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Theorem minveclem3a 18807
Description: Lemma for minvec 18816. 
D is a complete metric when restricted to  Y. (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
Hypotheses
Ref Expression
minvec.x  |-  X  =  ( Base `  U
)
minvec.m  |-  .-  =  ( -g `  U )
minvec.n  |-  N  =  ( norm `  U
)
minvec.u  |-  ( ph  ->  U  e.  CPreHil )
minvec.y  |-  ( ph  ->  Y  e.  ( LSubSp `  U ) )
minvec.w  |-  ( ph  ->  ( Us  Y )  e. CMetSp )
minvec.a  |-  ( ph  ->  A  e.  X )
minvec.j  |-  J  =  ( TopOpen `  U )
minvec.r  |-  R  =  ran  ( y  e.  Y  |->  ( N `  ( A  .-  y ) ) )
minvec.s  |-  S  =  sup ( R ,  RR ,  `'  <  )
minvec.d  |-  D  =  ( ( dist `  U
)  |`  ( X  X.  X ) )
Assertion
Ref Expression
minveclem3a  |-  ( ph  ->  ( D  |`  ( Y  X.  Y ) )  e.  ( CMet `  Y
) )
Distinct variable groups:    y,  .-    y, A    y, J    y, N    ph, y    y, R   
y, U    y, X    y, Y    y, D    y, S

Proof of Theorem minveclem3a
StepHypRef Expression
1 minvec.w . . 3  |-  ( ph  ->  ( Us  Y )  e. CMetSp )
2 eqid 2296 . . . 4  |-  ( Base `  ( Us  Y ) )  =  ( Base `  ( Us  Y ) )
3 eqid 2296 . . . 4  |-  ( (
dist `  ( Us  Y
) )  |`  (
( Base `  ( Us  Y
) )  X.  ( Base `  ( Us  Y ) ) ) )  =  ( ( dist `  ( Us  Y ) )  |`  ( ( Base `  ( Us  Y ) )  X.  ( Base `  ( Us  Y ) ) ) )
42, 3cmscmet 18784 . . 3  |-  ( ( Us  Y )  e. CMetSp  ->  ( ( dist `  ( Us  Y ) )  |`  ( ( Base `  ( Us  Y ) )  X.  ( Base `  ( Us  Y ) ) ) )  e.  ( CMet `  ( Base `  ( Us  Y ) ) ) )
51, 4syl 15 . 2  |-  ( ph  ->  ( ( dist `  ( Us  Y ) )  |`  ( ( Base `  ( Us  Y ) )  X.  ( Base `  ( Us  Y ) ) ) )  e.  ( CMet `  ( Base `  ( Us  Y ) ) ) )
6 minvec.d . . . . 5  |-  D  =  ( ( dist `  U
)  |`  ( X  X.  X ) )
76reseq1i 4967 . . . 4  |-  ( D  |`  ( Y  X.  Y
) )  =  ( ( ( dist `  U
)  |`  ( X  X.  X ) )  |`  ( Y  X.  Y
) )
8 minvec.y . . . . . . . 8  |-  ( ph  ->  Y  e.  ( LSubSp `  U ) )
9 minvec.x . . . . . . . . 9  |-  X  =  ( Base `  U
)
10 eqid 2296 . . . . . . . . 9  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
119, 10lssss 15710 . . . . . . . 8  |-  ( Y  e.  ( LSubSp `  U
)  ->  Y  C_  X
)
128, 11syl 15 . . . . . . 7  |-  ( ph  ->  Y  C_  X )
13 xpss12 4808 . . . . . . 7  |-  ( ( Y  C_  X  /\  Y  C_  X )  -> 
( Y  X.  Y
)  C_  ( X  X.  X ) )
1412, 12, 13syl2anc 642 . . . . . 6  |-  ( ph  ->  ( Y  X.  Y
)  C_  ( X  X.  X ) )
15 resabs1 5000 . . . . . 6  |-  ( ( Y  X.  Y ) 
