Theorem msxms 15323

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 2232	. . 3 |- ( TopOpen ` M ) = ( TopOpen ` M )
2		eqid 2232	. . 3 |- ( Base ` M ) = ( Base ` M )
3		eqid 2232	. . 3 |- ( ( dist ` M ) |` ( ( Base ` M ) X. ( Base ` M ) ) ) = ( ( dist ` M ) |` ( ( Base ` M ) X. ( Base ` M ) ) )
4	1, 2, 3	isms 15318	. 2 |- ( M e. MetSp <-> ( M e. *MetSp /\ ( ( dist ` M ) |` ( ( Base ` M ) X. ( Base ` M ) ) ) e. ( Met ` ( Base ` M ) ) ) )
5	4	simplbi 274	1 |- ( M e. MetSp -> M e. *MetSp )

Syntax hints:   -> wi 4   e. wcel 2203   X. cxp 4747   |` cres 4751   ` cfv 5352   Basecbs 13212   distcds 13299   TopOpenctopn 13453   Metcmet 14685   *MetSpcxms 15201   MetSpcms 15202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-rab 2529  df-v 2815  df-un 3215  df-in 3217  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-xp 4755  df-res 4761  df-iota 5312  df-fv 5360  df-ms 15205

This theorem is referenced by:  mstps  15324  cnfldxms  15402