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Theorem minveclem3a 18791
Description: Lemma for minvec 18800. 
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 2283 . . . 4  |-  ( Base `  ( Us  Y ) )  =  ( Base `  ( Us  Y ) )
3 eqid 2283 . . . 4  |-  ( (
dist `  ( Us  Y
) )  |`  (
( Base `  ( Us  Y
) )  X.  ( Base `  ( Us  Y ) ) ) )  =  ( ( dist `  ( Us  Y ) )  |`  ( ( Base `  ( Us  Y ) )  X.  ( Base `  ( Us  Y ) ) ) )
42, 3cmscmet 18768 . . 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 4951 . . . 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 2283 . . . . . . . . 9  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
119, 10lssss 15694 . . . . . . . 8  |-  ( Y  e.  ( LSubSp `  U
)  ->  Y  C_  X
)
128, 11syl 15 . . . . . . 7  |-  ( ph  ->  Y  C_  X )
13 xpss12 4792 . . . . . . 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 4984 . . . . . 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 2283 . . . . . . . 8  |-  ( Us  Y )  =  ( Us  Y )
18 eqid 2283 . . . . . . . 8  |-  ( dist `  U )  =  (
dist `  U )
1917, 18ressds 13318 . . . . . . 7  |-  ( Y  e.  ( LSubSp `  U
)  ->  ( dist `  U )  =  (
dist `  ( Us  Y
) ) )
208, 19syl 15 . . . . . 6  |-  ( ph  ->  ( dist `  U
)  =  ( dist `  ( Us  Y ) ) )
2117, 9ressbas2 13199 . . . . . . . 8  |-  ( Y 
C_  X  ->  Y  =  ( Base `  ( Us  Y ) ) )
2212, 21syl 15 . . . . . . 7  |-  ( ph  ->  Y  =  ( Base `  ( Us  Y ) ) )
2322, 22xpeq12d 4714 . . . . . 6  |-  ( ph  ->  ( Y  X.  Y
)  =  ( (
Base `  ( Us  Y
) )  X.  ( Base `  ( Us  Y ) ) ) )
2420, 23reseq12d 4956 . . . . 5  |-  ( ph  ->  ( ( dist `  U
)  |`  ( Y  X.  Y ) )  =  ( ( dist `  ( Us  Y ) )  |`  ( ( Base `  ( Us  Y ) )  X.  ( Base `  ( Us  Y ) ) ) ) )
2516, 24eqtrd 2315 . . . 4  |-  ( ph  ->  ( ( ( dist `  U )  |`  ( X  X.  X ) )  |`  ( Y  X.  Y
) )  =  ( ( dist `  ( Us  Y ) )  |`  ( ( Base `  ( Us  Y ) )  X.  ( Base `  ( Us  Y ) ) ) ) )
267, 25syl5eq 2327 . . 3  |-  ( ph  ->  ( D  |`  ( Y  X.  Y ) )  =  ( ( dist `  ( Us  Y ) )  |`  ( ( Base `  ( Us  Y ) )  X.  ( Base `  ( Us  Y ) ) ) ) )
2722fveq2d 5529 . . 3  |-  ( ph  ->  ( CMet `  Y
)  =  ( CMet `  ( Base `  ( Us  Y ) ) ) )
2826, 27eleq12d 2351 . 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 1623    e. wcel 1684    C_ wss 3152    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   ran crn 4690    |` cres 4691   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736    < clt 8867   Basecbs 13148   ↾s cress 13149   distcds 13217   TopOpenctopn 13326   -gcsg 14365   LSubSpclss 15689   normcnm 18099   CPreHilccph 18602   CMetcms 18680  CMetSpccms 18754
This theorem is referenced by:  minveclem3  18793  minveclem4a  18794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-ds 13230  df-lss 15690  df-cms 18757
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