Theorem iscms 23142

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

Hypotheses

Ref	Expression
iscms.1	|- X = ( Base ` M )
iscms.2	|- D = ( ( dist ` M ) |` ( X X. X ) )

Assertion

Ref	Expression
iscms	|- ( M e. CMetSp <-> ( M e. MetSp /\ D e. ( CMet ` X ) ) )

Proof of Theorem iscms

Dummy variables w b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvexd 6203	. . 3 |- ( w = M -> ( Base ` w ) e. _V )
2		fveq2 6191	. . . . . . 7 |- ( w = M -> ( dist ` w ) = ( dist ` M ) )
3	2	adantr 481	. . . . . 6 |- ( ( w = M /\ b = ( Base ` w ) ) -> ( dist ` w ) = ( dist ` M ) )
4		id 22	. . . . . . . 8 |- ( b = ( Base ` w ) -> b = ( Base ` w ) )
5		fveq2 6191	. . . . . . . . 9 |- ( w = M -> ( Base ` w ) = ( Base ` M ) )
6		iscms.1	. . . . . . . . 9 |- X = ( Base ` M )
7	5, 6	syl6eqr 2674	. . . . . . . 8 |- ( w = M -> ( Base ` w ) = X )
8	4, 7	sylan9eqr 2678	. . . . . . 7 |- ( ( w = M /\ b = ( Base ` w ) ) -> b = X )
9	8	sqxpeqd 5141	. . . . . 6 |- ( ( w = M /\ b = ( Base ` w ) ) -> ( b X. b ) = ( X X. X ) )
10	3, 9	reseq12d 5397	. . . . 5 |- ( ( w = M /\ b = ( Base ` w ) ) -> ( ( dist ` w ) |` ( b X. b ) ) = ( ( dist ` M ) |` ( X X. X ) ) )
11		iscms.2	. . . . 5 |- D = ( ( dist ` M ) |` ( X X. X ) )
12	10, 11	syl6eqr 2674	. . . 4 |- ( ( w = M /\ b = ( Base ` w ) ) -> ( ( dist ` w ) |` ( b X. b ) ) = D )
13	8	fveq2d 6195	. . . 4 |- ( ( w = M /\ b = ( Base ` w ) ) -> ( CMet ` b ) = ( CMet ` X ) )
14	12, 13	eleq12d 2695	. . 3 |- ( ( w = M /\ b = ( Base ` w ) ) -> ( ( ( dist ` w ) |` ( b X. b ) ) e. ( CMet ` b ) <-> D e. ( CMet ` X ) ) )
15	1, 14	sbcied 3472	. 2 |- ( w = M -> ( [. ( Base ` w ) / b ]. ( ( dist ` w ) |` ( b X. b ) ) e. ( CMet ` b ) <-> D e. ( CMet ` X ) ) )
16		df-cms 23132	. 2 |- CMetSp = { w e. MetSp | [. ( Base ` w ) / b ]. ( ( dist ` w ) |` ( b X. b ) ) e. ( CMet ` b ) }
17	15, 16	elrab2 3366	1 |- ( M e. CMetSp <-> ( M e. MetSp /\ D e. ( CMet ` X ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   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:  cmscmet  23143  cmsms  23145  cmspropd  23146  cmsss  23147  cncms  23151