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Theorem xpsmet 18395
Description: The direct product of two metric spaces. Definition 14-1.5 of [Gleason] p. 225. (Contributed by NM, 20-Jun-2007.) (Revised by Mario Carneiro, 20-Aug-2015.)
Hypotheses
Ref Expression
xpsds.t  |-  T  =  ( R  X.s  S )
xpsds.x  |-  X  =  ( Base `  R
)
xpsds.y  |-  Y  =  ( Base `  S
)
xpsds.1  |-  ( ph  ->  R  e.  V )
xpsds.2  |-  ( ph  ->  S  e.  W )
xpsds.p  |-  P  =  ( dist `  T
)
xpsds.m  |-  M  =  ( ( dist `  R
)  |`  ( X  X.  X ) )
xpsds.n  |-  N  =  ( ( dist `  S
)  |`  ( Y  X.  Y ) )
xpsmet.3  |-  ( ph  ->  M  e.  ( Met `  X ) )
xpsmet.4  |-  ( ph  ->  N  e.  ( Met `  Y ) )
Assertion
Ref Expression
xpsmet  |-  ( ph  ->  P  e.  ( Met `  ( X  X.  Y
) ) )

Proof of Theorem xpsmet
Dummy variables  x  k  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpsds.t . . 3  |-  T  =  ( R  X.s  S )
2 xpsds.x . . 3  |-  X  =  ( Base `  R
)
3 xpsds.y . . 3  |-  Y  =  ( Base `  S
)
4 xpsds.1 . . 3  |-  ( ph  ->  R  e.  V )
5 xpsds.2 . . 3  |-  ( ph  ->  S  e.  W )
6 eqid 2430 . . 3  |-  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )  =  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )
7 eqid 2430 . . 3  |-  (Scalar `  R )  =  (Scalar `  R )
8 eqid 2430 . . 3  |-  ( (Scalar `  R ) X_s `' ( { R }  +c  { S }
) )  =  ( (Scalar `  R ) X_s `' ( { R }  +c  { S } ) )
91, 2, 3, 4, 5, 6, 7, 8xpsval 13780 . 2  |-  ( ph  ->  T  =  ( `' ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )  "s  ( (Scalar `  R ) X_s `' ( { R }  +c  { S }
) ) ) )
101, 2, 3, 4, 5, 6, 7, 8xpslem 13781 . 2  |-  ( ph  ->  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )  =  ( Base `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) ) )
116xpsff1o2 13779 . . 3  |-  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) : ( X  X.  Y ) -1-1-onto-> ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )
12 f1ocnv 5673 . . 3  |-  ( ( x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) : ( X  X.  Y ) -1-1-onto-> ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) )  ->  `' (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) : ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) -1-1-onto-> ( X  X.  Y
) )
1311, 12mp1i 12 . 2  |-  ( ph  ->  `' ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) ) : ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) -1-1-onto-> ( X  X.  Y ) )
14 ovex 6092 . . 3  |-  ( (Scalar `  R ) X_s `' ( { R }  +c  { S }
) )  e.  _V
1514a1i 11 . 2  |-  ( ph  ->  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) )  e.  _V )
16 eqid 2430 . 2  |-  ( (
dist `  ( (Scalar `  R ) X_s `' ( { R }  +c  { S }
) ) )  |`  ( ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )  X.  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) ) )  =  ( ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  |`  ( ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )  X.  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) ) )
17 xpsds.p . 2  |-  P  =  ( dist `  T
)
18 eqid 2430 . . . . 5  |-  ( (Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) )  =  ( (Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) )
19 eqid 2430 . . . . 5  |-  ( Base `  ( (Scalar `  R
) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) )  =  ( Base `  (
(Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) )
20 eqid 2430 . . . . 5  |-  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
Base `  ( `' ( { R }  +c  { S } ) `  k ) )
21 eqid 2430 . . . . 5  |-  ( (
dist `  ( `' ( { R }  +c  { S } ) `  k ) )  |`  ( ( Base `  ( `' ( { R }  +c  { S }
) `  k )
)  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) ) )  =  ( ( dist `  ( `' ( { R }  +c  { S } ) `  k
) )  |`  (
( Base `  ( `' ( { R }  +c  { S } ) `  k ) )  X.  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) ) )
22 eqid 2430 . . . . 5  |-  ( dist `  ( (Scalar `  R
) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) )  =  ( dist `  (
(Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) )
23 fvex 5728 . . . . . 6  |-  (Scalar `  R )  e.  _V
2423a1i 11 . . . . 5  |-  ( ph  ->  (Scalar `  R )  e.  _V )
25 2onn 6869 . . . . . 6  |-  2o  e.  om
26 nnfi 7285 . . . . . 6  |-  ( 2o  e.  om  ->  2o  e.  Fin )
2725, 26mp1i 12 . . . . 5  |-  ( ph  ->  2o  e.  Fin )
28 fvex 5728 . . . . . 6  |-  ( `' ( { R }  +c  { S } ) `
 k )  e. 
