Theorem cmsms 23145

Description: A complete metric space is a metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)

Assertion

Ref	Expression
cmsms	|- ( G e. CMetSp -> G e. MetSp )

Proof of Theorem cmsms

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( Base ` G ) = ( Base ` G )
2		eqid 2622	. . 3 |- ( ( dist ` G ) |` ( ( Base ` G ) X. ( Base ` G ) ) ) = ( ( dist ` G ) |` ( ( Base ` G ) X. ( Base ` G ) ) )
3	1, 2	iscms 23142	. 2 |- ( G e. CMetSp <-> ( G e. MetSp /\ ( ( dist ` G ) |` ( ( Base ` G ) X. ( Base ` G ) ) ) e. ( CMet ` ( Base ` G ) ) ) )
4	3	simplbi 476	1 |- ( G e. CMetSp -> G e. MetSp )

Syntax hints:   -> wi 4   e. wcel 1990   X. cxp 5112   |` cres 5116   ` cfv 5888   Basecbs 15857   distcds 15950   MetSpcmt 22123   CMetcms 23052  CMetSpccms 23129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-res 5126  df-iota 5851  df-fv 5896  df-cms 23132

This theorem is referenced by:  cmsss  23147  cmetcusp1  23149  rlmbn  23157  rrhcn  30041  dya2icoseg2  30340  sitgclbn  30405