Theorem iscms 25461

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))

Assertion

Ref	Expression
iscms	⊢ (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (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 6886	. . 3 ⊢ (𝑤 = 𝑀 → (Base‘𝑤) ∈ V)
2		fveq2 6871	. . . . . . 7 ⊢ (𝑤 = 𝑀 → (dist‘𝑤) = (dist‘𝑀))
3	2	adantr 485	. . . . . 6 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → (dist‘𝑤) = (dist‘𝑀))
4		id 23	. . . . . . . 8 ⊢ (𝑏 = (Base‘𝑤) → 𝑏 = (Base‘𝑤))
5		fveq2 6871	. . . . . . . . 9 ⊢ (𝑤 = 𝑀 → (Base‘𝑤) = (Base‘𝑀))
6		iscms.1	. . . . . . . . 9 ⊢ 𝑋 = (Base‘𝑀)
7	5, 6	eqtr4di 2818	. . . . . . . 8 ⊢ (𝑤 = 𝑀 → (Base‘𝑤) = 𝑋)
8	4, 7	sylan9eqr 2822	. . . . . . 7 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → 𝑏 = 𝑋)
9	8	sqxpeqd 5683	. . . . . 6 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → (𝑏 × 𝑏) = (𝑋 × 𝑋))
10	3, 9	reseq12d 5969	. . . . 5 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → ((dist‘𝑤) ↾ (𝑏 × 𝑏)) = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))
11		iscms.2	. . . . 5 ⊢ 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))
12	10, 11	eqtr4di 2818	. . . 4 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → ((dist‘𝑤) ↾ (𝑏 × 𝑏)) = 𝐷)
13	8	fveq2d 6875	. . . 4 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → (CMet‘𝑏) = (CMet‘𝑋))
14	12, 13	eleq12d 2859	. . 3 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → (((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏) ↔ 𝐷 ∈ (CMet‘𝑋)))
15	1, 14	sbcied 3790	. 2 ⊢ (𝑤 = 𝑀 → ([(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏) ↔ 𝐷 ∈ (CMet‘𝑋)))
16		df-cms 25451	. 2 ⊢ CMetSp = {𝑤 ∈ MetSp ∣ [(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏)}
17	15, 16	elrab2 3657	1 ⊢ (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋)))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457  [wsbc 3747   × cxp 5649   ↾ cres 5653  ‘cfv 6525  Basecbs 17257  distcds 17307  MetSpcms 24432  CMetccmet 25370  CMetSpccms 25448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-nul 5260

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-opab 5167  df-xp 5657  df-res 5663  df-iota 6481  df-fv 6533  df-cms 25451

This theorem is referenced by:  cmscmet  25462  cmsms  25464  cmspropd  25465  cmssmscld  25466  cmsss  25467  cncms  25471  cmscsscms  25489  cssbn  25491