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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumaddf | Structured version Visualization version GIF version |
Description: Addition of infinite sums. (Contributed by Thierry Arnoux, 22-Jun-2017.) |
Ref | Expression |
---|---|
esumaddf.0 | ⊢ Ⅎ𝑘𝜑 |
esumaddf.a | ⊢ Ⅎ𝑘𝐴 |
esumaddf.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
esumaddf.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
esumaddf.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞)) |
Ref | Expression |
---|---|
esumaddf | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴(𝐵 +𝑒 𝐶) = (Σ*𝑘 ∈ 𝐴𝐵 +𝑒 Σ*𝑘 ∈ 𝐴𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | esumaddf.0 | . 2 ⊢ Ⅎ𝑘𝜑 | |
2 | esumaddf.a | . 2 ⊢ Ⅎ𝑘𝐴 | |
3 | esumaddf.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | esumaddf.2 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
5 | esumaddf.3 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞)) | |
6 | ge0xaddcl 13481 | . . 3 ⊢ ((𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → (𝐵 +𝑒 𝐶) ∈ (0[,]+∞)) | |
7 | 4, 5, 6 | syl2anc 582 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 +𝑒 𝐶) ∈ (0[,]+∞)) |
8 | xrge0base 32770 | . . . 4 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
9 | xrge0plusg 32772 | . . . 4 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
10 | xrge0cmn 21355 | . . . . 5 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
12 | xrge0tmd 33587 | . . . . 5 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopMnd | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopMnd) |
14 | nfcv 2899 | . . . . 5 ⊢ Ⅎ𝑘(0[,]+∞) | |
15 | eqid 2728 | . . . . 5 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
16 | 1, 2, 14, 4, 15 | fmptdF 32471 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
17 | eqid 2728 | . . . . 5 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐴 ↦ 𝐶) | |
18 | 1, 2, 14, 5, 17 | fmptdF 32471 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶(0[,]+∞)) |
19 | 1, 2, 3, 4 | esumel 33707 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))) |
20 | 1, 2, 3, 5 | esumel 33707 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))) |
21 | 8, 9, 11, 13, 3, 16, 18, 19, 20 | tsmsadd 24079 | . . 3 ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴𝐵 +𝑒 Σ*𝑘 ∈ 𝐴𝐶) ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘f +𝑒 (𝑘 ∈ 𝐴 ↦ 𝐶)))) |
22 | eqidd 2729 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
23 | eqidd 2729 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐴 ↦ 𝐶)) | |
24 | 1, 2, 3, 4, 5, 22, 23 | offval2f 7707 | . . . 4 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘f +𝑒 (𝑘 ∈ 𝐴 ↦ 𝐶)) = (𝑘 ∈ 𝐴 ↦ (𝐵 +𝑒 𝐶))) |
25 | 24 | oveq2d 7442 | . . 3 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘f +𝑒 (𝑘 ∈ 𝐴 ↦ 𝐶))) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ (𝐵 +𝑒 𝐶)))) |
26 | 21, 25 | eleqtrd 2831 | . 2 ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴𝐵 +𝑒 Σ*𝑘 ∈ 𝐴𝐶) ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ (𝐵 +𝑒 𝐶)))) |
27 | 1, 2, 3, 7, 26 | esumid 33704 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴(𝐵 +𝑒 𝐶) = (Σ*𝑘 ∈ 𝐴𝐵 +𝑒 Σ*𝑘 ∈ 𝐴𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 Ⅎwnfc 2879 ↦ cmpt 5235 (class class class)co 7426 ∘f cof 7690 0cc0 11148 +∞cpnf 11285 +𝑒 cxad 13132 [,]cicc 13369 ↾s cress 17218 ℝ*𝑠cxrs 17491 CMndccmn 19749 TopMndctmd 24002 tsums ctsu 24058 Σ*cesum 33687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-inf2 9674 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 ax-addf 11227 ax-mulf 11228 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 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 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-er 8733 df-map 8855 df-pm 8856 df-ixp 8925 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-fsupp 9396 df-fi 9444 df-sup 9475 df-inf 9476 df-oi 9543 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13134 df-xadd 13135 df-xmul 13136 df-ioo 13370 df-ioc 13371 df-ico 13372 df-icc 13373 df-fz 13527 df-fzo 13670 df-fl 13799 df-mod 13877 df-seq 14009 df-exp 14069 df-fac 14275 df-bc 14304 df-hash 14332 df-shft 15056 df-cj 15088 df-re 15089 df-im 15090 df-sqrt 15224 df-abs 15225 df-limsup 15457 df-clim 15474 df-rlim 15475 df-sum 15675 df-ef 16053 df-sin 16055 df-cos 16056 df-pi 16058 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-starv 17257 df-sca 17258 df-vsca 17259 df-ip 17260 df-tset 17261 df-ple 17262 df-ds 17264 df-unif 17265 df-hom 17266 df-cco 17267 df-rest 17413 df-topn 17414 df-0g 17432 df-gsum 17433 df-topgen 17434 df-pt 17435 df-prds 17438 df-ordt 17492 df-xrs 17493 df-qtop 17498 df-imas 17499 df-xps 17501 df-mre 17575 df-mrc 17576 df-acs 17578 df-ps 18567 df-tsr 18568 df-plusf 18608 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-mhm 18749 df-submnd 18750 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19038 df-subg 19092 df-cntz 19282 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-ring 20189 df-cring 20190 df-subrng 20497 df-subrg 20522 df-abv 20711 df-lmod 20759 df-scaf 20760 df-sra 21072 df-rgmod 21073 df-psmet 21285 df-xmet 21286 df-met 21287 df-bl 21288 df-mopn 21289 df-fbas 21290 df-fg 21291 df-cnfld 21294 df-top 22824 df-topon 22841 df-topsp 22863 df-bases 22877 df-cld 22951 df-ntr 22952 df-cls 22953 df-nei 23030 df-lp 23068 df-perf 23069 df-cn 23159 df-cnp 23160 df-haus 23247 df-tx 23494 df-hmeo 23687 df-fil 23778 df-fm 23870 df-flim 23871 df-flf 23872 df-tmd 24004 df-tgp 24005 df-tsms 24059 df-trg 24092 df-xms 24254 df-ms 24255 df-tms 24256 df-nm 24519 df-ngp 24520 df-nrg 24522 df-nlm 24523 df-ii 24825 df-cncf 24826 df-limc 25823 df-dv 25824 df-log 26518 df-esum 33688 |
This theorem is referenced by: esumlef 33722 omssubadd 33961 |
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