Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > esumcvg2 | Structured version Visualization version GIF version |
Description: Simpler version of esumcvg 31637. (Contributed by Thierry Arnoux, 5-Sep-2017.) |
Ref | Expression |
---|---|
esumcvg2.j | ⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) |
esumcvg2.a | ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞)) |
esumcvg2.l | ⊢ (𝑘 = 𝑙 → 𝐴 = 𝐵) |
esumcvg2.m | ⊢ (𝑘 = 𝑚 → 𝐴 = 𝐶) |
Ref | Expression |
---|---|
esumcvg2 | ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴)(⇝𝑡‘𝐽)Σ*𝑘 ∈ ℕ𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | esumcvg2.m | . . . . 5 ⊢ (𝑘 = 𝑚 → 𝐴 = 𝐶) | |
2 | 1 | cbvesumv 31594 | . . . 4 ⊢ Σ*𝑘 ∈ (1...𝑖)𝐴 = Σ*𝑚 ∈ (1...𝑖)𝐶 |
3 | oveq2 7191 | . . . . 5 ⊢ (𝑖 = 𝑛 → (1...𝑖) = (1...𝑛)) | |
4 | esumeq1 31585 | . . . . 5 ⊢ ((1...𝑖) = (1...𝑛) → Σ*𝑘 ∈ (1...𝑖)𝐴 = Σ*𝑘 ∈ (1...𝑛)𝐴) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝑖 = 𝑛 → Σ*𝑘 ∈ (1...𝑖)𝐴 = Σ*𝑘 ∈ (1...𝑛)𝐴) |
6 | 2, 5 | eqtr3id 2788 | . . 3 ⊢ (𝑖 = 𝑛 → Σ*𝑚 ∈ (1...𝑖)𝐶 = Σ*𝑘 ∈ (1...𝑛)𝐴) |
7 | 6 | cbvmptv 5143 | . 2 ⊢ (𝑖 ∈ ℕ ↦ Σ*𝑚 ∈ (1...𝑖)𝐶) = (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴) |
8 | esumcvg2.j | . . 3 ⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
9 | esumcvg2.a | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞)) | |
10 | esumcvg2.l | . . 3 ⊢ (𝑘 = 𝑙 → 𝐴 = 𝐵) | |
11 | 8, 7, 9, 10 | esumcvg 31637 | . 2 ⊢ (𝜑 → (𝑖 ∈ ℕ ↦ Σ*𝑚 ∈ (1...𝑖)𝐶)(⇝𝑡‘𝐽)Σ*𝑘 ∈ ℕ𝐴) |
12 | 7, 11 | eqbrtrrid 5076 | 1 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴)(⇝𝑡‘𝐽)Σ*𝑘 ∈ ℕ𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 class class class wbr 5040 ↦ cmpt 5120 ‘cfv 6350 (class class class)co 7183 0cc0 10628 1c1 10629 +∞cpnf 10763 ℕcn 11729 [,]cicc 12837 ...cfz 12994 ↾s cress 16600 TopOpenctopn 16811 ℝ*𝑠cxrs 16889 ⇝𝑡clm 21990 Σ*cesum 31578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-inf2 9190 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 ax-pre-sup 10706 ax-addf 10707 ax-mulf 10708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-iin 4894 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-of 7438 df-om 7613 df-1st 7727 df-2nd 7728 df-supp 7870 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-2o 8145 df-oadd 8148 df-er 8333 df-map 8452 df-pm 8453 df-ixp 8521 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-fsupp 8920 df-fi 8961 df-sup 8992 df-inf 8993 df-oi 9060 df-card 9454 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-div 11389 df-nn 11730 df-2 11792 df-3 11793 df-4 11794 df-5 11795 df-6 11796 df-7 11797 df-8 11798 df-9 11799 df-n0 11990 df-xnn0 12062 df-z 12076 df-dec 12193 df-uz 12338 df-q 12444 df-rp 12486 df-xneg 12603 df-xadd 12604 df-xmul 12605 df-ioo 12838 df-ioc 12839 df-ico 12840 df-icc 12841 df-fz 12995 df-fzo 13138 df-fl 13266 df-mod 13342 df-seq 13474 df-exp 13535 df-fac 13739 df-bc 13768 df-hash 13796 df-shft 14529 df-cj 14561 df-re 14562 df-im 14563 df-sqrt 14697 df-abs 14698 df-limsup 14931 df-clim 14948 df-rlim 14949 df-sum 15149 df-ef 15526 df-sin 15528 df-cos 15529 df-pi 15531 df-struct 16601 df-ndx 16602 df-slot 16603 df-base 16605 df-sets 16606 df-ress 16607 df-plusg 16694 df-mulr 16695 df-starv 16696 df-sca 16697 df-vsca 16698 df-ip 16699 df-tset 16700 df-ple 16701 df-ds 16703 df-unif 16704 df-hom 16705 df-cco 16706 df-rest 16812 df-topn 16813 df-0g 16831 df-gsum 16832 df-topgen 16833 df-pt 16834 df-prds 16837 df-ordt 16890 df-xrs 16891 df-qtop 16896 df-imas 16897 df-xps 16899 df-mre 16973 df-mrc 16974 df-acs 16976 df-ps 17939 df-tsr 17940 df-plusf 17980 df-mgm 17981 df-sgrp 18030 df-mnd 18041 df-mhm 18085 df-submnd 18086 df-grp 18235 df-minusg 18236 df-sbg 18237 df-mulg 18356 df-subg 18407 df-cntz 18578 df-cmn 19039 df-abl 19040 df-mgp 19372 df-ur 19384 df-ring 19431 df-cring 19432 df-subrg 19665 df-abv 19720 df-lmod 19768 df-scaf 19769 df-sra 20076 df-rgmod 20077 df-psmet 20222 df-xmet 20223 df-met 20224 df-bl 20225 df-mopn 20226 df-fbas 20227 df-fg 20228 df-cnfld 20231 df-top 21658 df-topon 21675 df-topsp 21697 df-bases 21710 df-cld 21783 df-ntr 21784 df-cls 21785 df-nei 21862 df-lp 21900 df-perf 21901 df-cn 21991 df-cnp 21992 df-lm 21993 df-haus 22079 df-tx 22326 df-hmeo 22519 df-fil 22610 df-fm 22702 df-flim 22703 df-flf 22704 df-tmd 22836 df-tgp 22837 df-tsms 22891 df-trg 22924 df-xms 23086 df-ms 23087 df-tms 23088 df-nm 23348 df-ngp 23349 df-nrg 23351 df-nlm 23352 df-ii 23642 df-cncf 23643 df-limc 24631 df-dv 24632 df-log 25313 df-esum 31579 |
This theorem is referenced by: meascnbl 31770 |
Copyright terms: Public domain | W3C validator |