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Theorem mspropd 23980
Description: Property deduction for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
xmspropd.1 (𝜑𝐵 = (Base‘𝐾))
xmspropd.2 (𝜑𝐵 = (Base‘𝐿))
xmspropd.3 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
xmspropd.4 (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
Assertion
Ref Expression
mspropd (𝜑 → (𝐾 ∈ MetSp ↔ 𝐿 ∈ MetSp))

Proof of Theorem mspropd
StepHypRef Expression
1 xmspropd.1 . . . 4 (𝜑𝐵 = (Base‘𝐾))
2 xmspropd.2 . . . 4 (𝜑𝐵 = (Base‘𝐿))
3 xmspropd.3 . . . 4 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
4 xmspropd.4 . . . 4 (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
51, 2, 3, 4xmspropd 23979 . . 3 (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐿 ∈ ∞MetSp))
61sqxpeqd 5709 . . . . . . 7 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
76reseq2d 5982 . . . . . 6 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
83, 7eqtr3d 2775 . . . . 5 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
92sqxpeqd 5709 . . . . . 6 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
109reseq2d 5982 . . . . 5 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
118, 10eqtr3d 2775 . . . 4 (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
121, 2eqtr3d 2775 . . . . 5 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
1312fveq2d 6896 . . . 4 (𝜑 → (Met‘(Base‘𝐾)) = (Met‘(Base‘𝐿)))
1411, 13eleq12d 2828 . . 3 (𝜑 → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾)) ↔ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) ∈ (Met‘(Base‘𝐿))))
155, 14anbi12d 632 . 2 (𝜑 → ((𝐾 ∈ ∞MetSp ∧ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾))) ↔ (𝐿 ∈ ∞MetSp ∧ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) ∈ (Met‘(Base‘𝐿)))))
16 eqid 2733 . . 3 (TopOpen‘𝐾) = (TopOpen‘𝐾)
17 eqid 2733 . . 3 (Base‘𝐾) = (Base‘𝐾)
18 eqid 2733 . . 3 ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
1916, 17, 18isms 23955 . 2 (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾))))
20 eqid 2733 . . 3 (TopOpen‘𝐿) = (TopOpen‘𝐿)
21 eqid 2733 . . 3 (Base‘𝐿) = (Base‘𝐿)
22 eqid 2733 . . 3 ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
2320, 21, 22isms 23955 . 2 (𝐿 ∈ MetSp ↔ (𝐿 ∈ ∞MetSp ∧ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) ∈ (Met‘(Base‘𝐿))))
2415, 19, 233bitr4g 314 1 (𝜑 → (𝐾 ∈ MetSp ↔ 𝐿 ∈ MetSp))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107   × cxp 5675  cres 5679  cfv 6544  Basecbs 17144  distcds 17206  TopOpenctopn 17367  Metcmet 20930  ∞MetSpcxms 23823  MetSpcms 23824
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 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-res 5689  df-iota 6496  df-fun 6546  df-fv 6552  df-top 22396  df-topon 22413  df-topsp 22435  df-xms 23826  df-ms 23827
This theorem is referenced by:  ngppropd  24146  cmspropd  24866  zhmnrg  32947
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