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Theorem mspropd 23084
 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 23083 . . 3 (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐿 ∈ ∞MetSp))
61sqxpeqd 5574 . . . . . . 7 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
76reseq2d 5840 . . . . . 6 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
83, 7eqtr3d 2861 . . . . 5 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
92sqxpeqd 5574 . . . . . 6 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
109reseq2d 5840 . . . . 5 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
118, 10eqtr3d 2861 . . . 4 (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
121, 2eqtr3d 2861 . . . . 5 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
1312fveq2d 6665 . . . 4 (𝜑 → (Met‘(Base‘𝐾)) = (Met‘(Base‘𝐿)))
1411, 13eleq12d 2910 . . 3 (𝜑 → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾)) ↔ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) ∈ (Met‘(Base‘𝐿))))
155, 14anbi12d 633 . 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 2824 . . 3 (TopOpen‘𝐾) = (TopOpen‘𝐾)
17 eqid 2824 . . 3 (Base‘𝐾) = (Base‘𝐾)
18 eqid 2824 . . 3 ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
1916, 17, 18isms 23059 . 2 (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾))))
20 eqid 2824 . . 3 (TopOpen‘𝐿) = (TopOpen‘𝐿)
21 eqid 2824 . . 3 (Base‘𝐿) = (Base‘𝐿)
22 eqid 2824 . . 3 ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
2320, 21, 22isms 23059 . 2 (𝐿 ∈ MetSp ↔ (𝐿 ∈ ∞MetSp ∧ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) ∈ (Met‘(Base‘𝐿))))
2415, 19, 233bitr4g 317 1 (𝜑 → (𝐾 ∈ MetSp ↔ 𝐿 ∈ MetSp))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115   × cxp 5540   ↾ cres 5544  ‘cfv 6343  Basecbs 16483  distcds 16574  TopOpenctopn 16695  Metcmet 20531  ∞MetSpcxms 22927  MetSpcms 22928 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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-res 5554  df-iota 6302  df-fun 6345  df-fv 6351  df-top 21502  df-topon 21519  df-topsp 21541  df-xms 22930  df-ms 22931 This theorem is referenced by:  ngppropd  23246  cmspropd  23956  zhmnrg  31265
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