![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > minveclem5 | Structured version Visualization version GIF version |
Description: Lemma for minvec 24816. Discharge the assumptions in minveclem4 24812. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) |
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, < ) |
minvec.d | β’ π· = ((distβπ) βΎ (π Γ π)) |
Ref | Expression |
---|---|
minveclem5 | β’ (π β βπ₯ β π βπ¦ β π (πβ(π΄ β π₯)) β€ (πβ(π΄ β π¦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | minvec.x | . 2 β’ π = (Baseβπ) | |
2 | minvec.m | . 2 β’ β = (-gβπ) | |
3 | minvec.n | . 2 β’ π = (normβπ) | |
4 | minvec.u | . 2 β’ (π β π β βPreHil) | |
5 | minvec.y | . 2 β’ (π β π β (LSubSpβπ)) | |
6 | minvec.w | . 2 β’ (π β (π βΎs π) β CMetSp) | |
7 | minvec.a | . 2 β’ (π β π΄ β π) | |
8 | minvec.j | . 2 β’ π½ = (TopOpenβπ) | |
9 | minvec.r | . 2 β’ π = ran (π¦ β π β¦ (πβ(π΄ β π¦))) | |
10 | minvec.s | . 2 β’ π = inf(π , β, < ) | |
11 | minvec.d | . 2 β’ π· = ((distβπ) βΎ (π Γ π)) | |
12 | oveq2 7370 | . . . . . . 7 β’ (π = π β ((πβ2) + π ) = ((πβ2) + π)) | |
13 | 12 | breq2d 5122 | . . . . . 6 β’ (π = π β (((π΄π·π§)β2) β€ ((πβ2) + π ) β ((π΄π·π§)β2) β€ ((πβ2) + π))) |
14 | 13 | rabbidv 3418 | . . . . 5 β’ (π = π β {π§ β π β£ ((π΄π·π§)β2) β€ ((πβ2) + π )} = {π§ β π β£ ((π΄π·π§)β2) β€ ((πβ2) + π)}) |
15 | oveq2 7370 | . . . . . . . 8 β’ (π§ = π¦ β (π΄π·π§) = (π΄π·π¦)) | |
16 | 15 | oveq1d 7377 | . . . . . . 7 β’ (π§ = π¦ β ((π΄π·π§)β2) = ((π΄π·π¦)β2)) |
17 | 16 | breq1d 5120 | . . . . . 6 β’ (π§ = π¦ β (((π΄π·π§)β2) β€ ((πβ2) + π) β ((π΄π·π¦)β2) β€ ((πβ2) + π))) |
18 | 17 | cbvrabv 3420 | . . . . 5 β’ {π§ β π β£ ((π΄π·π§)β2) β€ ((πβ2) + π)} = {π¦ β π β£ ((π΄π·π¦)β2) β€ ((πβ2) + π)} |
19 | 14, 18 | eqtrdi 2793 | . . . 4 β’ (π = π β {π§ β π β£ ((π΄π·π§)β2) β€ ((πβ2) + π )} = {π¦ β π β£ ((π΄π·π¦)β2) β€ ((πβ2) + π)}) |
20 | 19 | cbvmptv 5223 | . . 3 β’ (π β β+ β¦ {π§ β π β£ ((π΄π·π§)β2) β€ ((πβ2) + π )}) = (π β β+ β¦ {π¦ β π β£ ((π΄π·π¦)β2) β€ ((πβ2) + π)}) |
21 | 20 | rneqi 5897 | . 2 β’ ran (π β β+ β¦ {π§ β π β£ ((π΄π·π§)β2) β€ ((πβ2) + π )}) = ran (π β β+ β¦ {π¦ β π β£ ((π΄π·π¦)β2) β€ ((πβ2) + π)}) |
22 | eqid 2737 | . 2 β’ βͺ (π½ fLim (πfilGenran (π β β+ β¦ {π§ β π β£ ((π΄π·π§)β2) β€ ((πβ2) + π )}))) = βͺ (π½ fLim (πfilGenran (π β β+ β¦ {π§ β π β£ ((π΄π·π§)β2) β€ ((πβ2) + π )}))) | |
23 | eqid 2737 | . 2 β’ (((((π΄π·βͺ (π½ fLim (πfilGenran (π β β+ β¦ {π§ β π β£ ((π΄π·π§)β2) β€ ((πβ2) + π )})))) + π) / 2)β2) β (πβ2)) = (((((π΄π·βͺ (π½ fLim (πfilGenran (π β β+ β¦ {π§ β π β£ ((π΄π·π§)β2) β€ ((πβ2) + π )})))) + π) / 2)β2) β (πβ2)) | |
24 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 22, 23 | minveclem4 24812 | 1 β’ (π β βπ₯ β π βπ¦ β π (πβ(π΄ β π₯)) β€ (πβ(π΄ β π¦))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βwral 3065 βwrex 3074 {crab 3410 βͺ cuni 4870 class class class wbr 5110 β¦ cmpt 5193 Γ cxp 5636 ran crn 5639 βΎ cres 5640 βcfv 6501 (class class class)co 7362 infcinf 9384 βcr 11057 + caddc 11061 < clt 11196 β€ cle 11197 β cmin 11392 / cdiv 11819 2c2 12215 β+crp 12922 βcexp 13974 Basecbs 17090 βΎs cress 17119 distcds 17149 TopOpenctopn 17310 -gcsg 18757 LSubSpclss 20408 filGencfg 20801 fLim cflim 23301 normcnm 23948 βPreHilccph 24546 CMetSpccms 24712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 ax-addf 11137 ax-mulf 11138 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-tpos 8162 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fi 9354 df-sup 9385 df-inf 9386 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-q 12881 df-rp 12923 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-ico 13277 df-icc 13278 df-fz 13432 df-seq 13914 df-exp 13975 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-starv 17155 df-sca 17156 df-vsca 17157 df-ip 17158 df-tset 17159 df-ple 17160 df-ds 17162 df-unif 17163 df-rest 17311 df-0g 17330 df-topgen 17332 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-mhm 18608 df-grp 18758 df-minusg 18759 df-sbg 18760 df-mulg 18880 df-subg 18932 df-ghm 19013 df-cmn 19571 df-abl 19572 df-mgp 19904 df-ur 19921 df-ring 19973 df-cring 19974 df-oppr 20056 df-dvdsr 20077 df-unit 20078 df-invr 20108 df-dvr 20119 df-rnghom 20155 df-drng 20201 df-subrg 20236 df-staf 20320 df-srng 20321 df-lmod 20340 df-lss 20409 df-lmhm 20499 df-lvec 20580 df-sra 20649 df-rgmod 20650 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-fbas 20809 df-fg 20810 df-cnfld 20813 df-phl 21046 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-cld 22386 df-ntr 22387 df-cls 22388 df-nei 22465 df-haus 22682 df-fil 23213 df-flim 23306 df-xms 23689 df-ms 23690 df-nm 23954 df-ngp 23955 df-nlm 23958 df-clm 24442 df-cph 24548 df-cfil 24635 df-cmet 24637 df-cms 24715 |
This theorem is referenced by: minveclem7 24815 |
Copyright terms: Public domain | W3C validator |