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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 4166 | . 2 ⊢ Ⅎ𝑘(𝐴 ∪ 𝐵) |
5 | esumsplit.4 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) | |
6 | esumsplit.5 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) | |
7 | unexg 7739 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V) | |
8 | 5, 6, 7 | syl2anc 583 | . 2 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V) |
9 | elun 4149 | . . 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 955 | . . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵)) → 𝐶 ∈ (0[,]+∞)) |
13 | 9, 12 | sylan2b 593 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ 𝐵)) → 𝐶 ∈ (0[,]+∞)) |
14 | xrge0base 32450 | . . 3 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
15 | xrge0plusg 32452 | . . 3 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
16 | xrge0cmn 21188 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
18 | xrge0tmd 33220 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopMnd | |
19 | 18 | a1i 11 | . . 3 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopMnd) |
20 | nfcv 2902 | . . . 4 ⊢ Ⅎ𝑘(0[,]+∞) | |
21 | eqid 2731 | . . . 4 ⊢ (𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) = (𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) | |
22 | 1, 4, 20, 13, 21 | fmptdF 32145 | . . 3 ⊢ (𝜑 → (𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶):(𝐴 ∪ 𝐵)⟶(0[,]+∞)) |
23 | 1, 2, 5, 10 | esumel 33340 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))) |
24 | ssun1 4173 | . . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
25 | 4, 2 | resmptf 6040 | . . . . . 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 7428 | . . . 4 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) ↾ 𝐴)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))) |
28 | 23, 27 | eleqtrrd 2835 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) ↾ 𝐴))) |
29 | 1, 3, 6, 11 | esumel 33340 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐵𝐶 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐶))) |
30 | ssun2 4174 | . . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
31 | 4, 3 | resmptf 6040 | . . . . . 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 7428 | . . . 4 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) ↾ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐶))) |
34 | 29, 33 | eleqtrrd 2835 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐵𝐶 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) ↾ 𝐵))) |
35 | esumsplit.6 | . . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) | |
36 | eqidd 2732 | . . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) = (𝐴 ∪ 𝐵)) | |
37 | 14, 15, 17, 19, 8, 22, 28, 34, 35, 36 | tsmssplit 23877 | . 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 33337 | 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 844 = wceq 1540 Ⅎwnf 1784 ∈ wcel 2105 Ⅎwnfc 2882 Vcvv 3473 ∪ cun 3947 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 ↦ cmpt 5232 ↾ cres 5679 (class class class)co 7412 0cc0 11113 +∞cpnf 11250 +𝑒 cxad 13095 [,]cicc 13332 ↾s cress 17178 ℝ*𝑠cxrs 17451 CMndccmn 19690 TopMndctmd 23795 tsums ctsu 23851 Σ*cesum 33320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-inf2 9639 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191 ax-addf 11192 ax-mulf 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7673 df-om 7859 df-1st 7978 df-2nd 7979 df-supp 8150 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-2o 8470 df-er 8706 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9365 df-fi 9409 df-sup 9440 df-inf 9441 df-oi 9508 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ioo 13333 df-ioc 13334 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-fl 13762 df-mod 13840 df-seq 13972 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15019 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-limsup 15420 df-clim 15437 df-rlim 15438 df-sum 15638 df-ef 16016 df-sin 16018 df-cos 16019 df-pi 16021 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-ordt 17452 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-ps 18524 df-tsr 18525 df-plusf 18565 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18706 df-submnd 18707 df-grp 18859 df-minusg 18860 df-sbg 18861 df-mulg 18988 df-subg 19040 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-subrng 20435 df-subrg 20460 df-abv 20569 df-lmod 20617 df-scaf 20618 df-sra 20931 df-rgmod 20932 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-fbas 21142 df-fg 21143 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cld 22744 df-ntr 22745 df-cls 22746 df-nei 22823 df-lp 22861 df-perf 22862 df-cn 22952 df-cnp 22953 df-haus 23040 df-tx 23287 df-hmeo 23480 df-fil 23571 df-fm 23663 df-flim 23664 df-flf 23665 df-tmd 23797 df-tgp 23798 df-tsms 23852 df-trg 23885 df-xms 24047 df-ms 24048 df-tms 24049 df-nm 24312 df-ngp 24313 df-nrg 24315 df-nlm 24316 df-ii 24618 df-cncf 24619 df-limc 25616 df-dv 25617 df-log 26298 df-esum 33321 |
This theorem is referenced by: esummono 33347 esumpad 33348 esumpr 33359 esumrnmpt2 33361 esumfzf 33362 esumpmono 33372 hasheuni 33378 esum2dlem 33385 measvuni 33507 ddemeas 33529 carsgclctunlem1 33611 |
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