Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumsup | Structured version Visualization version GIF version |
Description: Express an extended sum as a supremum of extended sums. (Contributed by Thierry Arnoux, 24-May-2020.) |
Ref | Expression |
---|---|
esumsup.1 | ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) |
esumsup.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞)) |
Ref | Expression |
---|---|
esumsup | ⊢ (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = sup(ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴), ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | esumsup.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞)) | |
2 | 1 | fmpttd 6879 | . . 3 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶(0[,]+∞)) |
3 | nfmpt1 5164 | . . . 4 ⊢ Ⅎ𝑘(𝑘 ∈ ℕ ↦ 𝐴) | |
4 | 3 | esumfsup 31329 | . . 3 ⊢ ((𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶(0[,]+∞) → Σ*𝑘 ∈ ℕ((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = sup(ran seq1( +𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)), ℝ*, < )) |
5 | 2, 4 | syl 17 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ ℕ((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = sup(ran seq1( +𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)), ℝ*, < )) |
6 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ) | |
7 | eqid 2821 | . . . . 5 ⊢ (𝑘 ∈ ℕ ↦ 𝐴) = (𝑘 ∈ ℕ ↦ 𝐴) | |
8 | 7 | fvmpt2 6779 | . . . 4 ⊢ ((𝑘 ∈ ℕ ∧ 𝐴 ∈ (0[,]+∞)) → ((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = 𝐴) |
9 | 6, 1, 8 | syl2anc 586 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = 𝐴) |
10 | 9 | esumeq2dv 31297 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ ℕ((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = Σ*𝑘 ∈ ℕ𝐴) |
11 | 1z 12013 | . . . . . . . . 9 ⊢ 1 ∈ ℤ | |
12 | seqfn 13382 | . . . . . . . . 9 ⊢ (1 ∈ ℤ → seq1( +𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) Fn (ℤ≥‘1)) | |
13 | 11, 12 | ax-mp 5 | . . . . . . . 8 ⊢ seq1( +𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) Fn (ℤ≥‘1) |
14 | nnuz 12282 | . . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1) | |
15 | 14 | fneq2i 6451 | . . . . . . . 8 ⊢ (seq1( +𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) Fn ℕ ↔ seq1( +𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) Fn (ℤ≥‘1)) |
16 | 13, 15 | mpbir 233 | . . . . . . 7 ⊢ seq1( +𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) Fn ℕ |
17 | nfcv 2977 | . . . . . . . 8 ⊢ Ⅎ𝑛seq1( +𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) | |
18 | 17 | dffn5f 6736 | . . . . . . 7 ⊢ (seq1( +𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) Fn ℕ ↔ seq1( +𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) = (𝑛 ∈ ℕ ↦ (seq1( +𝑒 , (𝑘 ∈ ℕ ↦ 𝐴))‘𝑛))) |
19 | 16, 18 | mpbi 232 | . . . . . 6 ⊢ seq1( +𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) = (𝑛 ∈ ℕ ↦ (seq1( +𝑒 , (𝑘 ∈ ℕ ↦ 𝐴))‘𝑛)) |
20 | 19 | a1i 11 | . . . . 5 ⊢ (𝜑 → seq1( +𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) = (𝑛 ∈ ℕ ↦ (seq1( +𝑒 , (𝑘 ∈ ℕ ↦ 𝐴))‘𝑛))) |
21 | fz1ssnn 12939 | . . . . . . . . . . 11 ⊢ (1...𝑛) ⊆ ℕ | |
22 | 21 | a1i 11 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ) |
23 | 22 | sselda 3967 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ) |
24 | simpll 765 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑) | |
25 | 24, 23, 1 | syl2anc 586 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (0[,]+∞)) |
26 | 23, 25, 8 | syl2anc 586 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = 𝐴) |
27 | 26 | esumeq2dv 31297 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ*𝑘 ∈ (1...𝑛)((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = Σ*𝑘 ∈ (1...𝑛)𝐴) |
28 | 3 | esumfzf 31328 | . . . . . . . 8 ⊢ (((𝑘 ∈ ℕ ↦ 𝐴):ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → Σ*𝑘 ∈ (1...𝑛)((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = (seq1( +𝑒 , (𝑘 ∈ ℕ ↦ 𝐴))‘𝑛)) |
29 | 2, 28 | sylan 582 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ*𝑘 ∈ (1...𝑛)((𝑘 ∈ ℕ ↦ 𝐴)‘𝑘) = (seq1( +𝑒 , (𝑘 ∈ ℕ ↦ 𝐴))‘𝑛)) |
30 | 27, 29 | eqtr3d 2858 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ*𝑘 ∈ (1...𝑛)𝐴 = (seq1( +𝑒 , (𝑘 ∈ ℕ ↦ 𝐴))‘𝑛)) |
31 | 30 | mpteq2dva 5161 | . . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴) = (𝑛 ∈ ℕ ↦ (seq1( +𝑒 , (𝑘 ∈ ℕ ↦ 𝐴))‘𝑛))) |
32 | 20, 31 | eqtr4d 2859 | . . . 4 ⊢ (𝜑 → seq1( +𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) = (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴)) |
33 | 32 | rneqd 5808 | . . 3 ⊢ (𝜑 → ran seq1( +𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)) = ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴)) |
34 | 33 | supeq1d 8910 | . 2 ⊢ (𝜑 → sup(ran seq1( +𝑒 , (𝑘 ∈ ℕ ↦ 𝐴)), ℝ*, < ) = sup(ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴), ℝ*, < )) |
35 | 5, 10, 34 | 3eqtr3d 2864 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = sup(ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴), ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 ↦ cmpt 5146 ran crn 5556 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 supcsup 8904 0cc0 10537 1c1 10538 +∞cpnf 10672 ℝ*cxr 10674 < clt 10675 ℕcn 11638 ℤcz 11982 ℤ≥cuz 12244 +𝑒 cxad 12506 [,]cicc 12742 ...cfz 12893 seqcseq 13370 Σ*cesum 31286 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-sin 15423 df-cos 15424 df-pi 15426 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-ordt 16774 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-ps 17810 df-tsr 17811 df-plusf 17851 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-subg 18276 df-cntz 18447 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-subrg 19533 df-abv 19588 df-lmod 19636 df-scaf 19637 df-sra 19944 df-rgmod 19945 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-tmd 22680 df-tgp 22681 df-tsms 22735 df-trg 22768 df-xms 22930 df-ms 22931 df-tms 22932 df-nm 23192 df-ngp 23193 df-nrg 23195 df-nlm 23196 df-ii 23485 df-cncf 23486 df-limc 24464 df-dv 24465 df-log 25140 df-esum 31287 |
This theorem is referenced by: esumgect 31349 |
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