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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 3738 | . . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
2 | fsumsplit.2 | . . . . 5 ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵)) | |
3 | 1, 2 | syl5sseqr 3617 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝑈) |
4 | 3 | sselda 3568 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝑈) |
5 | fsumsplit.4 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐶 ∈ ℂ) | |
6 | 4, 5 | syldan 486 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
7 | 6 | ralrimiva 2949 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ ℂ) |
8 | fsumsplit.3 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ Fin) | |
9 | 8 | olcd 407 | . . . 4 ⊢ (𝜑 → (𝑈 ⊆ (ℤ≥‘0) ∨ 𝑈 ∈ Fin)) |
10 | sumss2 14304 | . . . 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 1317 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐴, 𝐶, 0)) |
12 | ssun2 3739 | . . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
13 | 12, 2 | syl5sseqr 3617 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝑈) |
14 | 13 | sselda 3568 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝑘 ∈ 𝑈) |
15 | 14, 5 | syldan 486 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ) |
16 | 15 | ralrimiva 2949 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ) |
17 | sumss2 14304 | . . . 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 1317 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐵, 𝐶, 0)) |
19 | 11, 18 | oveq12d 6567 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶) = (Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐴, 𝐶, 0) + Σ𝑘 ∈ 𝑈 if(𝑘 ∈ 𝐵, 𝐶, 0))) |
20 | 0cn 9911 | . . . 4 ⊢ 0 ∈ ℂ | |
21 | ifcl 4080 | . . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → if(𝑘 ∈ 𝐴, 𝐶, 0) ∈ ℂ) | |
22 | 5, 20, 21 | sylancl 693 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → if(𝑘 ∈ 𝐴, 𝐶, 0) ∈ ℂ) |
23 | ifcl 4080 | . . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℂ) → if(𝑘 ∈ 𝐵, 𝐶, 0) ∈ ℂ) | |
24 | 5, 20, 23 | sylancl 693 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → if(𝑘 ∈ 𝐵, 𝐶, 0) ∈ ℂ) |
25 | 8, 22, 24 | fsumadd 14317 | . 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 2673 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝑈 ↔ 𝑘 ∈ (𝐴 ∪ 𝐵))) |
27 | elun 3715 | . . . . . 6 ⊢ (𝑘 ∈ (𝐴 ∪ 𝐵) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵)) | |
28 | 26, 27 | syl6bb 275 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝑈 ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵))) |
29 | 28 | biimpa 500 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵)) |
30 | iftrue 4042 | . . . . . . . 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 3878 | . . . . . . . . . . 11 ⊢ ¬ 𝑘 ∈ ∅ | |
33 | elin 3758 | . . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝐴 ∩ 𝐵) ↔ (𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) | |
34 | fsumsplit.1 | . . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) | |
35 | 34 | eleq2d 2673 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝑘 ∈ (𝐴 ∩ 𝐵) ↔ 𝑘 ∈ ∅)) |
36 | 33, 35 | syl5rbbr 274 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ ∅ ↔ (𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵))) |
37 | 32, 36 | mtbii 315 | . . . . . . . . . 10 ⊢ (𝜑 → ¬ (𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) |
38 | imnan 437 | . . . . . . . . . 10 ⊢ ((𝑘 ∈ 𝐴 → ¬ 𝑘 ∈ 𝐵) ↔ ¬ (𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) | |
39 | 37, 38 | sylibr 223 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → ¬ 𝑘 ∈ 𝐵)) |
40 | 39 | imp 444 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 𝑘 ∈ 𝐵) |
41 | 40 | iffalsed 4047 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐵, 𝐶, 0) = 0) |
42 | 31, 41 | oveq12d 6567 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = (𝐶 + 0)) |
43 | 6 | addid1d 10115 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 + 0) = 𝐶) |
44 | 42, 43 | eqtrd 2644 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = 𝐶) |
45 | 39 | con2d 128 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝐵 → ¬ 𝑘 ∈ 𝐴)) |
46 | 45 | imp 444 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ¬ 𝑘 ∈ 𝐴) |
47 | 46 | iffalsed 4047 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝐴, 𝐶, 0) = 0) |
48 | iftrue 4042 | . . . . . . . 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 6567 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = (0 + 𝐶)) |
51 | 15 | addid2d 10116 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (0 + 𝐶) = 𝐶) |
52 | 50, 51 | eqtrd 2644 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = 𝐶) |
53 | 44, 52 | jaodan 822 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵)) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = 𝐶) |
54 | 29, 53 | syldan 486 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = 𝐶) |
55 | 54 | sumeq2dv 14281 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 (if(𝑘 ∈ 𝐴, 𝐶, 0) + if(𝑘 ∈ 𝐵, 𝐶, 0)) = Σ𝑘 ∈ 𝑈 𝐶) |
56 | 19, 25, 55 | 3eqtr2rd 2651 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ifcif 4036 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 ℂcc 9813 0cc0 9815 + caddc 9818 ℤ≥cuz 11563 Σcsu 14264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 |
This theorem is referenced by: sumpr 14321 sumtp 14322 fsumm1 14324 fsum1p 14326 fsumsplitsnun 14328 fsum2dlem 14343 fsumless 14369 fsumabs 14374 fsumrlim 14384 fsumo1 14385 o1fsum 14386 cvgcmpce 14391 fsumiun 14394 incexclem 14407 incexc 14408 isumltss 14419 climcndslem1 14420 climcndslem2 14421 mertenslem1 14455 bitsinv1 15002 bitsinvp1 15009 sylow2a 17857 fsumcn 22481 ovolfiniun 23076 volfiniun 23122 uniioombllem3 23159 itgfsum 23399 dvmptfsum 23542 vieta1lem2 23870 mtest 23962 birthdaylem2 24479 fsumharmonic 24538 ftalem5 24603 chtprm 24679 chtdif 24684 perfectlem2 24755 lgsquadlem2 24906 dchrisumlem1 24978 dchrisumlem2 24979 rpvmasum2 25001 dchrisum0lem1b 25004 dchrisum0lem3 25008 pntrsumbnd2 25056 pntrlog2bndlem6 25072 pntpbnd2 25076 pntlemf 25094 axlowdimlem16 25637 axlowdimlem17 25638 signsplypnf 29953 jm2.22 36580 jm2.23 36581 sumpair 38217 fsumsplitf 38634 sumnnodd 38697 stoweidlem11 38904 stoweidlem26 38919 stoweidlem44 38937 sge0resplit 39299 sge0split 39302 perfectALTVlem2 40165 fsumsplitsndif 40372 |
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