Theorem msxms 12818

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 2157	. . 3 |- ( TopOpen ` M ) = ( TopOpen ` M )
2		eqid 2157	. . 3 |- ( Base ` M ) = ( Base ` M )
3		eqid 2157	. . 3 |- ( ( dist ` M ) |` ( ( Base ` M ) X. ( Base ` M ) ) ) = ( ( dist ` M ) |` ( ( Base ` M ) X. ( Base ` M ) ) )
4	1, 2, 3	isms 12813	. 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 2128   X. cxp 4581   |` cres 4585   ` cfv 5167   Basecbs 12150   distcds 12221   TopOpenctopn 12312   Metcmet 12341   *MetSpcxms 12696   MetSpcms 12697

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-rab 2444  df-v 2714  df-un 3106  df-in 3108  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-xp 4589  df-res 4595  df-iota 5132  df-fv 5175  df-ms 12700

This theorem is referenced by:  mstps  12819