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Theorem mspropd 13645
Description: Property deduction for a 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
mspropd  |-  ( ph  ->  ( K  e.  MetSp  <->  L  e.  MetSp ) )

Proof of Theorem mspropd
StepHypRef Expression
1 xmspropd.1 . . . 4  |-  ( ph  ->  B  =  ( Base `  K ) )
2 xmspropd.2 . . . 4  |-  ( ph  ->  B  =  ( Base `  L ) )
3 xmspropd.3 . . . 4  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( B  X.  B ) ) )
4 xmspropd.4 . . . 4  |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L
) )
51, 2, 3, 4xmspropd 13644 . . 3  |-  ( ph  ->  ( K  e.  *MetSp  <-> 
L  e.  *MetSp ) )
61sqxpeqd 4649 . . . . . . 7  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
76reseq2d 4903 . . . . . 6  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
83, 7eqtr3d 2212 . . . . 5  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
92sqxpeqd 4649 . . . . . 6  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  L )  X.  ( Base `  L
) ) )
109reseq2d 4903 . . . . 5  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) )
118, 10eqtr3d 2212 . . . 4  |-  ( ph  ->  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  =  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) ) )
121, 2eqtr3d 2212 . . . . 5  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
1312fveq2d 5515 . . . 4  |-  ( ph  ->  ( Met `  ( Base `  K ) )  =  ( Met `  ( Base `  L ) ) )
1411, 13eleq12d 2248 . . 3  |-  ( ph  ->  ( ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )  e.  ( Met `  ( Base `  K ) )  <-> 
( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) )  e.  ( Met `  ( Base `  L
) ) ) )
155, 14anbi12d 473 . 2  |-  ( ph  ->  ( ( K  e. 
*MetSp  /\  ( ( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  e.  ( Met `  ( Base `  K ) ) )  <->  ( L  e. 
*MetSp  /\  ( ( dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) )  e.  ( Met `  ( 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, 18isms 13620 . 2  |-  ( K  e.  MetSp 
<->  ( K  e.  *MetSp  /\  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )  e.  ( Met `  ( 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, 22isms 13620 . 2  |-  ( L  e.  MetSp 
<->  ( L  e.  *MetSp  /\  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )  e.  ( Met `  ( 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 4621    |` cres 4625   ` cfv 5212   Basecbs 12445   distcds 12527   TopOpenctopn 12637   Metcmet 13148   *MetSpcxms 13503   MetSpcms 13504
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 4206  ax-un 4430  ax-cnex 7893  ax-resscn 7894  ax-1re 7896  ax-addrcl 7899
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 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-5 8970  df-6 8971  df-7 8972  df-8 8973  df-9 8974  df-ndx 12448  df-slot 12449  df-base 12451  df-tset 12537  df-rest 12638  df-topn 12639  df-top 13163  df-topon 13176  df-topsp 13196  df-xms 13506  df-ms 13507
This theorem is referenced by: (None)
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