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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 4180 | . 2 ⊢ Ⅎ𝑘(𝐴 ∪ 𝐵) |
5 | esumsplit.4 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) | |
6 | esumsplit.5 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) | |
7 | unexg 7762 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V) | |
8 | 5, 6, 7 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V) |
9 | elun 4163 | . . 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 32999 | . . 3 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
15 | xrge0plusg 33001 | . . 3 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
16 | xrge0cmn 21444 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
18 | xrge0tmd 33906 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopMnd | |
19 | 18 | a1i 11 | . . 3 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopMnd) |
20 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑘(0[,]+∞) | |
21 | eqid 2735 | . . . 4 ⊢ (𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) = (𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) | |
22 | 1, 4, 20, 13, 21 | fmptdF 32673 | . . 3 ⊢ (𝜑 → (𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶):(𝐴 ∪ 𝐵)⟶(0[,]+∞)) |
23 | 1, 2, 5, 10 | esumel 34028 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))) |
24 | ssun1 4188 | . . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
25 | 4, 2 | resmptf 6059 | . . . . . 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 7447 | . . . 4 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) ↾ 𝐴)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))) |
28 | 23, 27 | eleqtrrd 2842 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) ↾ 𝐴))) |
29 | 1, 3, 6, 11 | esumel 34028 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐵𝐶 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐶))) |
30 | ssun2 4189 | . . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
31 | 4, 3 | resmptf 6059 | . . . . . 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 7447 | . . . 4 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) ↾ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐶))) |
34 | 29, 33 | eleqtrrd 2842 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐵𝐶 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) ↾ 𝐵))) |
35 | esumsplit.6 | . . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) | |
36 | eqidd 2736 | . . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) = (𝐴 ∪ 𝐵)) | |
37 | 14, 15, 17, 19, 8, 22, 28, 34, 35, 36 | tsmssplit 24176 | . 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 34025 | 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 1537 Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 Vcvv 3478 ∪ cun 3961 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 ↦ cmpt 5231 ↾ cres 5691 (class class class)co 7431 0cc0 11153 +∞cpnf 11290 +𝑒 cxad 13150 [,]cicc 13387 ↾s cress 17274 ℝ*𝑠cxrs 17547 CMndccmn 19813 TopMndctmd 24094 tsums ctsu 24150 Σ*cesum 34008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 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 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-ordt 17548 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-ps 18624 df-tsr 18625 df-plusf 18665 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-subrng 20563 df-subrg 20587 df-abv 20827 df-lmod 20877 df-scaf 20878 df-sra 21190 df-rgmod 21191 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-tmd 24096 df-tgp 24097 df-tsms 24151 df-trg 24184 df-xms 24346 df-ms 24347 df-tms 24348 df-nm 24611 df-ngp 24612 df-nrg 24614 df-nlm 24615 df-ii 24917 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613 df-esum 34009 |
This theorem is referenced by: esummono 34035 esumpad 34036 esumpr 34047 esumrnmpt2 34049 esumfzf 34050 esumpmono 34060 hasheuni 34066 esum2dlem 34073 measvuni 34195 ddemeas 34217 carsgclctunlem1 34299 |
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