Theorem iscms 24509

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 6789	. . 3 ⊢ (𝑤 = 𝑀 → (Base‘𝑤) ∈ V)
2		fveq2 6774	. . . . . . 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 6774	. . . . . . . . 9 ⊢ (𝑤 = 𝑀 → (Base‘𝑤) = (Base‘𝑀))
6		iscms.1	. . . . . . . . 9 ⊢ 𝑋 = (Base‘𝑀)
7	5, 6	eqtr4di 2796	. . . . . . . 8 ⊢ (𝑤 = 𝑀 → (Base‘𝑤) = 𝑋)
8	4, 7	sylan9eqr 2800	. . . . . . 7 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → 𝑏 = 𝑋)
9	8	sqxpeqd 5621	. . . . . 6 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → (𝑏 × 𝑏) = (𝑋 × 𝑋))
10	3, 9	reseq12d 5892	. . . . 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 2796	. . . 4 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → ((dist‘𝑤) ↾ (𝑏 × 𝑏)) = 𝐷)
13	8	fveq2d 6778	. . . 4 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → (CMet‘𝑏) = (CMet‘𝑋))
14	12, 13	eleq12d 2833	. . 3 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → (((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏) ↔ 𝐷 ∈ (CMet‘𝑋)))
15	1, 14	sbcied 3761	. 2 ⊢ (𝑤 = 𝑀 → ([(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏) ↔ 𝐷 ∈ (CMet‘𝑋)))
16		df-cms 24499	. 2 ⊢ CMetSp = {𝑤 ∈ MetSp ∣ [(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏)}
17	15, 16	elrab2 3627	1 ⊢ (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋)))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432  [wsbc 3716   × cxp 5587   ↾ cres 5591  ‘cfv 6433  Basecbs 16912  distcds 16971  MetSpcms 23471  CMetccmet 24418  CMetSpccms 24496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-res 5601  df-iota 6391  df-fv 6441  df-cms 24499

This theorem is referenced by:  cmscmet  24510  cmsms  24512  cmspropd  24513  cmssmscld  24514  cmsss  24515  cncms  24519  cmscsscms  24537  cssbn  24539