 36 | mtbii 327 | . . . . . . . . . 10 ⊢ (𝜑 → ¬ (𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) |
38 | imnan 400 | . . . . . . . . . 10 ⊢ ((𝑘 ∈ 𝐴 → ¬ 𝑘 ∈ 𝐵) ↔ ¬ (𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) | |
39 | 37, 38 | sylibr 235 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → ¬ 𝑘 ∈ 𝐵)) |
40 | 39 | imp 407 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 𝑘 ∈ 𝐵) |
41 | 40 | iffalsed 4474 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐵, 𝐶, 0) = 0) |
42 | 31, 41 | oveq12d 7163 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = (𝐶 + 0)) |
43 | 6 | addid1d 10828 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 + 0) = 𝐶) |
44 | 42, 43 | eqtrd 2853 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = 𝐶) |
45 | 39 | con2d 136 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝐵 → ¬ 𝑘 ∈ 𝐴)) |
46 | 45 | imp 407 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ¬ 𝑘 ∈ 𝐴) |
47 | 46 | iffalsed 4474 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝐴, 𝐶, 0) = 0) |
48 | iftrue 4469 | . . . . . . . 8 ⊢ (𝑘 ∈ 𝐵 → if(𝑘 ∈ 𝐵, 𝐶, 0) = 𝐶) | |
49 | 48 | adantl 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝐵, 𝐶, 0) = 𝐶) |
50 | 47, 49 | oveq12d 7163 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = (0 + 𝐶)) |
51 | 15 | addid2d 10829 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (0 + 𝐶) = 𝐶) |
52 | 50, 51 | eqtrd 2853 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = 𝐶) |
53 | 44, 52 | jaodan 951 | . . . 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 15048 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = Σ𝑘 ∈ 𝑈 𝐶) |
56 | 19, 25, 55 | 3eqtr2rd 2860 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∪ cun 3931 ∩ cin 3932 ⊆ wss 3933 ∅c0 4288 ifcif 4463 ‘cfv 6348 (class class class)co 7145 Fincfn 8497 ℂcc 10523 0cc0 10525 + caddc 10528 ℤ≥cuz 12231 Σcsu 15030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-sum 15031 |
This theorem is referenced by: fsumsplitf 15086 sumpr 15091 sumtp 15092 fsumm1 15094 fsum1p 15096 fsumsplitsnun 15098 fsum2dlem 15113 fsumless 15139 fsumabs 15144 fsumrlim 15154 fsumo1 15155 o1fsum 15156 cvgcmpce 15161 fsumiun 15164 incexclem 15179 incexc 15180 isumltss 15191 climcndslem1 15192 climcndslem2 15193 mertenslem1 15228 bitsinv1 15779 bitsinvp1 15786 sylow2a 18673 fsumcn 23405 ovolfiniun 24029 volfiniun 24075 uniioombllem3 24113 itgfsum 24354 dvmptfsum 24499 vieta1lem2 24827 mtest 24919 birthdaylem2 25457 fsumharmonic 25516 ftalem5 25581 chtprm 25657 chtdif 25662 perfectlem2 25733 lgsquadlem2 25884 dchrisumlem1 25992 dchrisumlem2 25993 rpvmasum2 26015 dchrisum0lem1b 26018 dchrisum0lem3 26022 pntrsumbnd2 26070 pntrlog2bndlem6 26086 pntpbnd2 26090 pntlemf 26108 axlowdimlem16 26670 axlowdimlem17 26671 vtxdgoddnumeven 27262 indsumin 31180 signsplypnf 31719 fsum2dsub 31777 hgt750lemd 31818 tgoldbachgtde 31830 jm2.22 39470 jm2.23 39471 sumpair 41169 sumnnodd 41787 stoweidlem11 42173 stoweidlem26 42188 stoweidlem44 42206 sge0resplit 42565 sge0split 42568 fsumsplitsndif 43410 perfectALTVlem2 43764 |
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