Mathbox for Thierry Arnoux |
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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 4065 | . 2 ⊢ Ⅎ𝑘(𝐴 ∪ 𝐵) |
5 | esumsplit.4 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) | |
6 | esumsplit.5 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) | |
7 | unexg 7512 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V) | |
8 | 5, 6, 7 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V) |
9 | elun 4049 | . . 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 958 | . . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵)) → 𝐶 ∈ (0[,]+∞)) |
13 | 9, 12 | sylan2b 597 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ 𝐵)) → 𝐶 ∈ (0[,]+∞)) |
14 | xrge0base 30967 | . . 3 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
15 | xrge0plusg 30969 | . . 3 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
16 | xrge0cmn 20359 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
18 | xrge0tmd 31563 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopMnd | |
19 | 18 | a1i 11 | . . 3 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopMnd) |
20 | nfcv 2897 | . . . 4 ⊢ Ⅎ𝑘(0[,]+∞) | |
21 | eqid 2736 | . . . 4 ⊢ (𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) = (𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) | |
22 | 1, 4, 20, 13, 21 | fmptdF 30667 | . . 3 ⊢ (𝜑 → (𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶):(𝐴 ∪ 𝐵)⟶(0[,]+∞)) |
23 | 1, 2, 5, 10 | esumel 31681 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))) |
24 | ssun1 4072 | . . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
25 | 4, 2 | resmptf 5892 | . . . . . 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 7207 | . . . 4 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) ↾ 𝐴)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))) |
28 | 23, 27 | eleqtrrd 2834 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) ↾ 𝐴))) |
29 | 1, 3, 6, 11 | esumel 31681 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐵𝐶 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐶))) |
30 | ssun2 4073 | . . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
31 | 4, 3 | resmptf 5892 | . . . . . 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 7207 | . . . 4 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) ↾ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐶))) |
34 | 29, 33 | eleqtrrd 2834 | . . 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 23003 | . 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 31678 | 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 399 ∨ wo 847 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2112 Ⅎwnfc 2877 Vcvv 3398 ∪ cun 3851 ∩ cin 3852 ⊆ wss 3853 ∅c0 4223 ↦ cmpt 5120 ↾ cres 5538 (class class class)co 7191 0cc0 10694 +∞cpnf 10829 +𝑒 cxad 12667 [,]cicc 12903 ↾s cress 16667 ℝ*𝑠cxrs 16959 CMndccmn 19124 TopMndctmd 22921 tsums ctsu 22977 Σ*cesum 31661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 ax-addf 10773 ax-mulf 10774 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-er 8369 df-map 8488 df-pm 8489 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-fi 9005 df-sup 9036 df-inf 9037 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-q 12510 df-rp 12552 df-xneg 12669 df-xadd 12670 df-xmul 12671 df-ioo 12904 df-ioc 12905 df-ico 12906 df-icc 12907 df-fz 13061 df-fzo 13204 df-fl 13332 df-mod 13408 df-seq 13540 df-exp 13601 df-fac 13805 df-bc 13834 df-hash 13862 df-shft 14595 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-limsup 14997 df-clim 15014 df-rlim 15015 df-sum 15215 df-ef 15592 df-sin 15594 df-cos 15595 df-pi 15597 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-starv 16764 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-hom 16773 df-cco 16774 df-rest 16881 df-topn 16882 df-0g 16900 df-gsum 16901 df-topgen 16902 df-pt 16903 df-prds 16906 df-ordt 16960 df-xrs 16961 df-qtop 16966 df-imas 16967 df-xps 16969 df-mre 17043 df-mrc 17044 df-acs 17046 df-ps 18026 df-tsr 18027 df-plusf 18067 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-mulg 18443 df-subg 18494 df-cntz 18665 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-ring 19518 df-cring 19519 df-subrg 19752 df-abv 19807 df-lmod 19855 df-scaf 19856 df-sra 20163 df-rgmod 20164 df-psmet 20309 df-xmet 20310 df-met 20311 df-bl 20312 df-mopn 20313 df-fbas 20314 df-fg 20315 df-cnfld 20318 df-top 21745 df-topon 21762 df-topsp 21784 df-bases 21797 df-cld 21870 df-ntr 21871 df-cls 21872 df-nei 21949 df-lp 21987 df-perf 21988 df-cn 22078 df-cnp 22079 df-haus 22166 df-tx 22413 df-hmeo 22606 df-fil 22697 df-fm 22789 df-flim 22790 df-flf 22791 df-tmd 22923 df-tgp 22924 df-tsms 22978 df-trg 23011 df-xms 23172 df-ms 23173 df-tms 23174 df-nm 23434 df-ngp 23435 df-nrg 23437 df-nlm 23438 df-ii 23728 df-cncf 23729 df-limc 24717 df-dv 24718 df-log 25399 df-esum 31662 |
This theorem is referenced by: esummono 31688 esumpad 31689 esumpr 31700 esumrnmpt2 31702 esumfzf 31703 esumpmono 31713 hasheuni 31719 esum2dlem 31726 measvuni 31848 ddemeas 31870 carsgclctunlem1 31950 |
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