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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 4133 | . . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
2 | fsumsplit.2 | . . . . 5 ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵)) | |
3 | 1, 2 | sseqtrrid 3998 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝑈) |
4 | 3 | sselda 3945 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝑈) |
5 | fsumsplit.4 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐶 ∈ ℂ) | |
6 | 4, 5 | syldan 592 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
7 | 6 | ralrimiva 3140 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ ℂ) |
8 | fsumsplit.3 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ Fin) | |
9 | 8 | olcd 873 | . . . 4 ⊢ (𝜑 → (𝑈 ⊆ (ℤ≥‘0) ∨ 𝑈 ∈ Fin)) |
10 | sumss2 15616 | . . . 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 4134 | . . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
13 | 12, 2 | sseqtrrid 3998 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝑈) |
14 | 13 | sselda 3945 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝑘 ∈ 𝑈) |
15 | 14, 5 | syldan 592 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ) |
16 | 15 | ralrimiva 3140 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ) |
17 | sumss2 15616 | . . . 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 7376 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶) = (Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐴, 𝐶, 0) + Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐵, 𝐶, 0))) |
20 | 0cn 11152 | . . . 4 ⊢ 0 ∈ ℂ | |
21 | ifcl 4532 | . . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → if(𝑘 ∈ 𝐴, 𝐶, 0) ∈ ℂ) | |
22 | 5, 20, 21 | sylancl 587 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → if(𝑘 ∈ 𝐴, 𝐶, 0) ∈ ℂ) |
23 | ifcl 4532 | . . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → if(𝑘 ∈ 𝐵, 𝐶, 0) ∈ ℂ) | |
24 | 5, 20, 23 | sylancl 587 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → if(𝑘 ∈ 𝐵, 𝐶, 0) ∈ ℂ) |
25 | 8, 22, 24 | fsumadd 15630 | . 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 2820 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝑈 ↔ 𝑘 ∈ (𝐴 ∪ 𝐵))) |
27 | elun 4109 | . . . . . 6 ⊢ (𝑘 ∈ (𝐴 ∪ 𝐵) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵)) | |
28 | 26, 27 | bitrdi 287 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝑈 ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵))) |
29 | 28 | biimpa 478 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵)) |
30 | iftrue 4493 | . . . . . . . 8 ⊢ (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐶, 0) = 𝐶) | |
31 | 30 | adantl 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐶, 0) = 𝐶) |
32 | noel 4291 | . . . . . . . . . . 11 ⊢ ¬ 𝑘 ∈ ∅ | |
33 | fsumsplit.1 | . . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) | |
34 | 33 | eleq2d 2820 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝑘 ∈ (𝐴 ∩ 𝐵) ↔ 𝑘 ∈ ∅)) |
35 | elin 3927 | . . . . . . . . . . . 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 401 | . . . . . . . . . 10 ⊢ ((𝑘 ∈ 𝐴 → ¬ 𝑘 ∈ 𝐵) ↔ ¬ (𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) | |
39 | 37, 38 | sylibr 233 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → ¬ 𝑘 ∈ 𝐵)) |
40 | 39 | imp 408 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 𝑘 ∈ 𝐵) |
41 | 40 | iffalsed 4498 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐵, 𝐶, 0) = 0) |
42 | 31, 41 | oveq12d 7376 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = (𝐶 + 0)) |
43 | 6 | addid1d 11360 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 + 0) = 𝐶) |
44 | 42, 43 | eqtrd 2773 | . . . . 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 408 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ¬ 𝑘 ∈ 𝐴) |
47 | 46 | iffalsed 4498 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝐴, 𝐶, 0) = 0) |
48 | iftrue 4493 | . . . . . . . 8 ⊢ (𝑘 ∈ 𝐵 → if(𝑘 ∈ 𝐵, 𝐶, 0) = 𝐶) | |
49 | 48 | adantl 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝐵, 𝐶, 0) = 𝐶) |
50 | 47, 49 | oveq12d 7376 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = (0 + 𝐶)) |
51 | 15 | addid2d 11361 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (0 + 𝐶) = 𝐶) |
52 | 50, 51 | eqtrd 2773 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = 𝐶) |
53 | 44, 52 | jaodan 957 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵)) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = 𝐶) |
54 | 29, 53 | syldan 592 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = 𝐶) |
55 | 54 | sumeq2dv 15593 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = Σ𝑘 ∈ 𝑈 𝐶) |
56 | 19, 25, 55 | 3eqtr2rd 2780 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ∪ cun 3909 ∩ cin 3910 ⊆ wss 3911 ∅c0 4283 ifcif 4487 ‘cfv 6497 (class class class)co 7358 Fincfn 8886 ℂcc 11054 0cc0 11056 + caddc 11059 ℤ≥cuz 12768 Σcsu 15576 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-fz 13431 df-fzo 13574 df-seq 13913 df-exp 13974 df-hash 14237 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-clim 15376 df-sum 15577 |
This theorem is referenced by: fsumsplitf 15632 sumpr 15638 sumtp 15639 fsumm1 15641 fsum1p 15643 fsumsplitsnun 15645 fsum2dlem 15660 fsumless 15686 fsumabs 15691 fsumrlim 15701 fsumo1 15702 o1fsum 15703 cvgcmpce 15708 fsumiun 15711 incexclem 15726 incexc 15727 isumltss 15738 climcndslem1 15739 climcndslem2 15740 mertenslem1 15774 bitsinv1 16327 bitsinvp1 16334 sylow2a 19406 fsumcn 24249 ovolfiniun 24881 volfiniun 24927 uniioombllem3 24965 itgfsum 25207 dvmptfsum 25355 vieta1lem2 25687 mtest 25779 birthdaylem2 26318 fsumharmonic 26377 ftalem5 26442 chtprm 26518 chtdif 26523 perfectlem2 26594 lgsquadlem2 26745 dchrisumlem1 26853 dchrisumlem2 26854 rpvmasum2 26876 dchrisum0lem1b 26879 dchrisum0lem3 26883 pntrsumbnd2 26931 pntrlog2bndlem6 26947 pntpbnd2 26951 pntlemf 26969 axlowdimlem16 27948 axlowdimlem17 27949 vtxdgoddnumeven 28543 indsumin 32678 signsplypnf 33219 fsum2dsub 33277 hgt750lemd 33318 tgoldbachgtde 33330 sticksstones6 40605 sticksstones7 40606 jm2.22 41362 jm2.23 41363 sumpair 43328 sumnnodd 43957 stoweidlem11 44338 stoweidlem26 44353 stoweidlem44 44371 sge0resplit 44733 sge0split 44736 fsumsplitsndif 45651 perfectALTVlem2 46000 |
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