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Theorem xmspropd 20183
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 2497 . . . 4  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
4 xmspropd.4 . . . 4  |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L
) )
53, 4tpspropd 18680 . . 3  |-  ( ph  ->  ( K  e.  TopSp  <->  L  e.  TopSp ) )
6 xmspropd.3 . . . . . . 7  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( B  X.  B ) ) )
71, 1xpeq12d 4976 . . . . . . . 8  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
87reseq2d 5221 . . . . . . 7  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
96, 8eqtr3d 2497 . . . . . 6  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
102, 2xpeq12d 4976 . . . . . . 7  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  L )  X.  ( Base `  L
) ) )
1110reseq2d 5221 . . . . . 6  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) )
129, 11eqtr3d 2497 . . . . 5  |-  ( ph  ->  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  =  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) ) )
1312fveq2d 5806 . . . 4  |-  ( ph  ->  ( MetOpen `  ( ( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) )  =  ( MetOpen `  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) ) )
144, 13eqeq12d 2476 . . 3  |-  ( ph  ->  ( ( TopOpen `  K
)  =  ( MetOpen `  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )  <->  ( TopOpen `  L )  =  (
MetOpen `  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) ) ) ) )
155, 14anbi12d 710 . 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 2454 . . 3  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
17 eqid 2454 . . 3  |-  ( Base `  K )  =  (
Base `  K )
18 eqid 2454 . . 3  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
1916, 17, 18isxms 20157 . 2  |-  ( K  e.  *MetSp  <->  ( K  e.  TopSp  /\  ( TopOpen `  K )  =  (
MetOpen `  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) ) ) ) )
20 eqid 2454 . . 3  |-  ( TopOpen `  L )  =  (
TopOpen `  L )
21 eqid 2454 . . 3  |-  ( Base `  L )  =  (
Base `  L )
22 eqid 2454 . . 3  |-  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )
2320, 21, 22isxms 20157 . 2  |-  ( L  e.  *MetSp  <->  ( L  e.  TopSp  /\  ( TopOpen `  L )  =  (
MetOpen `  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) ) ) ) )
2415, 19, 233bitr4g 288 1  |-  ( ph  ->  ( K  e.  *MetSp  <-> 
L  e.  *MetSp ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    X. cxp 4949    |` cres 4953   ` cfv 5529   Basecbs 14295   distcds 14369   TopOpenctopn 14482   MetOpencmopn 17934   TopSpctps 18636   *MetSpcxme 20027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-res 4963  df-iota 5492  df-fun 5531  df-fv 5537  df-top 18638  df-topon 18641  df-topsp 18642  df-xms 20030
This theorem is referenced by:  mspropd  20184
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