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Theorem xpsmet 17940
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
) ) )
Dummy variables  x  k  y are mutually distinct and distinct from all other variables.

Proof of Theorem xpsmet
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 2284 . . 3  |-  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )  =  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )
7 eqid 2284 . . 3  |-  (Scalar `  R )  =  (Scalar `  R )
8 eqid 2284 . . 3  |-  ( (Scalar `  R ) X_s `' ( { R }  +c  { S }
) )  =  ( (Scalar `  R ) X_s `' ( { R }  +c  { S } ) )
91, 2, 3, 4, 5, 6, 7, 8xpsval 13468 . 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 13469 . 2  |-  ( ph  ->  ran  (  x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )  =  ( Base `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) ) )
116xpsff1o2 13467 . . 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 5450 . . 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 13 . 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 5844 . . 3  |-  ( (Scalar `  R ) X_s `' ( { R }  +c  { S }
) )  e.  _V
1514a1i 12 . 2  |-  ( ph  ->  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) )  e.  _V )
16 eqid 2284 . 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 2284 . . . . 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 2284 . . . . 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 2284 . . . . 5  |-  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
Base `  ( `' ( { R }  +c  { S } ) `  k ) )
21 eqid 2284 . . . . 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 2284 . . . . 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 5499 . . . . . 6  |-  (Scalar `  R )  e.  _V
2423a1i 12 . . . . 5  |-  ( ph  ->  (Scalar `  R )  e.  _V )
25 2onn 6633 . . . . . 6  |-  2o  e.  om
26 nnfi 7048 . . . . . 6  |-  ( 2o  e.  om  ->  2o  e.  Fin )
2725, 26mp1i 13 . . . . 5  |-  ( ph  ->  2o  e.  Fin )
28 fvex 5499 . . . . . 6  |-  ( `' ( { R }  +c  { S } ) `
 k )  e. 
_V
2928a1i 12 . . . . 5  |-  ( (
ph  /\  k  e.  2o )  ->  ( `' ( { R }  +c  { S } ) `
 k )  e. 
_V )
30 elpri 3661 . . . . . . 7  |-  ( k  e.  { (/) ,  1o }  ->  ( k  =  (/)  \/  k  =  1o ) )
31 df2o3 6487 . . . . . . 7  |-  2o  =  { (/) ,  1o }
3230, 31eleq2s 2376 . . . . . 6  |-  ( k  e.  2o  ->  (
k  =  (/)  \/  k  =  1o ) )
33 xpsmet.3 . . . . . . . . 9  |-  ( ph  ->  M  e.  ( Met `  X ) )
3433adantr 453 . . . . . . . 8  |-  ( (
ph  /\  k  =  (/) )  ->  M  e.  ( Met `  X ) )
35 fveq2 5485 . . . . . . . . . . . . 13  |-  ( k  =  (/)  ->  ( `' ( { R }  +c  { S } ) `
 k )  =  ( `' ( { R }  +c  { S } ) `  (/) ) )
36 xpsc0 13456 . . . . . . . . . . . . . 14  |-  ( R  e.  V  ->  ( `' ( { R }  +c  { S }
) `  (/) )  =  R )
374, 36syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' ( { R }  +c  { S } ) `  (/) )  =  R )
3835, 37sylan9eqr 2338 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  =  (/) )  ->  ( `' ( { R }  +c  { S } ) `  k )  =  R )
3938fveq2d 5489 . . . . . . . . . . 11  |-  ( (
ph  /\  k  =  (/) )  ->  ( dist `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
dist `  R )
)
4038fveq2d 5489 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  =  (/) )  ->  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
Base `  R )
)
4140, 2syl6eqr 2334 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  =  (/) )  ->  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  X )
4241, 41xpeq12d 4713 . . . . . . . . . . 11  |-  ( (
ph  /\  k  =  (/) )  ->  ( ( Base `  ( `' ( { R }  +c  { S } ) `  k ) )  X.  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) )  =  ( X  X.  X ) )
4339, 42reseq12d 4955 . . . . . . . . . 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 2334 . . . . . . . . 9  |-  ( (
ph  /\  k  =  (/) )  ->  ( ( dist `  ( `' ( { R }  +c  { S } ) `  k ) )  |`  ( ( Base `  ( `' ( { R }  +c  { S }
) `  k )
)  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) ) )  =  M )
4641fveq2d 5489 . . . . . . . . 9  |-  ( (
ph  /\  k  =  (/) )  ->  ( Met `  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) )  =  ( Met `  X ) )
4745, 46eleq12d 2352 . . . . . . . 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 225 . . . . . . 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 453 . . . . . . . 8  |-  ( (
ph  /\  k  =  1o )  ->  N  e.  ( Met `  Y
) )
51 fveq2 5485 . . . . . . . . . . . . 13  |-  ( k  =  1o  ->  ( `' ( { R }  +c  { S }
) `  k )  =  ( `' ( { R }  +c  { S } ) `  1o ) )
52 xpsc1 13457 . . . . . . . . . . . . . 14  |-  ( S  e.  W  ->  ( `' ( { R }  +c  { S }
) `  1o )  =  S )
535, 52syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' ( { R }  +c  { S } ) `  1o )  =  S )
5451, 53sylan9eqr 2338 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  =  1o )  ->  ( `' ( { R }  +c  { S } ) `