_V
2928a1i 11 . . . . 5  |-  ( (
ph  /\  k  e.  2o )  ->  ( `' ( { R }  +c  { S } ) `
 k )  e. 
_V )
30 elpri 3821 . . . . . . 7  |-  ( k  e.  { (/) ,  1o }  ->  ( k  =  (/)  \/  k  =  1o ) )
31 df2o3 6723 . . . . . . 7  |-  2o  =  { (/) ,  1o }
3230, 31eleq2s 2522 . . . . . 6  |-  ( k  e.  2o  ->  (
k  =  (/)  \/  k  =  1o ) )
33 xpsmet.3 . . . . . . . . 9  |-  ( ph  ->  M  e.  ( Met `  X ) )
3433adantr 452 . . . . . . . 8  |-  ( (
ph  /\  k  =  (/) )  ->  M  e.  ( Met `  X ) )
35 fveq2 5714 . . . . . . . . . . . 12  |-  ( k  =  (/)  ->  ( `' ( { R }  +c  { S } ) `
 k )  =  ( `' ( { R }  +c  { S } ) `  (/) ) )
36 xpsc0 13768 . . . . . . . . . . . . 13  |-  ( R  e.  V  ->  ( `' ( { R }  +c  { S }
) `  (/) )  =  R )
374, 36syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' ( { R }  +c  { S } ) `  (/) )  =  R )
3835, 37sylan9eqr 2484 . . . . . . . . . . 11  |-  ( (
ph  /\  k  =  (/) )  ->  ( `' ( { R }  +c  { S } ) `  k )  =  R )
3938fveq2d 5718 . . . . . . . . . 10  |-  ( (
ph  /\  k  =  (/) )  ->  ( dist `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
dist `  R )
)
4038fveq2d 5718 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  =  (/) )  ->  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
Base `  R )
)
4140, 2syl6eqr 2480 . . . . . . . . . . 11  |-  ( (
ph  /\  k  =  (/) )  ->  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  X )
4241, 41xpeq12d 4889 . . . . . . . . . 10  |-  ( (
ph  /\  k  =  (/) )  ->  ( ( Base `  ( `' ( { R }  +c  { S } ) `  k ) )  X.  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) )  =  ( X  X.  X ) )
4339, 42reseq12d 5133 . . . . . . . . 9  |-  ( (
ph  /\  k  =  (/) )  ->  ( ( dist `  ( `' ( { R }  +c  { S } ) `  k ) )  |`  ( ( Base `  ( `' ( { R }  +c  { S }
) `  k )
)  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) ) )  =  ( ( dist `  R )  |`  ( X  X.  X ) ) )
44 xpsds.m . . . . . . . . 9  |-  M  =  ( ( dist `  R
)  |`  ( X  X.  X ) )
4543, 44syl6eqr 2480 . . . . . . . 8  |-  ( (
ph  /\  k  =  (/) )  ->  ( ( dist `  ( `' ( { R }  +c  { S } ) `  k ) )  |`  ( ( Base `  ( `' ( { R }  +c  { S }
) `  k )
)  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) ) )  =  M )
4641fveq2d 5718 . . . . . . . 8  |-  ( (
ph  /\  k  =  (/) )  ->  ( Met `  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) )  =  ( Met `  X ) )
4734, 45, 463eltr4d 2511 . . . . . . 7  |-  ( (
ph  /\  k  =  (/) )  ->  ( ( dist `  ( `' ( { R }  +c  { S } ) `  k ) )  |`  ( ( Base `  ( `' ( { R }  +c  { S }
) `  k )
)  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) ) )  e.  ( Met `  ( Base `  ( `' ( { R }  +c  { S } ) `  k ) ) ) )
48 xpsmet.4 . . . . . . . . 9  |-  ( ph  ->  N  e.  ( Met `  Y ) )
4948adantr 452 . . . . . . . 8  |-  ( (
ph  /\  k  =  1o )  ->  N  e.  ( Met `  Y
) )
50 fveq2 5714 . . . . . . . . . . . 12  |-  ( k  =  1o  ->  ( `' ( { R }  +c  { S }
) `  k )  =  ( `' ( { R }  +c  { S } ) `  1o ) )
51 xpsc1 13769 . . . . . . . . . . . . 13  |-  ( S  e.  W  ->  ( `' ( { R }  +c  { S }
) `  1o )  =  S )
525, 51syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' ( { R }  +c  { S } ) `  1o )  =  S )
5350, 52sylan9eqr 2484 . . . . . . . . . . 11  |-  ( (
ph  /\  k  =  1o )  ->  ( `' ( { R }  +c  { S } ) `
 k )  =  S )
5453fveq2d 5718 . . . . . . . . . 10  |-  ( (
ph  /\  k  =  1o )  ->  ( dist `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
dist `  S )
)
5553fveq2d 5718 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  =  1o )  ->  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
Base `  S )
)
