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Mirrors > Home > MPE Home > Th. List > minveclem4b | Structured version Visualization version GIF version |
Description: Lemma for minvec 23015. The convergent point of the Cauchy sequence 𝐹 is a member of the base space. (Contributed by Mario Carneiro, 16-Jun-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
minvec.x | ⊢ 𝑋 = (Base‘𝑈) |
minvec.m | ⊢ − = (-g‘𝑈) |
minvec.n | ⊢ 𝑁 = (norm‘𝑈) |
minvec.u | ⊢ (𝜑 → 𝑈 ∈ ℂPreHil) |
minvec.y | ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) |
minvec.w | ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp) |
minvec.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
minvec.j | ⊢ 𝐽 = (TopOpen‘𝑈) |
minvec.r | ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) |
minvec.s | ⊢ 𝑆 = inf(𝑅, ℝ, < ) |
minvec.d | ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋)) |
minvec.f | ⊢ 𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) |
minvec.p | ⊢ 𝑃 = ∪ (𝐽 fLim (𝑋filGen𝐹)) |
Ref | Expression |
---|---|
minveclem4b | ⊢ (𝜑 → 𝑃 ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | minvec.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) | |
2 | minvec.x | . . . 4 ⊢ 𝑋 = (Base‘𝑈) | |
3 | eqid 2610 | . . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
4 | 2, 3 | lssss 18758 | . . 3 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋) |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
6 | inss2 3796 | . . 3 ⊢ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌) ⊆ 𝑌 | |
7 | minvec.m | . . . 4 ⊢ − = (-g‘𝑈) | |
8 | minvec.n | . . . 4 ⊢ 𝑁 = (norm‘𝑈) | |
9 | minvec.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ ℂPreHil) | |
10 | minvec.w | . . . 4 ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp) | |
11 | minvec.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
12 | minvec.j | . . . 4 ⊢ 𝐽 = (TopOpen‘𝑈) | |
13 | minvec.r | . . . 4 ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) | |
14 | minvec.s | . . . 4 ⊢ 𝑆 = inf(𝑅, ℝ, < ) | |
15 | minvec.d | . . . 4 ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋)) | |
16 | minvec.f | . . . 4 ⊢ 𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) | |
17 | minvec.p | . . . 4 ⊢ 𝑃 = ∪ (𝐽 fLim (𝑋filGen𝐹)) | |
18 | 2, 7, 8, 9, 1, 10, 11, 12, 13, 14, 15, 16, 17 | minveclem4a 23009 | . . 3 ⊢ (𝜑 → 𝑃 ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌)) |
19 | 6, 18 | sseldi 3566 | . 2 ⊢ (𝜑 → 𝑃 ∈ 𝑌) |
20 | 5, 19 | sseldd 3569 | 1 ⊢ (𝜑 → 𝑃 ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {crab 2900 ∩ cin 3539 ⊆ wss 3540 ∪ cuni 4372 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 ran crn 5039 ↾ cres 5040 ‘cfv 5804 (class class class)co 6549 infcinf 8230 ℝcr 9814 + caddc 9818 < clt 9953 ≤ cle 9954 2c2 10947 ℝ+crp 11708 ↑cexp 12722 Basecbs 15695 ↾s cress 15696 distcds 15777 TopOpenctopn 15905 -gcsg 17247 LSubSpclss 18753 filGencfg 19556 fLim cflim 21548 normcnm 22191 ℂPreHilccph 22774 CMetSpccms 22937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ico 12052 df-icc 12053 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-rest 15906 df-0g 15925 df-topgen 15927 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-ghm 17481 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-rnghom 18538 df-drng 18572 df-subrg 18601 df-staf 18668 df-srng 18669 df-lmod 18688 df-lss 18754 df-lmhm 18843 df-lvec 18924 df-sra 18993 df-rgmod 18994 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-phl 19790 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-ntr 20634 df-nei 20712 df-haus 20929 df-fil 21460 df-flim 21553 df-xms 21935 df-ms 21936 df-nm 22197 df-ngp 22198 df-nlm 22201 df-clm 22671 df-cph 22776 df-cfil 22861 df-cmet 22863 df-cms 22940 |
This theorem is referenced by: minveclem4 23011 |
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