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Theorem xpsmet 19957
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 2443 . . 3  |-  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )  =  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )
7 eqid 2443 . . 3  |-  (Scalar `  R )  =  (Scalar `  R )
8 eqid 2443 . . 3  |-  ( (Scalar `  R ) X_s `' ( { R }  +c  { S }
) )  =  ( (Scalar `  R ) X_s `' ( { R }  +c  { S } ) )
91, 2, 3, 4, 5, 6, 7, 8xpsval 14510 . 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 14511 . 2  |-  ( ph  ->  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )  =  ( Base `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) ) )
116xpsff1o2 14509 . . 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 5653 . . 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 6116 . . 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 2443 . 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 2443 . . . . 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 2443 . . . . 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 2443 . . . . 5  |-  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
Base `  ( `' ( { R }  +c  { S } ) `  k ) )
21 eqid 2443 . . . . 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 2443 . . . . 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 5701 . . . . . 6  |-  (Scalar `  R )  e.  _V
2423a1i 11 . . . . 5  |-  ( ph  ->  (Scalar `  R )  e.  _V )
25 2onn 7079 . . . . . 6  |-  2o  e.  om
26 nnfi 7503 . . . . . 6  |-  ( 2o  e.  om  ->  2o  e.  Fin )
2725, 26mp1i 12 . . . . 5  |-  ( ph  ->  2o  e.  Fin )
28 fvex 5701 . . . . . 6  |-  ( `' ( { R }  +c  { S } ) `
 k )  e. 
_V
2928a1i 11 . . . . 5  |-  ( (
ph  /\  k  e.  2o )  ->  ( `' ( { R }  +c  { S } ) `
 k )  e. 
_V )
30 elpri 3897 . . . . . . 7  |-  ( k  e.  { (/) ,  1o }  ->  ( k  =  (/)  \/  k  =  1o ) )
31 df2o3 6933 . . . . . . 7  |-  2o  =  { (/) ,  1o }
3230, 31eleq2s 2535 . . . . . 6  |-  ( k  e.  2o  ->  (
k  =  (/)  \/  k  =  1o ) )
33 xpsmet.3 . . . . . . . . 9  |-  ( ph  ->  M  e.  ( Met `  X ) )
3433adantr 465 . . . . . . . 8  |-  ( (
ph  /\  k  =  (/) )  ->  M  e.  ( Met `  X ) )
35 fveq2 5691 . . . . . . . . . . . 12  |-  ( k  =  (/)  ->  ( `' ( { R }  +c  { S } ) `
 k )  =  ( `' ( { R }  +c  { S } ) `  (/) ) )
36 xpsc0 14498 . . . . . . . . . . . . 13  |-  ( R  e.  V  ->  ( `' ( { R }  +c  { S }
) `  (/) )  =  R )
374, 36syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' ( { R }  +c  { S } ) `  (/) )  =  R )
3835, 37sylan9eqr 2497 . . . . . . . . . . 11  |-  ( (
ph  /\  k  =  (/) )  ->  ( `' ( { R }  +c  { S } ) `  k )  =  R )
3938fveq2d 5695 . . . . . . . . . 10  |-  ( (
ph  /\  k  =  (/) )  ->  ( dist `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
dist `  R )
)
4038fveq2d 5695 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  =  (/) )  ->  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
Base `  R )
)
4140, 2syl6eqr 2493 . . . . . . . . . . 11  |-  ( (
ph  /\  k  =  (/) )  ->  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  X )
4241, 41xpeq12d 4865 . . . . . . . . . 10  |-  ( (
ph  /\  k  =  (/) )  ->  ( ( Base `  ( `' ( { R }  +c  { S } ) `  k ) )  X.  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) )  =  ( X  X.  X ) )
4339, 42reseq12d 5111 . . . . . . . . 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 2493 . . . . . . . 8  |-  ( (
ph  /\  k  =  (/) )  ->  ( ( dist `  ( `' ( { R }  +c  { S } ) `  k ) )  |`  ( ( Base `  ( `' ( { R }  +c  { S }
) `  k )
)  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) ) )  =  M )
4641fveq2d 5695 . . . . . . . 8  |-  ( (
ph  /\  k  =  (/) )  ->  ( Met `  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) )  =  ( Met `  X ) )
4734, 45, 463eltr4d 2524 . . . . . . 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 465 . . . . . . . 8  |-  ( (
ph  /\  k  =  1o )  ->  N  e.  ( Met `  Y
) )
50 fveq2 5691 . . . . . . . . . . . 12  |-  ( k  =  1o  ->  ( `' ( { R }  +c  { S }
) `  k )  =  ( `' ( { R }  +c  { S } ) `  1o ) )
51 xpsc1 14499 . . . . . . . . . . . . 13  |-  ( S  e.  W  ->  ( `' ( { R }  +c  { S }
) `  1o )  =  S )
