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Mirrors > Home > MPE Home > Th. List > minveclem4c | Structured version Visualization version GIF version |
Description: Lemma for minvec 25286. The infimum of the distances to π΄ is a real number. (Contributed by Mario Carneiro, 16-Jun-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 3-Oct-2020.) |
Ref | Expression |
---|---|
minvec.x | β’ π = (Baseβπ) |
minvec.m | β’ β = (-gβπ) |
minvec.n | β’ π = (normβπ) |
minvec.u | β’ (π β π β βPreHil) |
minvec.y | β’ (π β π β (LSubSpβπ)) |
minvec.w | β’ (π β (π βΎs π) β CMetSp) |
minvec.a | β’ (π β π΄ β π) |
minvec.j | β’ π½ = (TopOpenβπ) |
minvec.r | β’ π = ran (π¦ β π β¦ (πβ(π΄ β π¦))) |
minvec.s | β’ π = inf(π , β, < ) |
Ref | Expression |
---|---|
minveclem4c | β’ (π β π β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | minvec.s | . 2 β’ π = inf(π , β, < ) | |
2 | minvec.x | . . . . 5 β’ π = (Baseβπ) | |
3 | minvec.m | . . . . 5 β’ β = (-gβπ) | |
4 | minvec.n | . . . . 5 β’ π = (normβπ) | |
5 | minvec.u | . . . . 5 β’ (π β π β βPreHil) | |
6 | minvec.y | . . . . 5 β’ (π β π β (LSubSpβπ)) | |
7 | minvec.w | . . . . 5 β’ (π β (π βΎs π) β CMetSp) | |
8 | minvec.a | . . . . 5 β’ (π β π΄ β π) | |
9 | minvec.j | . . . . 5 β’ π½ = (TopOpenβπ) | |
10 | minvec.r | . . . . 5 β’ π = ran (π¦ β π β¦ (πβ(π΄ β π¦))) | |
11 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | minveclem1 25274 | . . . 4 β’ (π β (π β β β§ π β β β§ βπ€ β π 0 β€ π€)) |
12 | 11 | simp1d 1139 | . . 3 β’ (π β π β β) |
13 | 11 | simp2d 1140 | . . 3 β’ (π β π β β ) |
14 | 0re 11213 | . . . 4 β’ 0 β β | |
15 | 11 | simp3d 1141 | . . . 4 β’ (π β βπ€ β π 0 β€ π€) |
16 | breq1 5141 | . . . . . 6 β’ (π¦ = 0 β (π¦ β€ π€ β 0 β€ π€)) | |
17 | 16 | ralbidv 3169 | . . . . 5 β’ (π¦ = 0 β (βπ€ β π π¦ β€ π€ β βπ€ β π 0 β€ π€)) |
18 | 17 | rspcev 3604 | . . . 4 β’ ((0 β β β§ βπ€ β π 0 β€ π€) β βπ¦ β β βπ€ β π π¦ β€ π€) |
19 | 14, 15, 18 | sylancr 586 | . . 3 β’ (π β βπ¦ β β βπ€ β π π¦ β€ π€) |
20 | infrecl 12193 | . . 3 β’ ((π β β β§ π β β β§ βπ¦ β β βπ€ β π π¦ β€ π€) β inf(π , β, < ) β β) | |
21 | 12, 13, 19, 20 | syl3anc 1368 | . 2 β’ (π β inf(π , β, < ) β β) |
22 | 1, 21 | eqeltrid 2829 | 1 β’ (π β π β β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wne 2932 βwral 3053 βwrex 3062 β wss 3940 β c0 4314 class class class wbr 5138 β¦ cmpt 5221 ran crn 5667 βcfv 6533 (class class class)co 7401 infcinf 9432 βcr 11105 0cc0 11106 < clt 11245 β€ cle 11246 Basecbs 17143 βΎs cress 17172 TopOpenctopn 17366 -gcsg 18855 LSubSpclss 20768 normcnm 24407 βPreHilccph 25016 CMetSpccms 25182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-n0 12470 df-z 12556 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-0g 17386 df-topgen 17388 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18856 df-minusg 18857 df-sbg 18858 df-lmod 20698 df-lss 20769 df-psmet 21220 df-xmet 21221 df-met 21222 df-bl 21223 df-mopn 21224 df-top 22718 df-topon 22735 df-topsp 22757 df-bases 22771 df-xms 24148 df-ms 24149 df-nm 24413 df-ngp 24414 df-nlm 24417 df-cph 25018 |
This theorem is referenced by: minveclem2 25276 minveclem3b 25278 minveclem4 25282 |
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