Theorem cmspropd 25402

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 24505	. . 3 ⊢ (𝜑 → (𝐾 ∈ MetSp ↔ 𝐿 ∈ MetSp))
6	1	sqxpeqd 5732	. . . . . . 7 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
7	6	reseq2d 6009	. . . . . 6 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
8	3, 7	eqtr3d 2782	. . . . 5 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
9	2	sqxpeqd 5732	. . . . . 6 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
10	9	reseq2d 6009	. . . . 5 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
11	8, 10	eqtr3d 2782	. . . 4 ⊢ (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
12	1, 2	eqtr3d 2782	. . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
13	12	fveq2d 6924	. . . 4 ⊢ (𝜑 → (CMet‘(Base‘𝐾)) = (CMet‘(Base‘𝐿)))
14	11, 13	eleq12d 2838	. . 3 ⊢ (𝜑 → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (CMet‘(Base‘𝐾)) ↔ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) ∈ (CMet‘(Base‘𝐿))))
15	5, 14	anbi12d 631	. 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 2740	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
17		eqid 2740	. . 3 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
18	16, 17	iscms 25398	. 2 ⊢ (𝐾 ∈ CMetSp ↔ (𝐾 ∈ MetSp ∧ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (CMet‘(Base‘𝐾))))
19		eqid 2740	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
20		eqid 2740	. . 3 ⊢ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
21	19, 20	iscms 25398	. 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 1537   ∈ wcel 2108   × cxp 5698   ↾ cres 5702  ‘cfv 6573  Basecbs 17258  distcds 17320  TopOpenctopn 17481  MetSpcms 24349  CMetccmet 25307  CMetSpccms 25385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-res 5712  df-iota 6525  df-fun 6575  df-fv 6581  df-top 22921  df-topon 22938  df-topsp 22960  df-xms 24351  df-ms 24352  df-cms 25388

This theorem is referenced by:  srabn  25413