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Theorem xmspropd 13870
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 2212 . . . 4 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
4 xmspropd.4 . . . 4 (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
53, 4tpspropd 13427 . . 3 (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))
6 xmspropd.3 . . . . . . 7 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
71sqxpeqd 4652 . . . . . . . 8 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
87reseq2d 4907 . . . . . . 7 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
96, 8eqtr3d 2212 . . . . . 6 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
102sqxpeqd 4652 . . . . . . 7 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
1110reseq2d 4907 . . . . . 6 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
129, 11eqtr3d 2212 . . . . 5 (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
1312fveq2d 5519 . . . 4 (𝜑 → (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) = (MetOpen‘((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))))
144, 13eqeq12d 2192 . . 3 (𝜑 → ((TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) ↔ (TopOpen‘𝐿) = (MetOpen‘((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))))
155, 14anbi12d 473 . 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 2177 . . 3 (TopOpen‘𝐾) = (TopOpen‘𝐾)
17 eqid 2177 . . 3 (Base‘𝐾) = (Base‘𝐾)
18 eqid 2177 . . 3 ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
1916, 17, 18isxms 13844 . 2 (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))))
20 eqid 2177 . . 3 (TopOpen‘𝐿) = (TopOpen‘𝐿)
21 eqid 2177 . . 3 (Base‘𝐿) = (Base‘𝐿)
22 eqid 2177 . . 3 ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
2320, 21, 22isxms 13844 . 2 (𝐿 ∈ ∞MetSp ↔ (𝐿 ∈ TopSp ∧ (TopOpen‘𝐿) = (MetOpen‘((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))))
2415, 19, 233bitr4g 223 1 (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐿 ∈ ∞MetSp))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148   × cxp 4624  cres 4628  cfv 5216  Basecbs 12456  distcds 12539  TopOpenctopn 12679  MetOpencmopn 13336  TopSpctps 13421  ∞MetSpcxms 13729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-cnex 7901  ax-resscn 7902  ax-1re 7904  ax-addrcl 7907
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-inn 8918  df-2 8976  df-3 8977  df-4 8978  df-5 8979  df-6 8980  df-7 8981  df-8 8982  df-9 8983  df-ndx 12459  df-slot 12460  df-base 12462  df-tset 12549  df-rest 12680  df-topn 12681  df-top 13389  df-topon 13402  df-topsp 13422  df-xms 13732
This theorem is referenced by:  mspropd  13871
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