5655, 3syl6eqr 2480 . . . . . . . . . . 11  |-  ( (
ph  /\  k  =  1o )  ->  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  Y )
5756, 56xpeq12d 4889 . . . . . . . . . 10  |-  ( (
ph  /\  k  =  1o )  ->  ( (
Base `  ( `' ( { R }  +c  { S } ) `  k ) )  X.  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) )  =  ( Y  X.  Y ) )
5854, 57reseq12d 5133 . . . . . . . . 9  |-  ( (
ph  /\  k  =  1o )  ->  ( (
dist `  ( `' ( { R }  +c  { S } ) `  k ) )  |`  ( ( Base `  ( `' ( { R }  +c  { S }
) `  k )
)  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) ) )  =  ( ( dist `  S )  |`  ( Y  X.  Y ) ) )
59 xpsds.n . . . . . . . . 9  |-  N  =  ( ( dist `  S
)  |`  ( Y  X.  Y ) )
6058, 59syl6eqr 2480 . . . . . . . 8  |-  ( (
ph  /\  k  =  1o )  ->  ( (
dist `  ( `' ( { R }  +c  { S } ) `  k ) )  |`  ( ( Base `  ( `' ( { R }  +c  { S }
) `  k )
)  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) ) )  =  N )
6156fveq2d 5718 . . . . . . . 8  |-  ( (
ph  /\  k  =  1o )  ->  ( Met `  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) )  =  ( Met `  Y ) )
6249, 60, 613eltr4d 2511 . . . . . . 7  |-  ( (
ph  /\  k  =  1o )  ->  ( (
dist `  ( `' ( { R }  +c  { S } ) `  k ) )  |`  ( ( Base `  ( `' ( { R }  +c  { S }
) `  k )
)  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) ) )  e.  ( Met `  ( Base `  ( `' ( { R }  +c  { S } ) `  k ) ) ) )
6347, 62jaodan 761 . . . . . 6  |-  ( (
ph  /\  ( k  =  (/)  \/  k  =  1o ) )  -> 
( ( dist `  ( `' ( { R }  +c  { S }
) `  k )
)  |`  ( ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k ) ) ) )  e.  ( Met `  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) ) )
6432, 63sylan2 461 . . . . 5  |-  ( (
ph  /\  k  e.  2o )  ->  ( (
dist `  ( `' ( { R }  +c  { S } ) `  k ) )  |`  ( ( Base `  ( `' ( { R }  +c  { S }
) `  k )
)  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) ) )  e.  ( Met `  ( Base `  ( `' ( { R }  +c  { S } ) `  k ) ) ) )
6518, 19, 20, 21, 22, 24, 27, 29, 64prdsmet 18383 . . . 4  |-  ( ph  ->  ( dist `  (
(Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) )  e.  ( Met `  ( Base `  ( (Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) ) ) )
66 xpscfn 13767 . . . . . . . 8  |-  ( ( R  e.  V  /\  S  e.  W )  ->  `' ( { R }  +c  { S }
)  Fn  2o )
674, 5, 66syl2anc 643 . . . . . . 7  |-  ( ph  ->  `' ( { R }  +c  { S }
)  Fn  2o )
68 dffn5 5758 . . . . . . 7  |-  ( `' ( { R }  +c  { S } )  Fn  2o  <->  `' ( { R }  +c  { S } )  =  ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) )
6967, 68sylib 189 . . . . . 6  |-  ( ph  ->  `' ( { R }  +c  { S }
)  =  ( k  e.  2o  |->  ( `' ( { R }  +c  { S } ) `
 k ) ) )
7069oveq2d 6083 . . . . 5  |-  ( ph  ->  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) )  =  ( (Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) )
7170fveq2d 5718 . . . 4  |-  ( ph  ->  ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  =  (
dist `  ( (Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) ) )
7270fveq2d 5718 . . . . . 6  |-  ( ph  ->  ( Base `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  =  (
Base `  ( (Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) ) )
7310, 72eqtrd 2462 . . . . 5  |-  ( ph  ->  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )  =  ( Base `  (
(Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) ) )
7473fveq2d 5718 . . . 4  |-  ( ph  ->  ( Met `  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) )  =  ( Met `  ( Base `  ( (Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) ) ) )
