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Theorem xmspropd 12683
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 2175 . . . 4  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
4 xmspropd.4 . . . 4  |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L
) )
53, 4tpspropd 12240 . . 3  |-  ( ph  ->  ( K  e.  TopSp  <->  L  e.  TopSp ) )
6 xmspropd.3 . . . . . . 7  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( B  X.  B ) ) )
71sqxpeqd 4572 . . . . . . . 8  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
87reseq2d 4826 . . . . . . 7  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
96, 8eqtr3d 2175 . . . . . 6  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
102sqxpeqd 4572 . . . . . . 7  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  L )  X.  ( Base `  L
) ) )
1110reseq2d 4826 . . . . . 6  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) )
129, 11eqtr3d 2175 . . . . 5  |-  ( ph  ->  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  =  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) ) )
1312fveq2d 5432 . . . 4  |-  ( ph  ->  ( MetOpen `  ( ( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) )  =  ( MetOpen `  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) ) )
144, 13eqeq12d 2155 . . 3  |-  ( ph  ->  ( ( TopOpen `  K
)  =  ( MetOpen `  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )  <->  ( TopOpen `  L )  =  (
MetOpen `  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) ) ) ) )
155, 14anbi12d 465 . 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 2140 . . 3  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
17 eqid 2140 . . 3  |-  ( Base `  K )  =  (
Base `  K )
18 eqid 2140 . . 3  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
1916, 17, 18isxms 12657 . 2  |-  ( K  e.  *MetSp  <->  ( K  e.  TopSp  /\  ( TopOpen `  K )  =  (
MetOpen `  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) ) ) ) )
20 eqid 2140 . . 3  |-  ( TopOpen `  L )  =  (
TopOpen `  L )
21 eqid 2140 . . 3  |-  ( Base `  L )  =  (
Base `  L )
22 eqid 2140 . . 3  |-  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )
2320, 21, 22isxms 12657 . 2  |-  ( L  e.  *MetSp  <->  ( L  e.  TopSp  /\  ( TopOpen `  L )  =  (
MetOpen `  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) ) ) ) )
2415, 19, 233bitr4g 222 1  |-  ( ph  ->  ( K  e.  *MetSp  <-> 
L  e.  *MetSp ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1332    e. wcel 1481    X. cxp 4544    |` cres 4548   ` cfv 5130   Basecbs 11996   distcds 12067   TopOpenctopn 12158   MetOpencmopn 12191   TopSpctps 12234   *MetSpcxms 12542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-cnex 7734  ax-resscn 7735  ax-1re 7737  ax-addrcl 7740
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-inn 8744  df-2 8802  df-3 8803  df-4 8804  df-5 8805  df-6 8806  df-7 8807  df-8 8808  df-9 8809  df-ndx 11999  df-slot 12000  df-base 12002  df-tset 12077  df-rest 12159  df-topn 12160  df-top 12202  df-topon 12215  df-topsp 12235  df-xms 12545
This theorem is referenced by:  mspropd  12684
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