 k )  =  S )
5554fveq2d 5489 . . . . . . . . . . 11  |-  ( (
ph  /\  k  =  1o )  ->  ( dist `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
dist `  S )
)
5654fveq2d 5489 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  =  1o )  ->  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
Base `  S )
)
5756, 3syl6eqr 2334 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  =  1o )  ->  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  Y )
5857, 57xpeq12d 4713 . . . . . . . . . . 11  |-  ( (
ph  /\  k  =  1o )  ->  ( (
Base `  ( `' ( { R }  +c  { S } ) `  k ) )  X.  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) )  =  ( Y  X.  Y ) )
5955, 58reseq12d 4955 . . . . . . . . . 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 2334 . . . . . . . . 9  |-  ( (
ph  /\  k  =  1o )  ->  ( (
dist `  ( `' ( { R }  +c  { S } ) `  k ) )  |`  ( ( Base `  ( `' ( { R }  +c  { S }
) `  k )
)  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) ) )  =  N )
6257fveq2d 5489 . . . . . . . . 9  |-  ( (
ph  /\  k  =  1o )  ->  ( Met `  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) )  =  ( Met `  Y ) )
6361, 62eleq12d 2352 . . . . . . . 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 225 . . . . . . 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 762 . . . . . 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 462 . . . . 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 17928 . . . 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 13455 . . . . . . . . 9  |-  ( ( R  e.  V  /\  S  e.  W )  ->  `' ( { R }  +c  { S }
)  Fn  2o )
694, 5, 68syl2anc 644 . . . . . . . 8  |-  ( ph  ->  `' ( { R }  +c  { S }
)  Fn  2o )
70 dffn5 5529 . . . . . . . 8  |-  ( `' ( { R }  +c  { S } )  Fn  2o  <->  `' ( { R }  +c  { S } )  =  ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) )
7169, 70sylib 190 . . . . . . 7  |-  ( ph  ->  `' ( { R }  +c  { S }
)  =  ( k  e.  2o  |->  ( `' ( { R }  +c  { S } ) `
 k ) ) )
7271oveq2d 5835 . . . . . 6  |-  ( ph  ->  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) )  =  ( (Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) )
7372fveq2d 5489 . . . . 5  |-  ( ph  ->  ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  =  (
dist `  ( (Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) ) )
7472fveq2d 5489 . . . . . . 7  |-  ( ph  ->  ( Base `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  =  (
Base `  ( (Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) ) )
7510, 74eqtrd 2316 . . . . . 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 5489 . . . . 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 2352 . . . 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 225 . . 3  |-  ( ph  ->  ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  e.  ( Met `  ran  (  x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) ) )
79 ssid 3198 . . 3  |-  ran  (  x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) )  C_  ran  (  x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )
80 metres2 17921 . . 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 645 . 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 17934 1  |-  ( ph  ->  P  e.  ( Met `  ( X  X.  Y
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    \/ wo 359    /\ wa 360    = wceq 1624    e. wcel 1685   _Vcvv 2789    C_ wss 3153   (/)c0 3456   {csn 3641   {cpr 3642    e. cmpt 4078   omcom 4655    X. cxp 4686   `'ccnv 4687   ran crn 4689    |` cres 4690    Fn wfn 5216   -1-1-onto->wf1o 5220   ` cfv 5221  (class class class)co 5819    e. cmpt2 5821   1oc1o 6467   2oc2o 6468   Fincfn 6858    +c ccda 7788   Basecbs 13142  Scalarcsca 13205   distcds 13211   X_scprds 13340    X.s cxps 13403   Metcme 16364
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-of 6039  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6655  df-map 6769  df-ixp 6813  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-sup 7189  df-oi 7220  df-card 7567  df-cda 7789  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-4 9801  df-5 9802  df-6 9803  df-7 9804  df-8 9805  df-9 9806  df-10 9807  df-n0 9961  df-z 10020  df-dec 10120  df-uz 10226  df-rp 10350  df-xneg 10447  df-xadd 10448  df-xmul 10449  df-icc 10657  df-fz 10777  df-fzo 10865  df-seq 11041  df-hash 11332  df-struct 13144  df-ndx 13145  df-slot 13146  df-base 13147  df-sets 13148  df-ress 13149  df-plusg 13215  df-mulr 13216  df-sca 13218  df-vsca 13219  df-tset 13221  df-ple 13222  df-ds 13224  df-hom 13226  df-cco 13227  df-prds 13342  df-xrs 13397  df-0g 13398  df-gsum 13399  df-imas 13405  df-xps 13407  df-mre 13482  df-mrc 13483  df-acs 13485  df-mnd 14361  df-submnd 14410  df-mulg 14486  df-cntz 14787  df-cmn 15085  df-xmet 16367  df-met 16368
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