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Theorem xmspropd 23010
Description: Property deduction for an extended 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
xmspropd (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐿 ∈ ∞MetSp))

Proof of Theorem xmspropd
StepHypRef Expression
1 xmspropd.1 . . . . 5 (𝜑𝐵 = (Base‘𝐾))
2 xmspropd.2 . . . . 5 (𝜑𝐵 = (Base‘𝐿))
31, 2eqtr3d 2855 . . . 4 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
4 xmspropd.4 . . . 4 (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
53, 4tpspropd 21474 . . 3 (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))
6 xmspropd.3 . . . . . . 7 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
71sqxpeqd 5580 . . . . . . . 8 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
87reseq2d 5846 . . . . . . 7 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
96, 8eqtr3d 2855 . . . . . 6 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
102sqxpeqd 5580 . . . . . . 7 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
1110reseq2d 5846 . . . . . 6 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
129, 11eqtr3d 2855 . . . . 5 (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
1312fveq2d 6667 . . . 4 (𝜑 → (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) = (MetOpen‘((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))))
144, 13eqeq12d 2834 . . 3 (𝜑 → ((TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) ↔ (TopOpen‘𝐿) = (MetOpen‘((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))))
155, 14anbi12d 630 . 2 (𝜑 → ((𝐾 ∈ TopSp ∧ (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))) ↔ (𝐿 ∈ TopSp ∧ (TopOpen‘𝐿) = (MetOpen‘((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))))))
16 eqid 2818 . . 3 (TopOpen‘𝐾) = (TopOpen‘𝐾)
17 eqid 2818 . . 3 (Base‘𝐾) = (Base‘𝐾)
18 eqid 2818 . . 3 ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
1916, 17, 18isxms 22984 . 2 (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))))
20 eqid 2818 . . 3 (TopOpen‘𝐿) = (TopOpen‘𝐿)
21 eqid 2818 . . 3 (Base‘𝐿) = (Base‘𝐿)
22 eqid 2818 . . 3 ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
2320, 21, 22isxms 22984 . 2 (𝐿 ∈ ∞MetSp ↔ (𝐿 ∈ TopSp ∧ (TopOpen‘𝐿) = (MetOpen‘((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))))
2415, 19, 233bitr4g 315 1 (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐿 ∈ ∞MetSp))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105   × cxp 5546  cres 5550  cfv 6348  Basecbs 16471  distcds 16562  TopOpenctopn 16683  MetOpencmopn 20463  TopSpctps 21468  ∞MetSpcxms 22854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-res 5560  df-iota 6307  df-fun 6350  df-fv 6356  df-top 21430  df-topon 21447  df-topsp 21469  df-xms 22857
This theorem is referenced by:  mspropd  23011
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