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Metamath Proof Explorer

Theorem setsms

Description: The constructed metric space is a metric space iff the provided distance function is a metric. (Contributed by Mario Carneiro, 28-Aug-2015)

Ref Expression
Hypotheses setsms.x 
|- ( ph -> X = ( Base ` M ) )
setsms.d 
|- ( ph -> D = ( ( dist ` M ) |` ( X X. X ) ) )
setsms.k 
|- ( ph -> K = ( M sSet <. ( TopSet ` ndx ) , ( MetOpen ` D ) >. ) )
setsms.m 
|- ( ph -> M e. V )
Assertion setsms 
|- ( ph -> ( K e. MetSp <-> D e. ( Met ` X ) ) )

Proof

Step Hyp Ref Expression
1 setsms.x 
 |-  ( ph -> X = ( Base ` M ) )
2 setsms.d 
 |-  ( ph -> D = ( ( dist ` M ) |` ( X X. X ) ) )
3 setsms.k 
 |-  ( ph -> K = ( M sSet <. ( TopSet ` ndx ) , ( MetOpen ` D ) >. ) )
4 setsms.m 
 |-  ( ph -> M e. V )
5 1 2 3 4 setsxms 
 |-  ( ph -> ( K e. *MetSp <-> D e. ( *Met ` X ) ) )
6 1 2 3 setsmsds 
 |-  ( ph -> ( dist ` M ) = ( dist ` K ) )
7 1 2 3 setsmsbas 
 |-  ( ph -> X = ( Base ` K ) )
8 7 sqxpeqd 
 |-  ( ph -> ( X X. X ) = ( ( Base ` K ) X. ( Base ` K ) ) )
9 6 8 reseq12d 
 |-  ( ph -> ( ( dist ` M ) |` ( X X. X ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
10 2 9 eqtr2d 
 |-  ( ph -> ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = D )
11 7 fveq2d 
 |-  ( ph -> ( Met ` X ) = ( Met ` ( Base ` K ) ) )
12 11 eqcomd 
 |-  ( ph -> ( Met ` ( Base ` K ) ) = ( Met ` X ) )
13 10 12 eleq12d 
 |-  ( ph -> ( ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) e. ( Met ` ( Base ` K ) ) <-> D e. ( Met ` X ) ) )
14 5 13 anbi12d 
 |-  ( ph -> ( ( K e. *MetSp /\ ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) e. ( Met ` ( Base ` K ) ) ) <-> ( D e. ( *Met ` X ) /\ D e. ( Met ` X ) ) ) )
15 eqid 
 |-  ( TopOpen ` K ) = ( TopOpen ` K )
16 eqid 
 |-  ( Base ` K ) = ( Base ` K )
17 eqid 
 |-  ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) )
18 15 16 17 isms 
 |-  ( K e. MetSp <-> ( K e. *MetSp /\ ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) e. ( Met ` ( Base ` K ) ) ) )
19 metxmet 
 |-  ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
20 19 pm4.71ri 
 |-  ( D e. ( Met ` X ) <-> ( D e. ( *Met ` X ) /\ D e. ( Met ` X ) ) )
21 14 18 20 3bitr4g 
 |-  ( ph -> ( K e. MetSp <-> D e. ( Met ` X ) ) )