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Mirrors > Home > MPE Home > Th. List > minvecolem4a | Structured version Visualization version GIF version |
Description: Lemma for minveco 29868. 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 29798 | . . . . . 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 3927 | . . . . . . 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 496 | . . . . 5 β’ (π β π β (SubSpβπ)) |
8 | minveco.y | . . . . . 6 β’ π = (BaseSetβπ) | |
9 | minveco.d | . . . . . 6 β’ π· = (IndMetβπ) | |
10 | eqid 2733 | . . . . . 6 β’ (IndMetβπ) = (IndMetβπ) | |
11 | eqid 2733 | . . . . . 6 β’ (SubSpβπ) = (SubSpβπ) | |
12 | 8, 9, 10, 11 | sspims 29723 | . . . . 5 β’ ((π β NrmCVec β§ π β (SubSpβπ)) β (IndMetβπ) = (π· βΎ (π Γ π))) |
13 | 3, 7, 12 | syl2anc 585 | . . . 4 β’ (π β (IndMetβπ) = (π· βΎ (π Γ π))) |
14 | 6 | simprd 497 | . . . . 5 β’ (π β π β CBan) |
15 | eqid 2733 | . . . . . 6 β’ (BaseSetβπ) = (BaseSetβπ) | |
16 | 15, 10 | cbncms 29849 | . . . . 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 2835 | . . 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 29860 | . . . 4 β’ (π β πΉ β (Cauβπ·)) |
29 | 19, 9 | imsmet 29675 | . . . . . . 7 β’ (π β NrmCVec β π· β (Metβπ)) |
30 | 1, 2, 29 | 3syl 18 | . . . . . 6 β’ (π β π· β (Metβπ)) |
31 | metxmet 23703 | . . . . . 6 β’ (π· β (Metβπ) β π· β (βMetβπ)) | |
32 | 30, 31 | syl 17 | . . . . 5 β’ (π β π· β (βMetβπ)) |
33 | causs 24678 | . . . . 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 585 | . . . 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 2733 | . . . 4 β’ (MetOpenβ(π· βΎ (π Γ π))) = (MetOpenβ(π· βΎ (π Γ π))) | |
37 | 36 | cmetcau 24669 | . . 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 585 | . 2 β’ (π β πΉ β dom (βπ‘β(MetOpenβ(π· βΎ (π Γ π))))) |
39 | xmetres 23733 | . . . 4 β’ (π· β (βMetβπ) β (π· βΎ (π Γ π)) β (βMetβ(π β© π))) | |
40 | 36 | methaus 23892 | . . . 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 22748 | . . 3 β’ ((MetOpenβ(π· βΎ (π Γ π))) β Haus β Fun (βπ‘β(MetOpenβ(π· βΎ (π Γ π))))) | |
43 | funfvbrb 7002 | . . 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 397 = wceq 1542 β wcel 2107 β© cin 3910 class class class wbr 5106 β¦ cmpt 5189 Γ cxp 5632 dom cdm 5634 ran crn 5635 βΎ cres 5636 Fun wfun 6491 βΆwf 6493 βcfv 6497 (class class class)co 7358 infcinf 9382 βcr 11055 1c1 11057 + caddc 11059 < clt 11194 β€ cle 11195 / cdiv 11817 βcn 12158 2c2 12213 βcexp 13973 βMetcxmet 20797 Metcmet 20798 MetOpencmopn 20802 βπ‘clm 22593 Hauscha 22675 Cauccau 24633 CMetccmet 24634 NrmCVeccnv 29568 BaseSetcba 29570 βπ£ cnsb 29573 normCVcnmcv 29574 IndMetcims 29575 SubSpcss 29705 CPreHilOLDccphlo 29796 CBanccbn 29846 |
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 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 ax-addf 11135 ax-mulf 11136 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-map 8770 df-pm 8771 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-n0 12419 df-z 12505 df-uz 12769 df-q 12879 df-rp 12921 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ico 13276 df-icc 13277 df-fl 13703 df-seq 13913 df-exp 13974 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-rest 17309 df-topgen 17330 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-fbas 20809 df-fg 20810 df-top 22259 df-topon 22276 df-bases 22312 df-ntr 22387 df-nei 22465 df-lm 22596 df-haus 22682 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 df-cfil 24635 df-cau 24636 df-cmet 24637 df-grpo 29477 df-gid 29478 df-ginv 29479 df-gdiv 29480 df-ablo 29529 df-vc 29543 df-nv 29576 df-va 29579 df-ba 29580 df-sm 29581 df-0v 29582 df-vs 29583 df-nmcv 29584 df-ims 29585 df-ssp 29706 df-ph 29797 df-cbn 29847 |
This theorem is referenced by: minvecolem4b 29862 minvecolem4 29864 |
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