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Mirrors > Home > MPE Home > Th. List > minvecolem4a | Structured version Visualization version GIF version |
Description: Lemma for minveco 30733. πΉ is convergent in the subspace topology on π. (Contributed by Mario Carneiro, 7-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
minveco.x | β’ π = (BaseSetβπ) |
minveco.m | β’ π = ( βπ£ βπ) |
minveco.n | β’ π = (normCVβπ) |
minveco.y | β’ π = (BaseSetβπ) |
minveco.u | β’ (π β π β CPreHilOLD) |
minveco.w | β’ (π β π β ((SubSpβπ) β© CBan)) |
minveco.a | β’ (π β π΄ β π) |
minveco.d | β’ π· = (IndMetβπ) |
minveco.j | β’ π½ = (MetOpenβπ·) |
minveco.r | β’ π = ran (π¦ β π β¦ (πβ(π΄ππ¦))) |
minveco.s | β’ π = inf(π , β, < ) |
minveco.f | β’ (π β πΉ:ββΆπ) |
minveco.1 | β’ ((π β§ π β β) β ((π΄π·(πΉβπ))β2) β€ ((πβ2) + (1 / π))) |
Ref | Expression |
---|---|
minvecolem4a | β’ (π β πΉ(βπ‘β(MetOpenβ(π· βΎ (π Γ π))))((βπ‘β(MetOpenβ(π· βΎ (π Γ π))))βπΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | minveco.u | . . . . . 6 β’ (π β π β CPreHilOLD) | |
2 | phnv 30663 | . . . . . 6 β’ (π β CPreHilOLD β π β NrmCVec) | |
3 | 1, 2 | syl 17 | . . . . 5 β’ (π β π β NrmCVec) |
4 | minveco.w | . . . . . . 7 β’ (π β π β ((SubSpβπ) β© CBan)) | |
5 | elin 3957 | . . . . . . 7 β’ (π β ((SubSpβπ) β© CBan) β (π β (SubSpβπ) β§ π β CBan)) | |
6 | 4, 5 | sylib 217 | . . . . . 6 β’ (π β (π β (SubSpβπ) β§ π β CBan)) |
7 | 6 | simpld 493 | . . . . 5 β’ (π β π β (SubSpβπ)) |
8 | minveco.y | . . . . . 6 β’ π = (BaseSetβπ) | |
9 | minveco.d | . . . . . 6 β’ π· = (IndMetβπ) | |
10 | eqid 2725 | . . . . . 6 β’ (IndMetβπ) = (IndMetβπ) | |
11 | eqid 2725 | . . . . . 6 β’ (SubSpβπ) = (SubSpβπ) | |
12 | 8, 9, 10, 11 | sspims 30588 | . . . . 5 β’ ((π β NrmCVec β§ π β (SubSpβπ)) β (IndMetβπ) = (π· βΎ (π Γ π))) |
13 | 3, 7, 12 | syl2anc 582 | . . . 4 β’ (π β (IndMetβπ) = (π· βΎ (π Γ π))) |
14 | 6 | simprd 494 | . . . . 5 β’ (π β π β CBan) |
15 | eqid 2725 | . . . . . 6 β’ (BaseSetβπ) = (BaseSetβπ) | |
16 | 15, 10 | cbncms 30714 | . . . . 5 β’ (π β CBan β (IndMetβπ) β (CMetβ(BaseSetβπ))) |
17 | 14, 16 | syl 17 | . . . 4 β’ (π β (IndMetβπ) β (CMetβ(BaseSetβπ))) |
18 | 13, 17 | eqeltrrd 2826 | . . 3 β’ (π β (π· βΎ (π Γ π)) β (CMetβ(BaseSetβπ))) |
19 | minveco.x | . . . . 5 β’ π = (BaseSetβπ) | |
20 | minveco.m | . . . . 5 β’ π = ( βπ£ βπ) | |
21 | minveco.n | . . . . 5 β’ π = (normCVβπ) | |
22 | minveco.a | . . . . 5 β’ (π β π΄ β π) | |
23 | minveco.j | . . . . 5 β’ π½ = (MetOpenβπ·) | |
24 | minveco.r | . . . . 5 β’ π = ran (π¦ β π β¦ (πβ(π΄ππ¦))) | |
25 | minveco.s | . . . . 5 β’ π = inf(π , β, < ) | |
26 | minveco.f | . . . . 5 β’ (π β πΉ:ββΆπ) | |
27 | minveco.1 | . . . . 5 β’ ((π β§ π β β) β ((π΄π·(πΉβπ))β2) β€ ((πβ2) + (1 / π))) | |
28 | 19, 20, 21, 8, 1, 4, 22, 9, 23, 24, 25, 26, 27 | minvecolem3 30725 | . . . 4 β’ (π β πΉ β (Cauβπ·)) |
29 | 19, 9 | imsmet 30540 | . . . . . . 7 β’ (π β NrmCVec β π· β (Metβπ)) |
30 | 1, 2, 29 | 3syl 18 | . . . . . 6 β’ (π β π· β (Metβπ)) |
31 | metxmet 24253 | . . . . . 6 β’ (π· β (Metβπ) β π· β (βMetβπ)) | |
32 | 30, 31 | syl 17 | . . . . 5 β’ (π β π· β (βMetβπ)) |
