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| Mirrors > Home > MPE Home > Th. List > Mathboxes > esumsplit | Structured version Visualization version GIF version | ||
| Description: Split an extended sum into two parts. (Contributed by Thierry Arnoux, 9-May-2017.) | 
| Ref | Expression | 
|---|---|
| esumsplit.1 | ⊢ Ⅎ𝑘𝜑 | 
| esumsplit.2 | ⊢ Ⅎ𝑘𝐴 | 
| esumsplit.3 | ⊢ Ⅎ𝑘𝐵 | 
| esumsplit.4 | ⊢ (𝜑 → 𝐴 ∈ V) | 
| esumsplit.5 | ⊢ (𝜑 → 𝐵 ∈ V) | 
| esumsplit.6 | ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) | 
| esumsplit.7 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞)) | 
| esumsplit.8 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞)) | 
| Ref | Expression | 
|---|---|
| esumsplit | ⊢ (𝜑 → Σ*𝑘 ∈ (𝐴 ∪ 𝐵)𝐶 = (Σ*𝑘 ∈ 𝐴𝐶 +𝑒 Σ*𝑘 ∈ 𝐵𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | esumsplit.1 | . 2 ⊢ Ⅎ𝑘𝜑 | |
| 2 | esumsplit.2 | . . 3 ⊢ Ⅎ𝑘𝐴 | |
| 3 | esumsplit.3 | . . 3 ⊢ Ⅎ𝑘𝐵 | |
| 4 | 2, 3 | nfun 4169 | . 2 ⊢ Ⅎ𝑘(𝐴 ∪ 𝐵) | 
| 5 | esumsplit.4 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) | |
| 6 | esumsplit.5 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) | |
| 7 | unexg 7764 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V) | |
| 8 | 5, 6, 7 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V) | 
| 9 | elun 4152 | . . 3 ⊢ (𝑘 ∈ (𝐴 ∪ 𝐵) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵)) | |
| 10 | esumsplit.7 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞)) | |
| 11 | esumsplit.8 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞)) | |
| 12 | 10, 11 | jaodan 959 | . . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵)) → 𝐶 ∈ (0[,]+∞)) | 
| 13 | 9, 12 | sylan2b 594 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ 𝐵)) → 𝐶 ∈ (0[,]+∞)) | 
| 14 | xrge0base 33017 | . . 3 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
| 15 | xrge0plusg 33019 | . . 3 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
| 16 | xrge0cmn 21427 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
| 17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) | 
| 18 | xrge0tmd 33945 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopMnd | |
| 19 | 18 | a1i 11 | . . 3 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopMnd) | 
| 20 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑘(0[,]+∞) | |
| 21 | eqid 2736 | . . . 4 ⊢ (𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) = (𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) | |
| 22 | 1, 4, 20, 13, 21 | fmptdF 32667 | . . 3 ⊢ (𝜑 → (𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶):(𝐴 ∪ 𝐵)⟶(0[,]+∞)) | 
| 23 | 1, 2, 5, 10 | esumel 34049 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))) | 
| 24 | ssun1 4177 | . . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
| 25 | 4, 2 | resmptf 6056 | . . . . . 6 ⊢ (𝐴 ⊆ (𝐴 ∪ 𝐵) → ((𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) ↾ 𝐴) = (𝑘 ∈ 𝐴 ↦ 𝐶)) | 
| 26 | 24, 25 | mp1i 13 | . . . . 5 ⊢ (𝜑 → ((𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) ↾ 𝐴) = (𝑘 ∈ 𝐴 ↦ 𝐶)) | 
| 27 | 26 | oveq2d 7448 | . . . 4 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) ↾ 𝐴)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))) | 
| 28 | 23, 27 | eleqtrrd 2843 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) ↾ 𝐴))) | 
| 29 | 1, 3, 6, 11 | esumel 34049 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐵𝐶 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐶))) | 
| 30 | ssun2 4178 | . . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
| 31 | 4, 3 | resmptf 6056 | . . . . . 6 ⊢ (𝐵 ⊆ (𝐴 ∪ 𝐵) → ((𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) ↾ 𝐵) = (𝑘 ∈ 𝐵 ↦ 𝐶)) | 
| 32 | 30, 31 | mp1i 13 | . . . . 5 ⊢ (𝜑 → ((𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) ↾ 𝐵) = (𝑘 ∈ 𝐵 ↦ 𝐶)) | 
| 33 | 32 | oveq2d 7448 | . . . 4 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) ↾ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐶))) | 
| 34 | 29, 33 | eleqtrrd 2843 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐵𝐶 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) ↾ 𝐵))) | 
| 35 | esumsplit.6 | . . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) | |
| 36 | eqidd 2737 | . . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) = (𝐴 ∪ 𝐵)) | |
| 37 | 14, 15, 17, 19, 8, 22, 28, 34, 35, 36 | tsmssplit 24161 | . 2 ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴𝐶 +𝑒 Σ*𝑘 ∈ 𝐵𝐶) ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶))) | 
| 38 | 1, 4, 8, 13, 37 | esumid 34046 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ (𝐴 ∪ 𝐵)𝐶 = (Σ*𝑘 ∈ 𝐴𝐶 +𝑒 Σ*𝑘 ∈ 𝐵𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1539 Ⅎwnf 1782 ∈ wcel 2107 Ⅎwnfc 2889 Vcvv 3479 ∪ cun 3948 ∩ cin 3949 ⊆ wss 3950 ∅c0 4332 ↦ cmpt 5224 ↾ cres 5686 (class class class)co 7432 0cc0 11156 +∞cpnf 11293 +𝑒 cxad 13153 [,]cicc 13391 ↾s cress 17275 ℝ*𝑠cxrs 17546 CMndccmn 19799 TopMndctmd 24079 tsums ctsu 24135 Σ*cesum 34029 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 ax-mulf 11236 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-exp 14104 df-fac 14314 df-bc 14343 df-hash 14371 df-shft 15107 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-limsup 15508 df-clim 15525 df-rlim 15526 df-sum 15724 df-ef 16104 df-sin 16106 df-cos 16107 df-pi 16109 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-ordt 17547 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-ps 18612 df-tsr 18613 df-plusf 18653 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-mulg 19087 df-subg 19142 df-cntz 19336 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-cring 20234 df-subrng 20547 df-subrg 20571 df-abv 20811 df-lmod 20861 df-scaf 20862 df-sra 21173 df-rgmod 21174 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-cnfld 21366 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-lp 23145 df-perf 23146 df-cn 23236 df-cnp 23237 df-haus 23324 df-tx 23571 df-hmeo 23764 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-tmd 24081 df-tgp 24082 df-tsms 24136 df-trg 24169 df-xms 24331 df-ms 24332 df-tms 24333 df-nm 24596 df-ngp 24597 df-nrg 24599 df-nlm 24600 df-ii 24904 df-cncf 24905 df-limc 25902 df-dv 25903 df-log 26599 df-esum 34030 | 
| This theorem is referenced by: esummono 34056 esumpad 34057 esumpr 34068 esumrnmpt2 34070 esumfzf 34071 esumpmono 34081 hasheuni 34087 esum2dlem 34094 measvuni 34216 ddemeas 34238 carsgclctunlem1 34320 | 
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