Theorem iscms 25297

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 6891	. . 3 ⊢ (𝑤 = 𝑀 → (Base‘𝑤) ∈ V)
2		fveq2 6876	. . . . . . 7 ⊢ (𝑤 = 𝑀 → (dist‘𝑤) = (dist‘𝑀))
3	2	adantr 480	. . . . . 6 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → (dist‘𝑤) = (dist‘𝑀))
4		id 22	. . . . . . . 8 ⊢ (𝑏 = (Base‘𝑤) → 𝑏 = (Base‘𝑤))
5		fveq2 6876	. . . . . . . . 9 ⊢ (𝑤 = 𝑀 → (Base‘𝑤) = (Base‘𝑀))
6		iscms.1	. . . . . . . . 9 ⊢ 𝑋 = (Base‘𝑀)
7	5, 6	eqtr4di 2788	. . . . . . . 8 ⊢ (𝑤 = 𝑀 → (Base‘𝑤) = 𝑋)
8	4, 7	sylan9eqr 2792	. . . . . . 7 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → 𝑏 = 𝑋)
9	8	sqxpeqd 5686	. . . . . 6 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → (𝑏 × 𝑏) = (𝑋 × 𝑋))
10	3, 9	reseq12d 5967	. . . . 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 2788	. . . 4 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → ((dist‘𝑤) ↾ (𝑏 × 𝑏)) = 𝐷)
13	8	fveq2d 6880	. . . 4 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → (CMet‘𝑏) = (CMet‘𝑋))
14	12, 13	eleq12d 2828	. . 3 ⊢ ((𝑤 = 𝑀 ∧ 𝑏 = (Base‘𝑤)) → (((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏) ↔ 𝐷 ∈ (CMet‘𝑋)))
15	1, 14	sbcied 3809	. 2 ⊢ (𝑤 = 𝑀 → ([(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏) ↔ 𝐷 ∈ (CMet‘𝑋)))
16		df-cms 25287	. 2 ⊢ CMetSp = {𝑤 ∈ MetSp ∣ [(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏)}
17	15, 16	elrab2 3674	1 ⊢ (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3459  [wsbc 3765   × cxp 5652   ↾ cres 5656  ‘cfv 6531  Basecbs 17228  distcds 17280  MetSpcms 24257  CMetccmet 25206  CMetSpccms 25284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-nul 5276

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-xp 5660  df-res 5666  df-iota 6484  df-fv 6539  df-cms 25287

This theorem is referenced by:  cmscmet  25298  cmsms  25300  cmspropd  25301  cmssmscld  25302  cmsss  25303  cncms  25307  cmscsscms  25325  cssbn  25327