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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumfsupre | Structured version Visualization version GIF version |
Description: Formulating an extended sum over integers using the recursive sequence builder. This version is limited to real-valued functions. (Contributed by Thierry Arnoux, 19-Oct-2017.) |
Ref | Expression |
---|---|
esumfsup.1 | ⊢ Ⅎ𝑘𝐹 |
Ref | Expression |
---|---|
esumfsupre | ⊢ (𝐹:ℕ⟶(0[,)+∞) → Σ*𝑘 ∈ ℕ(𝐹‘𝑘) = sup(ran seq1( + , 𝐹), ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | icossicc 13360 | . . . 4 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
2 | fss 6690 | . . . 4 ⊢ ((𝐹:ℕ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐹:ℕ⟶(0[,]+∞)) | |
3 | 1, 2 | mpan2 690 | . . 3 ⊢ (𝐹:ℕ⟶(0[,)+∞) → 𝐹:ℕ⟶(0[,]+∞)) |
4 | esumfsup.1 | . . . 4 ⊢ Ⅎ𝑘𝐹 | |
5 | 4 | esumfsup 32709 | . . 3 ⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ*𝑘 ∈ ℕ(𝐹‘𝑘) = sup(ran seq1( +𝑒 , 𝐹), ℝ*, < )) |
6 | 3, 5 | syl 17 | . 2 ⊢ (𝐹:ℕ⟶(0[,)+∞) → Σ*𝑘 ∈ ℕ(𝐹‘𝑘) = sup(ran seq1( +𝑒 , 𝐹), ℝ*, < )) |
7 | 1zzd 12541 | . . . . 5 ⊢ (𝐹:ℕ⟶(0[,)+∞) → 1 ∈ ℤ) | |
8 | elnnuz 12814 | . . . . . 6 ⊢ (𝑥 ∈ ℕ ↔ 𝑥 ∈ (ℤ≥‘1)) | |
9 | ffvelcdm 7037 | . . . . . 6 ⊢ ((𝐹:ℕ⟶(0[,)+∞) ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ (0[,)+∞)) | |
10 | 8, 9 | sylan2br 596 | . . . . 5 ⊢ ((𝐹:ℕ⟶(0[,)+∞) ∧ 𝑥 ∈ (ℤ≥‘1)) → (𝐹‘𝑥) ∈ (0[,)+∞)) |
11 | ge0addcl 13384 | . . . . . 6 ⊢ ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 + 𝑦) ∈ (0[,)+∞)) | |
12 | 11 | adantl 483 | . . . . 5 ⊢ ((𝐹:ℕ⟶(0[,)+∞) ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 + 𝑦) ∈ (0[,)+∞)) |
13 | rge0ssre 13380 | . . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ | |
14 | simprl 770 | . . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,)+∞) ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → 𝑥 ∈ (0[,)+∞)) | |
15 | 13, 14 | sselid 3947 | . . . . . 6 ⊢ ((𝐹:ℕ⟶(0[,)+∞) ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → 𝑥 ∈ ℝ) |
16 | simprr 772 | . . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,)+∞) ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → 𝑦 ∈ (0[,)+∞)) | |
17 | 13, 16 | sselid 3947 | . . . . . 6 ⊢ ((𝐹:ℕ⟶(0[,)+∞) ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → 𝑦 ∈ ℝ) |
18 | rexadd 13158 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 +𝑒 𝑦) = (𝑥 + 𝑦)) | |
19 | 18 | eqcomd 2743 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) = (𝑥 +𝑒 𝑦)) |
20 | 15, 17, 19 | syl2anc 585 | . . . . 5 ⊢ ((𝐹:ℕ⟶(0[,)+∞) ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 + 𝑦) = (𝑥 +𝑒 𝑦)) |
21 | 7, 10, 12, 20 | seqfeq3 13965 | . . . 4 ⊢ (𝐹:ℕ⟶(0[,)+∞) → seq1( + , 𝐹) = seq1( +𝑒 , 𝐹)) |
22 | 21 | rneqd 5898 | . . 3 ⊢ (𝐹:ℕ⟶(0[,)+∞) → ran seq1( + , 𝐹) = ran seq1( +𝑒 , 𝐹)) |
23 | 22 | supeq1d 9389 | . 2 ⊢ (𝐹:ℕ⟶(0[,)+∞) → sup(ran seq1( + , 𝐹), ℝ*, < ) = sup(ran seq1( +𝑒 , 𝐹), ℝ*, < )) |
24 | 6, 23 | eqtr4d 2780 | 1 ⊢ (𝐹:ℕ⟶(0[,)+∞) → Σ*𝑘 ∈ ℕ(𝐹‘𝑘) = sup(ran seq1( + , 𝐹), ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Ⅎwnfc 2888 ⊆ wss 3915 ran crn 5639 ⟶wf 6497 ‘cfv 6501 (class class class)co 7362 supcsup 9383 ℝcr 11057 0cc0 11058 1c1 11059 + caddc 11061 +∞cpnf 11193 ℝ*cxr 11195 < clt 11196 ℕcn 12160 ℤ≥cuz 12770 +𝑒 cxad 13038 [,)cico 13273 [,]cicc 13274 seqcseq 13913 Σ*cesum 32666 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 ax-addf 11137 ax-mulf 11138 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-er 8655 df-map 8774 df-pm 8775 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-fi 9354 df-sup 9385 df-inf 9386 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-q 12881 df-rp 12923 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-ioo 13275 df-ioc 13276 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-fl 13704 df-mod 13782 df-seq 13914 df-exp 13975 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14959 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-limsup 15360 df-clim 15377 df-rlim 15378 df-sum 15578 df-ef 15957 df-sin 15959 df-cos 15960 df-pi 15962 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-starv 17155 df-sca 17156 df-vsca 17157 df-ip 17158 df-tset 17159 df-ple 17160 df-ds 17162 df-unif 17163 df-hom 17164 df-cco 17165 df-rest 17311 df-topn 17312 df-0g 17330 df-gsum 17331 df-topgen 17332 df-pt 17333 df-prds 17336 df-ordt 17390 df-xrs 17391 df-qtop 17396 df-imas 17397 df-xps 17399 df-mre 17473 df-mrc 17474 df-acs 17476 df-ps 18462 df-tsr 18463 df-plusf 18503 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-mhm 18608 df-submnd 18609 df-grp 18758 df-minusg 18759 df-sbg 18760 df-mulg 18880 df-subg 18932 df-cntz 19104 df-cmn 19571 df-abl 19572 df-mgp 19904 df-ur 19921 df-ring 19973 df-cring 19974 df-subrg 20236 df-abv 20292 df-lmod 20340 df-scaf 20341 df-sra 20649 df-rgmod 20650 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-fbas 20809 df-fg 20810 df-cnfld 20813 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-cld 22386 df-ntr 22387 df-cls 22388 df-nei 22465 df-lp 22503 df-perf 22504 df-cn 22594 df-cnp 22595 df-haus 22682 df-tx 22929 df-hmeo 23122 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 df-tmd 23439 df-tgp 23440 df-tsms 23494 df-trg 23527 df-xms 23689 df-ms 23690 df-tms 23691 df-nm 23954 df-ngp 23955 df-nrg 23957 df-nlm 23958 df-ii 24256 df-cncf 24257 df-limc 25246 df-dv 25247 df-log 25928 df-esum 32667 |
This theorem is referenced by: voliune 32868 |
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