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Theorem cmspropd 24716
Description: Property deduction for a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
Hypotheses
Ref Expression
cmspropd.1 (𝜑𝐵 = (Base‘𝐾))
cmspropd.2 (𝜑𝐵 = (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
StepHypRef Expression
1 cmspropd.1 . . . 4 (𝜑𝐵 = (Base‘𝐾))
2 cmspropd.2 . . . 4 (𝜑𝐵 = (Base‘𝐿))
3 cmspropd.3 . . . 4 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
4 cmspropd.4 . . . 4 (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
51, 2, 3, 4mspropd 23830 . . 3 (𝜑 → (𝐾 ∈ MetSp ↔ 𝐿 ∈ MetSp))
61sqxpeqd 5666 . . . . . . 7 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
76reseq2d 5938 . . . . . 6 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
83, 7eqtr3d 2779 . . . . 5 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
92sqxpeqd 5666 . . . . . 6 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
109reseq2d 5938 . . . . 5 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
118, 10eqtr3d 2779 . . . 4 (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
121, 2eqtr3d 2779 . . . . 5 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
1312fveq2d 6847 . . . 4 (𝜑 → (CMet‘(Base‘𝐾)) = (CMet‘(Base‘𝐿)))
1411, 13eleq12d 2832 . . 3 (𝜑 → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (CMet‘(Base‘𝐾)) ↔ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) ∈ (CMet‘(Base‘𝐿))))
155, 14anbi12d 632 . 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 2737 . . 3 (Base‘𝐾) = (Base‘𝐾)
17 eqid 2737 . . 3 ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
1816, 17iscms 24712 . 2 (𝐾 ∈ CMetSp ↔ (𝐾 ∈ MetSp ∧ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (CMet‘(Base‘𝐾))))
19 eqid 2737 . . 3 (Base‘𝐿) = (Base‘𝐿)
20 eqid 2737 . . 3 ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
2119, 20iscms 24712 . 2 (𝐿 ∈ CMetSp ↔ (𝐿 ∈ MetSp ∧ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) ∈ (CMet‘(Base‘𝐿))))
2215, 18, 213bitr4g 314 1 (𝜑 → (𝐾 ∈ CMetSp ↔ 𝐿 ∈ CMetSp))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107   × cxp 5632  cres 5636  cfv 6497  Basecbs 17084  distcds 17143  TopOpenctopn 17304  MetSpcms 23674  CMetccmet 24621  CMetSpccms 24699
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-res 5646  df-iota 6449  df-fun 6499  df-fv 6505  df-top 22246  df-topon 22263  df-topsp 22285  df-xms 23676  df-ms 23677  df-cms 24702
This theorem is referenced by:  srabn  24727
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