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Theorem xpsmet 17946
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 2283 . . 3  |-  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )  =  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )
7 eqid 2283 . . 3  |-  (Scalar `  R )  =  (Scalar `  R )
8 eqid 2283 . . 3  |-  ( (Scalar `  R ) X_s `' ( { R }  +c  { S }
) )  =  ( (Scalar `  R ) X_s `' ( { R }  +c  { S } ) )
91, 2, 3, 4, 5, 6, 7, 8xpsval 13474 . 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 13475 . 2  |-  ( ph  ->  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )  =  ( Base `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) ) )
116xpsff1o2 13473 . . 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 5485 . . 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 11 . 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 5883 . . 3  |-  ( (Scalar `  R ) X_s `' ( { R }  +c  { S }
) )  e.  _V
1514a1i 10 . 2  |-  ( ph  ->  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) )  e.  _V )
16 eqid 2283 . 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 2283 . . . . 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 2283 . . . . 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 2283 . . . . 5  |-  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
Base `  ( `' ( { R }  +c  { S } ) `  k ) )
21 eqid 2283 . . . . 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 2283 . . . . 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 5539 . . . . . 6  |-  (Scalar `  R )  e.  _V
2423a1i 10 . . . . 5  |-  ( ph  ->  (Scalar `  R )  e.  _V )
25 2onn 6638 . . . . . 6  |-  2o  e.  om
26 nnfi 7053 . . . . . 6  |-  ( 2o  e.  om  ->  2o  e.  Fin )
2725, 26mp1i 11 . . . . 5  |-  ( ph  ->  2o  e.  Fin )
28 fvex 5539 . . . . . 6  |-  ( `' ( { R }  +c  { S } ) `
 k )  e. 
_V
2928a1i 10 . . . . 5  |-  ( (
ph  /\  k  e.  2o )  ->  ( `' ( { R }  +c  { S } ) `
 k )  e. 
_V )
30 elpri 3660 . . . . . . 7  |-  ( k  e.  { (/) ,  1o }  ->  ( k  =  (/)  \/  k  =  1o ) )
31 df2o3 6492 . . . . . . 7  |-  2o  =  { (/) ,  1o }
3230, 31eleq2s 2375 . . . . . 6  |-  ( k  e.  2o  ->  (
k  =  (/)  \/  k  =  1o ) )
33 xpsmet.3 . . . . . . . . 9  |-  ( ph  ->  M  e.  ( Met `  X ) )
3433adantr 451 . . . . . . . 8  |-  ( (
ph  /\  k  =  (/) )  ->  M  e.  ( Met `  X ) )
35 fveq2 5525 . . . . . . . . . . . . 13  |-  ( k  =  (/)  ->  ( `' ( { R }  +c  { S } ) `
 k )  =  ( `' ( { R }  +c  { S } ) `  (/) ) )
36 xpsc0 13462 . . . . . . . . . . . . . 14  |-  ( R  e.  V  ->  ( `' ( { R }  +c  { S }
) `  (/) )  =  R )
374, 36syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' ( { R }  +c  { S } ) `  (/) )  =  R )
3835, 37sylan9eqr 2337 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  =  (/) )  ->  ( `' ( { R }  +c  { S } ) `  k )  =  R )
3938fveq2d 5529 . . . . . . . . . . 11  |-  ( (
ph  /\  k  =  (/) )  ->  ( dist `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
dist `  R )
)
4038fveq2d 5529 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  =  (/) )  ->  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
Base `  R )
)
4140, 2syl6eqr 2333 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  =  (/) )  ->  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  X )
4241, 41xpeq12d 4714 . . . . . . . . . . 11  |-  ( (
ph  /\  k  =  (/) )  ->  ( ( Base `  ( `' ( { R }  +c  { S } ) `  k ) )  X.  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) )  =  ( X  X.  X ) )
4339, 42reseq12d 4956 . . . . . . . . . 10  |-  ( (
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 . . . . . . . . . 10  |-  M  =  ( ( dist `  R
)  |`  ( X  X.  X ) )
4543, 44syl6eqr 2333 . . . . . . . . 9  |-  ( (
ph  /\  k  =  (/) )  ->  ( ( dist `  ( `' ( { R }  +c  { S } ) `  k ) )  |`  ( ( Base `  ( `' ( { R }  +c  { S }
) `  k )
)  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) ) )  =  M )
4641fveq2d 5529 . . . . . . . . 9  |-  ( (
ph  /\  k  =  (/) )  ->  ( Met `  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) )  =  ( Met `  X ) )
4745, 46eleq12d 2351 . . . . . . . 8  |-  ( (
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 ) ) )  <-> 
M  e.  ( Met `  X ) ) )
4834, 47mpbird 223 . . . . . . 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 ) ) ) )
49 xpsmet.4 . . . . . . . . 9  |-  ( ph  ->  N  e.  ( Met `  Y ) )
5049adantr 451 . . . . . . . 8  |-  ( (
ph  /\  k  =  1o )  ->  N  e.  ( Met `  Y
) )
51 fveq2 5525 . . . . . . . . . . . . 13  |-  ( k  =  1o  ->  ( `' ( { R }  +c  { S }
) `  k )  =  ( `' ( { R }  +c  { S } ) `  1o ) )
52 xpsc1 13463 . . . . . . . . . . . . . 14  |-  ( S  e.  W  ->  ( `' ( { R }  +c  { S }
) `  1o )  =  S )
535, 52syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' ( { R }  +c  { S } ) `  1o )  =  S )
5451, 53sylan9eqr 2337 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  =  1o )  ->  ( `' ( { R }  +c  { S } ) `
 k )  =  S )
5554fveq2d 5529 . . . . . . . . . . 11  |-  ( (