525, 51syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' ( { R }  +c  { S } ) `  1o )  =  S )
5350, 52sylan9eqr 2497 . . . . . . . . . . 11  |-  ( (
ph  /\  k  =  1o )  ->  ( `' ( { R }  +c  { S } ) `
 k )  =  S )
5453fveq2d 5695 . . . . . . . . . 10  |-  ( (
ph  /\  k  =  1o )  ->  ( dist `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
dist `  S )
)
5553fveq2d 5695 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  =  1o )  ->  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
Base `  S )
)
5655, 3syl6eqr 2493 . . . . . . . . . . 11  |-  ( (
ph  /\  k  =  1o )  ->  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  Y )
5756, 56xpeq12d 4865 . . . . . . . . . 10  |-  ( (
ph  /\  k  =  1o )  ->  ( (
Base `  ( `' ( { R }  +c  { S } ) `  k ) )  X.  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) )  =  ( Y  X.  Y ) )
5854, 57reseq12d 5111 . . . . . . . . 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 2493 . . . . . . . 8  |-  ( (
ph  /\  k  =  1o )  ->  ( (
dist `  ( `' ( { R }  +c  { S } ) `  k ) )  |`  ( ( Base `  ( `' ( { R }  +c  { S }
) `  k )
)  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) ) )  =  N )
6156fveq2d 5695 . . . . . . . 8  |-  ( (
ph  /\  k  =  1o )  ->  ( Met `  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) )  =  ( Met `  Y ) )
6249, 60, 613eltr4d 2524 . . . . . . 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 783 . . . . . 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 474 . . . . 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 19945 . . . 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 14497 . . . . . . . 8  |-  ( ( R  e.  V  /\  S  e.  W )  ->  `' ( { R }  +c  { S }
)  Fn  2o )
674, 5, 66syl2anc 661 . . . . . . 7  |-  ( ph  ->  `' ( { R }  +c  { S }
)  Fn  2o )
68 dffn5 5737 . . . . . . 7  |-  ( `' ( { R }  +c  { S } )  Fn  2o  <->  `' ( { R }  +c  { S } )  =  ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) )
6967, 68sylib 196 . . . . . 6  |-  ( ph  ->  `' ( { R }  +c  { S }
)  =  ( k  e.  2o  |->  ( `' ( { R }  +c  { S } ) `
 k ) ) )
7069oveq2d 6107 . . . . 5  |-  ( ph  ->  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) )  =  ( (Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) )
7170fveq2d 5695 . . . 4  |-  ( ph  ->  ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  =  (
dist `  ( (Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) ) )
7270fveq2d 5695 . . . . . 6  |-  ( ph  ->  ( Base `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  =  (
Base `  ( (Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) ) )
7310, 72eqtrd 2475 . . . . 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 5695 . . . 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 2524 . . 3  |-  ( ph  ->  ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  e.  ( Met `  ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) ) )
76 ssid 3375 . . 3  |-  ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) )  C_  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )
77 metres2 19938 . . 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 662 . 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 19951 1  |-  ( ph  ->  P  e.  ( Met `  ( X  X.  Y
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2972    C_ wss 3328   (/)c0 3637   {csn 3877   {cpr 3879    e. cmpt 4350    X. cxp 4838   `'ccnv 4839   ran crn 4841    |` cres 4842    Fn wfn 5413   -1-1-onto->wf1o 5417   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   omcom 6476   1oc1o 6913   2oc2o 6914   Fincfn 7310    +c ccda 8336   Basecbs 14174  Scalarcsca 14241   distcds 14247   X_scprds 14384    X.s cxps 14444   Metcme 17802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-icc 11307  df-fz 11438  df-fzo 11549  df-seq 11807  df-hash 12104  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-hom 14262  df-cco 14263  df-0g 14380  df-gsum 14381  df-prds 14386  df-xrs 14440  df-imas 14446  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-mulg 15548  df-cntz 15835  df-cmn 16279  df-xmet 17810  df-met 17811
This theorem is referenced by: (None)
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