This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem mopnex

Description: The topology generated by an extended metric can also be generated by a true metric. Thus, "metrizable topologies" can equivalently be defined in terms of metrics or extended metrics. (Contributed by Mario Carneiro, 26-Aug-2015)

Ref Expression
Hypothesis mopnex.1 
|- J = ( MetOpen ` D )
Assertion mopnex 
|- ( D e. ( *Met ` X ) -> E. d e. ( Met ` X ) J = ( MetOpen ` d ) )

Proof

Step Hyp Ref Expression
1 mopnex.1 
 |-  J = ( MetOpen ` D )
2 1rp 
 |-  1 e. RR+
3 eqid 
 |-  ( x e. X , y e. X |-> if ( ( x D y ) <_ 1 , ( x D y ) , 1 ) ) = ( x e. X , y e. X |-> if ( ( x D y ) <_ 1 , ( x D y ) , 1 ) )
4 3 stdbdmet 
 |-  ( ( D e. ( *Met ` X ) /\ 1 e. RR+ ) -> ( x e. X , y e. X |-> if ( ( x D y ) <_ 1 , ( x D y ) , 1 ) ) e. ( Met ` X ) )
5 2 4 mpan2 
 |-  ( D e. ( *Met ` X ) -> ( x e. X , y e. X |-> if ( ( x D y ) <_ 1 , ( x D y ) , 1 ) ) e. ( Met ` X ) )
6 1xr 
 |-  1 e. RR*
7 0lt1 
 |-  0 < 1
8 3 1 stdbdmopn 
 |-  ( ( D e. ( *Met ` X ) /\ 1 e. RR* /\ 0 < 1 ) -> J = ( MetOpen ` ( x e. X , y e. X |-> if ( ( x D y ) <_ 1 , ( x D y ) , 1 ) ) ) )
9 6 7 8 mp3an23 
 |-  ( D e. ( *Met ` X ) -> J = ( MetOpen ` ( x e. X , y e. X |-> if ( ( x D y ) <_ 1 , ( x D y ) , 1 ) ) ) )
10 fveq2 
 |-  ( d = ( x e. X , y e. X |-> if ( ( x D y ) <_ 1 , ( x D y ) , 1 ) ) -> ( MetOpen ` d ) = ( MetOpen ` ( x e. X , y e. X |-> if ( ( x D y ) <_ 1 , ( x D y ) , 1 ) ) ) )
11 10 rspceeqv 
 |-  ( ( ( x e. X , y e. X |-> if ( ( x D y ) <_ 1 , ( x D y ) , 1 ) ) e. ( Met ` X ) /\ J = ( MetOpen ` ( x e. X , y e. X |-> if ( ( x D y ) <_ 1 , ( x D y ) , 1 ) ) ) ) -> E. d e. ( Met ` X ) J = ( MetOpen ` d ) )
12 5 9 11 syl2anc 
 |-  ( D e. ( *Met ` X ) -> E. d e. ( Met ` X ) J = ( MetOpen ` d ) )