C_  ( X  X.  X )  ->  (
( ( dist `  U
)  |`  ( X  X.  X ) )  |`  ( Y  X.  Y
) )  =  ( ( dist `  U
)  |`  ( Y  X.  Y ) ) )
1614, 15syl 15 . . . . 5  |-  ( ph  ->  ( ( ( dist `  U )  |`  ( X  X.  X ) )  |`  ( Y  X.  Y
) )  =  ( ( dist `  U
)  |`  ( Y  X.  Y ) ) )
17 eqid 2296 . . . . . . . 8  |-  ( Us  Y )  =  ( Us  Y )
18 eqid 2296 . . . . . . . 8  |-  ( dist `  U )  =  (
dist `  U )
1917, 18ressds 13334 . . . . . . 7  |-  ( Y  e.  ( LSubSp `  U
)  ->  ( dist `  U )  =  (
dist `  ( Us  Y
) ) )
208, 19syl 15 . . . . . 6  |-  ( ph  ->  ( dist `  U
)  =  ( dist `  ( Us  Y ) ) )
2117, 9ressbas2 13215 . . . . . . . 8  |-  ( Y 
C_  X  ->  Y  =  ( Base `  ( Us  Y ) ) )
2212, 21syl 15 . . . . . . 7  |-  ( ph  ->  Y  =  ( Base `  ( Us  Y ) ) )
2322, 22xpeq12d 4730 . . . . . 6  |-  ( ph  ->  ( Y  X.  Y
)  =  ( (
Base `  ( Us  Y
) )  X.  ( Base `  ( Us  Y ) ) ) )
2420, 23reseq12d 4972 . . . . 5  |-  ( ph  ->  ( ( dist `  U
)  |`  ( Y  X.  Y ) )  =  ( ( dist `  ( Us  Y ) )  |`  ( ( Base `  ( Us  Y ) )  X.  ( Base `  ( Us  Y ) ) ) ) )
2516, 24eqtrd 2328 . . . 4  |-  ( ph  ->  ( ( ( dist `  U )  |`  ( X  X.  X ) )  |`  ( Y  X.  Y
) )  =  ( ( dist `  ( Us  Y ) )  |`  ( ( Base `  ( Us  Y ) )  X.  ( Base `  ( Us  Y ) ) ) ) )
267, 25syl5eq 2340 . . 3  |-  ( ph  ->  ( D  |`  ( Y  X.  Y ) )  =  ( ( dist `  ( Us  Y ) )  |`  ( ( Base `  ( Us  Y ) )  X.  ( Base `  ( Us  Y ) ) ) ) )
2722fveq2d 5545 . . 3  |-  ( ph  ->  ( CMet `  Y
)  =  ( CMet `  ( Base `  ( Us  Y ) ) ) )
2826, 27eleq12d 2364 . 2  |-  ( ph  ->  ( ( D  |`  ( Y  X.  Y
) )  e.  (
CMet `  Y )  <->  ( ( dist `  ( Us  Y ) )  |`  ( ( Base `  ( Us  Y ) )  X.  ( Base `  ( Us  Y ) ) ) )  e.  ( CMet `  ( Base `  ( Us  Y ) ) ) ) )
295, 28mpbird 223 1  |-  ( ph  ->  ( D  |`  ( Y  X.  Y ) )  e.  ( CMet `  Y
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    C_ wss 3165    e. cmpt 4093    X. cxp 4703   `'ccnv 4704   ran crn 4706    |` cres 4707   ` cfv 5271  (class class class)co 5874   supcsup 7209   RRcr 8752    < clt 8883   Basecbs 13164   ↾s cress 13165   distcds 13233   TopOpenctopn 13342   -gcsg 14381   LSubSpclss 15705   normcnm 18115   CPreHilccph 18618   CMetcms 18696  CMetSpccms 18770
This theorem is referenced by:  minveclem3  18809  minveclem4a  18810
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-ds 13246  df-lss 15706  df-cms 18773
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