Theorem cmspropd 25316

Description: Property deduction for a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)

Hypotheses

Ref	Expression
cmspropd.1	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
cmspropd.2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
cmspropd.3	⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
cmspropd.4	⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))

Assertion

Ref	Expression
cmspropd	⊢ (𝜑 → (𝐾 ∈ CMetSp ↔ 𝐿 ∈ CMetSp))

Proof of Theorem cmspropd

Step	Hyp	Ref	Expression
1		cmspropd.1	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
2		cmspropd.2	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
3		cmspropd.3	. . . 4 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
4		cmspropd.4	. . . 4 ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
5	1, 2, 3, 4	mspropd 24439	. . 3 ⊢ (𝜑 → (𝐾 ∈ MetSp ↔ 𝐿 ∈ MetSp))
6	1	sqxpeqd 5663	. . . . . . 7 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
7	6	reseq2d 5944	. . . . . 6 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
8	3, 7	eqtr3d 2773	. . . . 5 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
9	2	sqxpeqd 5663	. . . . . 6 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
10	9	reseq2d 5944	. . . . 5 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
11	8, 10	eqtr3d 2773	. . . 4 ⊢ (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
12	1, 2	eqtr3d 2773	. . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
13	12	fveq2d 6844	. . . 4 ⊢ (𝜑 → (CMet‘(Base‘𝐾)) = (CMet‘(Base‘𝐿)))
14	11, 13	eleq12d 2830	. . 3 ⊢ (𝜑 → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (CMet‘(Base‘𝐾)) ↔ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) ∈ (CMet‘(Base‘𝐿))))
15	5, 14	anbi12d 633	. 2 ⊢ (𝜑 → ((𝐾 ∈ MetSp ∧ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (CMet‘(Base‘𝐾))) ↔ (𝐿 ∈ MetSp ∧ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) ∈ (CMet‘(Base‘𝐿)))))
16		eqid 2736	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
17		eqid 2736	. . 3 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
18	16, 17	iscms 25312	. 2 ⊢ (𝐾 ∈ CMetSp ↔ (𝐾 ∈ MetSp ∧ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (CMet‘(Base‘𝐾))))
19		eqid 2736	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
20		eqid 2736	. . 3 ⊢ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
21	19, 20	iscms 25312	. 2 ⊢ (𝐿 ∈ CMetSp ↔ (𝐿 ∈ MetSp ∧ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) ∈ (CMet‘(Base‘𝐿))))
22	15, 18, 21	3bitr4g 314	1 ⊢ (𝜑 → (𝐾 ∈ CMetSp ↔ 𝐿 ∈ CMetSp))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   × cxp 5629   ↾ cres 5633  ‘cfv 6498  Basecbs 17179  distcds 17229  TopOpenctopn 17384  MetSpcms 24283  CMetccmet 25221  CMetSpccms 25299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-res 5643  df-iota 6454  df-fun 6500  df-fv 6506  df-top 22859  df-topon 22876  df-topsp 22898  df-xms 24285  df-ms 24286  df-cms 25302

This theorem is referenced by:  srabn  25327