Theorem msxms 15559

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 2238	. . 3 |- ( TopOpen ` M ) = ( TopOpen ` M )
2		eqid 2238	. . 3 |- ( Base ` M ) = ( Base ` M )
3		eqid 2238	. . 3 |- ( ( dist ` M ) |` ( ( Base ` M ) X. ( Base ` M ) ) ) = ( ( dist ` M ) |` ( ( Base ` M ) X. ( Base ` M ) ) )
4	1, 2, 3	isms 15554	. 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 )

This proof depends on syntax axioms:   -> wi 4   e. wcel 2209   X. cxp 4772   |` cres 4776   ` cfv 5377   Basecbs 13352   distcds 13440   TopOpenctopn 13594   Metcmet 14874   *MetSpcxms 15437   MetSpcms 15438

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-in 3226  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-res 4786  df-iota 5337  df-fv 5385  df-ms 15441

This theorem is used by:  mstps  15560  cnfldxms  15638