33 | causs 25239 | . . . . 5 β’ ((π· β (βMetβπ) β§ πΉ:ββΆπ) β (πΉ β (Cauβπ·) β πΉ β (Cauβ(π· βΎ (π Γ π))))) | |
34 | 32, 26, 33 | syl2anc 582 | . . . 4 β’ (π β (πΉ β (Cauβπ·) β πΉ β (Cauβ(π· βΎ (π Γ π))))) |
35 | 28, 34 | mpbid 231 | . . 3 β’ (π β πΉ β (Cauβ(π· βΎ (π Γ π)))) |
36 | eqid 2725 | . . . 4 β’ (MetOpenβ(π· βΎ (π Γ π))) = (MetOpenβ(π· βΎ (π Γ π))) | |
37 | 36 | cmetcau 25230 | . . 3 β’ (((π· βΎ (π Γ π)) β (CMetβ(BaseSetβπ)) β§ πΉ β (Cauβ(π· βΎ (π Γ π)))) β πΉ β dom (βπ‘β(MetOpenβ(π· βΎ (π Γ π))))) |
38 | 18, 35, 37 | syl2anc 582 | . 2 β’ (π β πΉ β dom (βπ‘β(MetOpenβ(π· βΎ (π Γ π))))) |
39 | xmetres 24283 | . . . 4 β’ (π· β (βMetβπ) β (π· βΎ (π Γ π)) β (βMetβ(π β© π))) | |
40 | 36 | methaus 24442 | . . . 4 β’ ((π· βΎ (π Γ π)) β (βMetβ(π β© π)) β (MetOpenβ(π· βΎ (π Γ π))) β Haus) |
41 | 32, 39, 40 | 3syl 18 | . . 3 β’ (π β (MetOpenβ(π· βΎ (π Γ π))) β Haus) |
42 | lmfun 23298 | . . 3 β’ ((MetOpenβ(π· βΎ (π Γ π))) β Haus β Fun (βπ‘β(MetOpenβ(π· βΎ (π Γ π))))) | |
43 | funfvbrb 7053 | . . 3 β’ (Fun (βπ‘β(MetOpenβ(π· βΎ (π Γ π)))) β (πΉ β dom (βπ‘β(MetOpenβ(π· βΎ (π Γ π)))) β πΉ(βπ‘β(MetOpenβ(π· βΎ (π Γ π))))((βπ‘β(MetOpenβ(π· βΎ (π Γ π))))βπΉ))) | |
44 | 41, 42, 43 | 3syl 18 | . 2 β’ (π β (πΉ β dom (βπ‘β(MetOpenβ(π· βΎ (π Γ π)))) β πΉ(βπ‘β(MetOpenβ(π· βΎ (π Γ π))))((βπ‘β(MetOpenβ(π· βΎ (π Γ π))))βπΉ))) |
45 | 38, 44 | mpbid 231 | 1 β’ (π β πΉ(βπ‘β(MetOpenβ(π· βΎ (π Γ π))))((βπ‘β(MetOpenβ(π· βΎ (π Γ π))))βπΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 β© cin 3940 class class class wbr 5144 β¦ cmpt 5227 Γ cxp 5671 dom cdm 5673 ran crn 5674 βΎ cres 5675 Fun wfun 6537 βΆwf 6539 βcfv 6543 (class class class)co 7413 infcinf 9459 βcr 11132 1c1 11134 + caddc 11136 < clt 11273 β€ cle 11274 / cdiv 11896 βcn 12237 2c2 12292 βcexp 14053 βMetcxmet 21263 Metcmet 21264 MetOpencmopn 21268 βπ‘clm 23143 Hauscha 23225 Cauccau 25194 CMetccmet 25195 NrmCVeccnv 30433 BaseSetcba 30435 βπ£ cnsb 30438 normCVcnmcv 30439 IndMetcims 30440 SubSpcss 30570 CPreHilOLDccphlo 30661 CBanccbn 30711 |
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 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 ax-addf 11212 ax-mulf 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-map 8840 df-pm 8841 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9460 df-inf 9461 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-n0 12498 df-z 12584 df-uz 12848 df-q 12958 df-rp 13002 df-xneg 13119 df-xadd 13120 df-xmul 13121 df-ico 13357 df-icc 13358 df-fl 13784 df-seq 13994 df-exp 14054 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-rest 17398 df-topgen 17419 df-psmet 21270 df-xmet 21271 df-met 21272 df-bl 21273 df-mopn 21274 df-fbas 21275 df-fg 21276 df-top 22809 df-topon 22826 df-bases 22862 df-ntr 22937 df-nei 23015 df-lm 23146 df-haus 23232 df-fil 23763 df-fm 23855 df-flim 23856 df-flf 23857 df-cfil 25196 df-cau 25197 df-cmet 25198 df-grpo 30342 df-gid 30343 df-ginv 30344 df-gdiv 30345 df-ablo 30394 df-vc 30408 df-nv 30441 df-va 30444 df-ba 30445 df-sm 30446 df-0v 30447 df-vs 30448 df-nmcv 30449 df-ims 30450 df-ssp 30571 df-ph 30662 df-cbn 30712 |
This theorem is referenced by: minvecolem4b 30727 minvecolem4 30729 |
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