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Theorem mspropd 21411
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 21410 . . 3  |-  ( ph  ->  ( K  e.  *MetSp  <-> 
L  e.  *MetSp ) )
61sqxpeqd 4880 . . . . . . 7  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
76reseq2d 5125 . . . . . 6  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
83, 7eqtr3d 2472 . . . . 5  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
92sqxpeqd 4880 . . . . . 6  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  L )  X.  ( Base `  L
) ) )
109reseq2d 5125 . . . . 5  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) )
118, 10eqtr3d 2472 . . . 4  |-  ( ph  ->  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  =  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) ) )
121, 2eqtr3d 2472 . . . . 5  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
1312fveq2d 5885 . . . 4  |-  ( ph  ->  ( Met `  ( Base `  K ) )  =  ( Met `  ( Base `  L ) ) )
1411, 13eleq12d 2511 . . 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 715 . 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 2429 . . 3  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
17 eqid 2429 . . 3  |-  ( Base `  K )  =  (
Base `  K )
18 eqid 2429 . . 3  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
1916, 17, 18isms 21386 . 2  |-  ( K  e.  MetSp 
<->  ( K  e.  *MetSp  /\  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )  e.  ( Met `  ( Base `  K ) ) ) )
20 eqid 2429 . . 3  |-  ( TopOpen `  L )  =  (
TopOpen `  L )
21 eqid 2429 . . 3  |-  ( Base `  L )  =  (
Base `  L )
22 eqid 2429 . . 3  |-  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )
2320, 21, 22isms 21386 . 2  |-  ( L  e.  MetSp 
<->  ( L  e.  *MetSp  /\  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )  e.  ( Met `  ( Base `  L ) ) ) )
2415, 19, 233bitr4g 291 1  |-  ( ph  ->  ( K  e.  MetSp  <->  L  e.  MetSp ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870    X. cxp 4852    |` cres 4856   ` cfv 5601   Basecbs 15075   distcds 15152   TopOpenctopn 15270   Metcme 18882   *MetSpcxme 21254   MetSpcmt 21255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-res 4866  df-iota 5565  df-fun 5603  df-fv 5609  df-top 19843  df-topon 19845  df-topsp 19846  df-xms 21257  df-ms 21258
This theorem is referenced by:  ngppropd  21567  cmspropd  22201  zhmnrg  28601
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