7565, 71, 743eltr4d 2511 . . 3  |-  ( ph  ->  ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  e.  ( Met `  ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) ) )
76 ssid 3354 . . 3  |-  ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) )  C_  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )
77 metres2 18376 . . 3  |-  ( ( ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  e.  ( Met `  ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) )  /\  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )  C_  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) )  -> 
( ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  |`  ( ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )  X.  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) ) )  e.  ( Met `  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) ) )
7875, 76, 77sylancl 644 . 2  |-  ( ph  ->  ( ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  |`  ( ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )  X.  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) ) )  e.  ( Met `  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) ) )
799, 10, 13, 15, 16, 17, 78imasf1omet 18389 1  |-  ( ph  ->  P  e.  ( Met `  ( X  X.  Y
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2943    C_ wss 3307   (/)c0 3615   {csn 3801   {cpr 3802    e. cmpt 4253   omcom 4831    X. cxp 4862   `'ccnv 4863   ran crn 4865    |` cres 4866    Fn wfn 5435   -1-1-onto->wf1o 5439   ` cfv 5440  (class class class)co 6067    e. cmpt2 6069   1oc1o 6703   2oc2o 6704   Fincfn 7095    +c ccda 8031   Basecbs 13452  Scalarcsca 13515   distcds 13521   X_scprds 13652    X.s cxps 13715   Metcme 16670
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-inf2 7580  ax-cnex 9030  ax-resscn 9031  ax-1cn 9032  ax-icn 9033  ax-addcl 9034  ax-addrcl 9035  ax-mulcl 9036  ax-mulrcl 9037  ax-mulcom 9038  ax-addass 9039  ax-mulass 9040  ax-distr 9041  ax-i2m1 9042  ax-1ne0 9043  ax-1rid 9044  ax-rnegex 9045  ax-rrecex 9046  ax-cnre 9047  ax-pre-lttri 9048  ax-pre-lttrn 9049  ax-pre-ltadd 9050  ax-pre-mulgt0 9051  ax-pre-sup 9052
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-int 4038  df-iun 4082  df-iin 4083  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-se 4529  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-isom 5449  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-of 6291  df-1st 6335  df-2nd 6336  df-riota 6535  df-recs 6619  df-rdg 6654  df-1o 6710  df-2o 6711  df-oadd 6714  df-er 6891  df-map 7006  df-ixp 7050  df-en 7096  df-dom 7097  df-sdom 7098  df-fin 7099  df-sup 7432  df-oi 7463  df-card 7810  df-cda 8032  df-pnf 9106  df-mnf 9107  df-xr 9108  df-ltxr 9109  df-le 9110  df-sub 9277  df-neg 9278  df-div 9662  df-nn 9985  df-2 10042  df-3 10043  df-4 10044  df-5 10045  df-6 10046  df-7 10047  df-8 10048  df-9 10049  df-10 10050  df-n0 10206  df-z 10267  df-dec 10367  df-uz 10473  df-rp 10597  df-xneg 10694  df-xadd 10695  df-xmul 10696  df-icc 10907  df-fz 11028  df-fzo 11119  df-seq 11307  df-hash 11602  df-struct 13454  df-ndx 13455  df-slot 13456  df-base 13457  df-sets 13458  df-ress 13459  df-plusg 13525  df-mulr 13526  df-sca 13528  df-vsca 13529  df-tset 13531  df-ple 13532  df-ds 13534  df-hom 13536  df-cco 13537  df-prds 13654  df-xrs 13709  df-0g 13710  df-gsum 13711  df-imas 13717  df-xps 13719  df-mre 13794  df-mrc 13795  df-acs 13797  df-mnd 14673  df-submnd 14722  df-mulg 14798  df-cntz 15099  df-cmn 15397  df-xmet 16678  df-met 16679
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