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Theorem xmspropd 13610
Description: Property deduction for an extended metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
xmspropd.1  |-  ( ph  ->  B  =  ( Base `  K ) )
xmspropd.2  |-  ( ph  ->  B  =  ( Base `  L ) )
xmspropd.3  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( B  X.  B ) ) )
xmspropd.4  |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L
) )
Assertion
Ref Expression
xmspropd  |-  ( ph  ->  ( K  e.  *MetSp  <-> 
L  e.  *MetSp ) )

Proof of Theorem xmspropd
StepHypRef Expression
1 xmspropd.1 . . . . 5  |-  ( ph  ->  B  =  ( Base `  K ) )
2 xmspropd.2 . . . . 5  |-  ( ph  ->  B  =  ( Base `  L ) )
31, 2eqtr3d 2212 . . . 4  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
4 xmspropd.4 . . . 4  |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L
) )
53, 4tpspropd 13167 . . 3  |-  ( ph  ->  ( K  e.  TopSp  <->  L  e.  TopSp ) )
6 xmspropd.3 . . . . . . 7  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( B  X.  B ) ) )
71sqxpeqd 4648 . . . . . . . 8  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
87reseq2d 4902 . . . . . . 7  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
96, 8eqtr3d 2212 . . . . . 6  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
102sqxpeqd 4648 . . . . . . 7  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  L )  X.  ( Base `  L
) ) )
1110reseq2d 4902 . . . . . 6  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) )
129, 11eqtr3d 2212 . . . . 5  |-  ( ph  ->  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  =  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) ) )
1312fveq2d 5514 . . . 4  |-  ( ph  ->  ( MetOpen `  ( ( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) )  =  ( MetOpen `  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) ) )
144, 13eqeq12d 2192 . . 3  |-  ( ph  ->  ( ( TopOpen `  K
)  =  ( MetOpen `  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )  <->  ( TopOpen `  L )  =  (
MetOpen `  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) ) ) ) )
155, 14anbi12d 473 . 2  |-  ( ph  ->  ( ( K  e. 
TopSp  /\  ( TopOpen `  K
)  =  ( MetOpen `  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) ) )  <->  ( L  e.  TopSp  /\  ( TopOpen `  L )  =  (
MetOpen `  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) ) ) ) ) )
16 eqid 2177 . . 3  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
17 eqid 2177 . . 3  |-  ( Base `  K )  =  (
Base `  K )
18 eqid 2177 . . 3  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
1916, 17, 18isxms 13584 . 2  |-  ( K  e.  *MetSp  <->  ( K  e.  TopSp  /\  ( TopOpen `  K )  =  (
MetOpen `  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) ) ) ) )
20 eqid 2177 . . 3  |-  ( TopOpen `  L )  =  (
TopOpen `  L )
21 eqid 2177 . . 3  |-  ( Base `  L )  =  (
Base `  L )
22 eqid 2177 . . 3  |-  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )
2320, 21, 22isxms 13584 . 2  |-  ( L  e.  *MetSp  <->  ( L  e.  TopSp  /\  ( TopOpen `  L )  =  (
MetOpen `  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) ) ) ) )
2415, 19, 233bitr4g 223 1  |-  ( ph  ->  ( K  e.  *MetSp  <-> 
L  e.  *MetSp ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148    X. cxp 4620    |` cres 4624   ` cfv 5211   Basecbs 12432   distcds 12514   TopOpenctopn 12624   MetOpencmopn 13118   TopSpctps 13161   *MetSpcxms 13469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-inn 8896  df-2 8954  df-3 8955  df-4 8956  df-5 8957  df-6 8958  df-7 8959  df-8 8960  df-9 8961  df-ndx 12435  df-slot 12436  df-base 12438  df-tset 12524  df-rest 12625  df-topn 12626  df-top 13129  df-topon 13142  df-topsp 13162  df-xms 13472
This theorem is referenced by:  mspropd  13611
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