Theorem msxms 12627

Description: A metric space is an extended metric space. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
msxms	|- ( M e. MetSp -> M e. *MetSp )

Proof of Theorem msxms

Step	Hyp	Ref	Expression
1		eqid 2139	. . 3 |- ( TopOpen ` M ) = ( TopOpen ` M )
2		eqid 2139	. . 3 |- ( Base ` M ) = ( Base ` M )
3		eqid 2139	. . 3 |- ( ( dist ` M ) |` ( ( Base ` M ) X. ( Base ` M ) ) ) = ( ( dist ` M ) |` ( ( Base ` M ) X. ( Base ` M ) ) )
4	1, 2, 3	isms 12622	. 2 |- ( M e. MetSp <-> ( M e. *MetSp /\ ( ( dist ` M ) |` ( ( Base ` M ) X. ( Base ` M ) ) ) e. ( Met ` ( Base ` M ) ) ) )
5	4	simplbi 272	1 |- ( M e. MetSp -> M e. *MetSp )

Syntax hints:   -> wi 4   e. wcel 1480   X. cxp 4537   |` cres 4541   ` cfv 5123   Basecbs 11959   distcds 12030   TopOpenctopn 12121   Metcmet 12150   *MetSpcxms 12505   MetSpcms 12506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-rab 2425  df-v 2688  df-un 3075  df-in 3077  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-res 4551  df-iota 5088  df-fv 5131  df-ms 12509

This theorem is referenced by:  mstps  12628