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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 13420 | . . . 4 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
2 | fss 6734 | . . . 4 ⊢ ((𝐹:ℕ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐹:ℕ⟶(0[,]+∞)) | |
3 | 1, 2 | mpan2 688 | . . 3 ⊢ (𝐹:ℕ⟶(0[,)+∞) → 𝐹:ℕ⟶(0[,]+∞)) |
4 | esumfsup.1 | . . . 4 ⊢ Ⅎ𝑘𝐹 | |
5 | 4 | esumfsup 33381 | . . 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 12600 | . . . . 5 ⊢ (𝐹:ℕ⟶(0[,)+∞) → 1 ∈ ℤ) | |
8 | elnnuz 12873 | . . . . . 6 ⊢ (𝑥 ∈ ℕ ↔ 𝑥 ∈ (ℤ≥‘1)) | |
9 | ffvelcdm 7083 | . . . . . 6 ⊢ ((𝐹:ℕ⟶(0[,)+∞) ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ (0[,)+∞)) | |
10 | 8, 9 | sylan2br 594 | . . . . 5 ⊢ ((𝐹:ℕ⟶(0[,)+∞) ∧ 𝑥 ∈ (ℤ≥‘1)) → (𝐹‘𝑥) ∈ (0[,)+∞)) |
11 | ge0addcl 13444 | . . . . . 6 ⊢ ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 + 𝑦) ∈ (0[,)+∞)) | |
12 | 11 | adantl 481 | . . . . 5 ⊢ ((𝐹:ℕ⟶(0[,)+∞) ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 + 𝑦) ∈ (0[,)+∞)) |
13 | rge0ssre 13440 | . . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ | |
14 | simprl 768 | . . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,)+∞) ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → 𝑥 ∈ (0[,)+∞)) | |
15 | 13, 14 | sselid 3980 | . . . . . 6 ⊢ ((𝐹:ℕ⟶(0[,)+∞) ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → 𝑥 ∈ ℝ) |
16 | simprr 770 | . . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,)+∞) ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → 𝑦 ∈ (0[,)+∞)) | |
17 | 13, 16 | sselid 3980 | . . . . . 6 ⊢ ((𝐹:ℕ⟶(0[,)+∞) ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → 𝑦 ∈ ℝ) |
18 | rexadd 13218 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 +𝑒 𝑦) = (𝑥 + 𝑦)) | |
19 | 18 | eqcomd 2737 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) = (𝑥 +𝑒 𝑦)) |
20 | 15, 17, 19 | syl2anc 583 | . . . . 5 ⊢ ((𝐹:ℕ⟶(0[,)+∞) ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 + 𝑦) = (𝑥 +𝑒 𝑦)) |
21 | 7, 10, 12, 20 | seqfeq3 14025 | . . . 4 ⊢ (𝐹:ℕ⟶(0[,)+∞) → seq1( + , 𝐹) = seq1( +𝑒 , 𝐹)) |
22 | 21 | rneqd 5937 | . . 3 ⊢ (𝐹:ℕ⟶(0[,)+∞) → ran seq1( + , 𝐹) = ran seq1( +𝑒 , 𝐹)) |
23 | 22 | supeq1d 9447 | . 2 ⊢ (𝐹:ℕ⟶(0[,)+∞) → sup(ran seq1( + , 𝐹), ℝ*, < ) = sup(ran seq1( +𝑒 , 𝐹), ℝ*, < )) |
24 | 6, 23 | eqtr4d 2774 | 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 395 = wceq 1540 ∈ wcel 2105 Ⅎwnfc 2882 ⊆ wss 3948 ran crn 5677 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 supcsup 9441 ℝcr 11115 0cc0 11116 1c1 11117 + caddc 11119 +∞cpnf 11252 ℝ*cxr 11254 < clt 11255 ℕcn 12219 ℤ≥cuz 12829 +𝑒 cxad 13097 [,)cico 13333 [,]cicc 13334 seqcseq 13973 Σ*cesum 33338 |
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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 ax-mulf 11196 |
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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-er 8709 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-fi 9412 df-sup 9443 df-inf 9444 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ioo 13335 df-ioc 13336 df-ico 13337 df-icc 13338 df-fz 13492 df-fzo 13635 df-fl 13764 df-mod 13842 df-seq 13974 df-exp 14035 df-fac 14241 df-bc 14270 df-hash 14298 df-shft 15021 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-limsup 15422 df-clim 15439 df-rlim 15440 df-sum 15640 df-ef 16018 df-sin 16020 df-cos 16021 df-pi 16023 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-pt 17397 df-prds 17400 df-ordt 17454 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-ps 18526 df-tsr 18527 df-plusf 18567 df-mgm 18568 df-sgrp 18647 df-mnd 18663 df-mhm 18708 df-submnd 18709 df-grp 18861 df-minusg 18862 df-sbg 18863 df-mulg 18991 df-subg 19043 df-cntz 19226 df-cmn 19695 df-abl 19696 df-mgp 20033 df-rng 20051 df-ur 20080 df-ring 20133 df-cring 20134 df-subrng 20438 df-subrg 20463 df-abv 20572 df-lmod 20620 df-scaf 20621 df-sra 20934 df-rgmod 20935 df-psmet 21140 df-xmet 21141 df-met 21142 df-bl 21143 df-mopn 21144 df-fbas 21145 df-fg 21146 df-cnfld 21149 df-top 22629 df-topon 22646 df-topsp 22668 df-bases 22682 df-cld 22756 df-ntr 22757 df-cls 22758 df-nei 22835 df-lp 22873 df-perf 22874 df-cn 22964 df-cnp 22965 df-haus 23052 df-tx 23299 df-hmeo 23492 df-fil 23583 df-fm 23675 df-flim 23676 df-flf 23677 df-tmd 23809 df-tgp 23810 df-tsms 23864 df-trg 23897 df-xms 24059 df-ms 24060 df-tms 24061 df-nm 24324 df-ngp 24325 df-nrg 24327 df-nlm 24328 df-ii 24630 df-cncf 24631 df-limc 25628 df-dv 25629 df-log 26316 df-esum 33339 |
This theorem is referenced by: voliune 33540 |
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