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Theorem cmspropd 24494
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 23608 . . 3 (𝜑 → (𝐾 ∈ MetSp ↔ 𝐿 ∈ MetSp))
61sqxpeqd 5620 . . . . . . 7 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
76reseq2d 5888 . . . . . 6 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
83, 7eqtr3d 2781 . . . . 5 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
92sqxpeqd 5620 . . . . . 6 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
109reseq2d 5888 . . . . 5 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
118, 10eqtr3d 2781 . . . 4 (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
121, 2eqtr3d 2781 . . . . 5 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
1312fveq2d 6772 . . . 4 (𝜑 → (CMet‘(Base‘𝐾)) = (CMet‘(Base‘𝐿)))
1411, 13eleq12d 2834 . . 3 (𝜑 → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (CMet‘(Base‘𝐾)) ↔ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) ∈ (CMet‘(Base‘𝐿))))
155, 14anbi12d 630 . 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 2739 . . 3 (Base‘𝐾) = (Base‘𝐾)
17 eqid 2739 . . 3 ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
1816, 17iscms 24490 . 2 (𝐾 ∈ CMetSp ↔ (𝐾 ∈ MetSp ∧ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (CMet‘(Base‘𝐾))))
19 eqid 2739 . . 3 (Base‘𝐿) = (Base‘𝐿)
20 eqid 2739 . . 3 ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
2119, 20iscms 24490 . 2 (𝐿 ∈ CMetSp ↔ (𝐿 ∈ MetSp ∧ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) ∈ (CMet‘(Base‘𝐿))))
2215, 18, 213bitr4g 313 1 (𝜑 → (𝐾 ∈ CMetSp ↔ 𝐿 ∈ CMetSp))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1541  wcel 2109   × cxp 5586  cres 5590  cfv 6430  Basecbs 16893  distcds 16952  TopOpenctopn 17113  MetSpcms 23452  CMetccmet 24399  CMetSpccms 24477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-res 5600  df-iota 6388  df-fun 6432  df-fv 6438  df-top 22024  df-topon 22041  df-topsp 22063  df-xms 23454  df-ms 23455  df-cms 24480
This theorem is referenced by:  srabn  24505
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