ph  /\  k  =  1o )  ->  ( dist `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
dist `  S )
)
5654fveq2d 5529 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  =  1o )  ->  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
Base `  S )
)
5756, 3syl6eqr 2333 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  =  1o )  ->  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  Y )
5857, 57xpeq12d 4714 . . . . . . . . . . 11  |-  ( (
ph  /\  k  =  1o )  ->  ( (
Base `  ( `' ( { R }  +c  { S } ) `  k ) )  X.  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) )  =  ( Y  X.  Y ) )
5955, 58reseq12d 4956 . . . . . . . . . 10  |-  ( (
ph  /\  k  =  1o )  ->  ( (
dist `  ( `' ( { R }  +c  { S } ) `  k ) )  |`  ( ( Base `  ( `' ( { R }  +c  { S }
) `  k )
)  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) ) )  =  ( ( dist `  S )  |`  ( Y  X.  Y ) ) )
60 xpsds.n . . . . . . . . . 10  |-  N  =  ( ( dist `  S
)  |`  ( Y  X.  Y ) )
6159, 60syl6eqr 2333 . . . . . . . . 9  |-  ( (
ph  /\  k  =  1o )  ->  ( (
dist `  ( `' ( { R }  +c  { S } ) `  k ) )  |`  ( ( Base `  ( `' ( { R }  +c  { S }
) `  k )
)  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) ) )  =  N )
6257fveq2d 5529 . . . . . . . . 9  |-  ( (
ph  /\  k  =  1o )  ->  ( Met `  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) )  =  ( Met `  Y ) )
6361, 62eleq12d 2351 . . . . . . . 8  |-  ( (
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 )
) )  <->  N  e.  ( Met `  Y ) ) )
6450, 63mpbird 223 . . . . . . 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 ) ) ) )
6548, 64jaodan 760 . . . . . 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 )
) ) )
6632, 65sylan2 460 . . . . 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 ) ) ) )
6718, 19, 20, 21, 22, 24, 27, 29, 66prdsmet 17934 . . . 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 )
) ) ) ) )
68 xpscfn 13461 . . . . . . . . 9  |-  ( ( R  e.  V  /\  S  e.  W )  ->  `' ( { R }  +c  { S }
)  Fn  2o )
694, 5, 68syl2anc 642 . . . . . . . 8  |-  ( ph  ->  `' ( { R }  +c  { S }
)  Fn  2o )
70 dffn5 5568 . . . . . . . 8  |-  ( `' ( { R }  +c  { S } )  Fn  2o  <->  `' ( { R }  +c  { S } )  =  ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) )
7169, 70sylib 188 . . . . . . 7  |-  ( ph  ->  `' ( { R }  +c  { S }
)  =  ( k  e.  2o  |->  ( `' ( { R }  +c  { S } ) `
 k ) ) )
7271oveq2d 5874 . . . . . 6  |-  ( ph  ->  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) )  =  ( (Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) )
7372fveq2d 5529 . . . . 5  |-  ( ph  ->  ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  =  (
dist `  ( (Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) ) )
7472fveq2d 5529 . . . . . . 7  |-  ( ph  ->  ( Base `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  =  (
Base `  ( (Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) ) )
7510, 74eqtrd 2315 . . . . . 6  |-  ( ph  ->  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )  =  ( Base `  (
(Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) ) )
7675fveq2d 5529 . . . . 5  |-  ( 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 )
) ) ) ) )
7773, 76eleq12d 2351 . . . 4  |-  ( ph  ->  ( ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  e.  ( Met `  ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) )  <->  ( 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 )
) ) ) ) ) )
7867, 77mpbird 223 . . 3  |-  ( ph  ->  ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  e.  ( Met `  ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) ) )
79 ssid 3197 . . 3  |-  ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) )  C_  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )
80 metres2 17927 . . 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 } ) ) ) )
8178, 79, 80sylancl 643 . 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 } ) ) ) )
829, 10, 13, 15, 16, 17, 81imasf1omet 17940 1  |-  ( ph  ->  P  e.  ( Met `  ( X  X.  Y
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   (/)c0 3455   {csn 3640   {cpr 3641    e. cmpt 4077   omcom 4656    X. cxp 4687   `'ccnv 4688   ran crn 4690    |` cres 4691    Fn wfn 5250   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1oc1o 6472   2oc2o 6473   Fincfn 6863    +c ccda 7793   Basecbs 13148  Scalarcsca 13211   distcds 13217   X_scprds 13346    X.s cxps 13409   Metcme 16370
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  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  ax-pre-sup 8815
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-rmo 2551  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-int 3863  df-iun 3907  df-iin 3908  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-se 4353  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-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  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-uz 10231  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374
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