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Theorem pstmval 30066
Description: Value of the metric induced by a pseudometric 𝐷. (Contributed by Thierry Arnoux, 7-Feb-2018.)
Hypothesis
Ref Expression
pstmval.1 = (~Met𝐷)
Assertion
Ref Expression
pstmval (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑎 ∈ (𝑋 / ), 𝑏 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)}))
Distinct variable groups:   𝑎,𝑏,𝑥,𝑦,𝑧,𝐷   𝑋,𝑎,𝑏,𝑥,𝑦,𝑧   ,𝑎,𝑏,𝑥,𝑦,𝑧

Proof of Theorem pstmval
Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pstm 30060 . . 3 pstoMet = (𝑑 ran PsMet ↦ (𝑎 ∈ (dom dom 𝑑 / (~Met𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met𝑑)) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦)}))
21a1i 11 . 2 (𝐷 ∈ (PsMet‘𝑋) → pstoMet = (𝑑 ran PsMet ↦ (𝑎 ∈ (dom dom 𝑑 / (~Met𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met𝑑)) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦)})))
3 psmetdmdm 22157 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
43adantr 480 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑋 = dom dom 𝐷)
5 dmeq 5356 . . . . . . . . 9 (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
65dmeqd 5358 . . . . . . . 8 (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
76adantl 481 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = dom dom 𝐷)
84, 7eqtr4d 2688 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑋 = dom dom 𝑑)
9 qseq1 7839 . . . . . 6 (𝑋 = dom dom 𝑑 → (𝑋 / ) = (dom dom 𝑑 / ))
108, 9syl 17 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑋 / ) = (dom dom 𝑑 / ))
11 fveq2 6229 . . . . . . . 8 (𝑑 = 𝐷 → (~Met𝑑) = (~Met𝐷))
12 pstmval.1 . . . . . . . 8 = (~Met𝐷)
1311, 12syl6reqr 2704 . . . . . . 7 (𝑑 = 𝐷 = (~Met𝑑))
14 qseq2 7840 . . . . . . 7 ( = (~Met𝑑) → (dom dom 𝑑 / ) = (dom dom 𝑑 / (~Met𝑑)))
1513, 14syl 17 . . . . . 6 (𝑑 = 𝐷 → (dom dom 𝑑 / ) = (dom dom 𝑑 / (~Met𝑑)))
1615adantl 481 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (dom dom 𝑑 / ) = (dom dom 𝑑 / (~Met𝑑)))
1710, 16eqtr2d 2686 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (dom dom 𝑑 / (~Met𝑑)) = (𝑋 / ))
18 mpt2eq12 6757 . . . 4 (((dom dom 𝑑 / (~Met𝑑)) = (𝑋 / ) ∧ (dom dom 𝑑 / (~Met𝑑)) = (𝑋 / )) → (𝑎 ∈ (dom dom 𝑑 / (~Met𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met𝑑)) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦)}) = (𝑎 ∈ (𝑋 / ), 𝑏 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦)}))
1917, 17, 18syl2anc 694 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑎 ∈ (dom dom 𝑑 / (~Met𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met𝑑)) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦)}) = (𝑎 ∈ (𝑋 / ), 𝑏 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦)}))
20 simp1r 1106 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ) ∧ 𝑏 ∈ (𝑋 / )) → 𝑑 = 𝐷)
2120oveqd 6707 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ) ∧ 𝑏 ∈ (𝑋 / )) → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
2221eqeq2d 2661 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ) ∧ 𝑏 ∈ (𝑋 / )) → (𝑧 = (𝑥𝑑𝑦) ↔ 𝑧 = (𝑥𝐷𝑦)))
23222rexbidv 3086 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ) ∧ 𝑏 ∈ (𝑋 / )) → (∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦) ↔ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)))
2423abbidv 2770 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ) ∧ 𝑏 ∈ (𝑋 / )) → {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦)} = {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)})
2524unieqd 4478 . . . 4 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ) ∧ 𝑏 ∈ (𝑋 / )) → {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦)} = {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)})
2625mpt2eq3dva 6761 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑎 ∈ (𝑋 / ), 𝑏 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦)}) = (𝑎 ∈ (𝑋 / ), 𝑏 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)}))
2719, 26eqtrd 2685 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑎 ∈ (dom dom 𝑑 / (~Met𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met𝑑)) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦)}) = (𝑎 ∈ (𝑋 / ), 𝑏 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)}))
28 elfvdm 6258 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ dom PsMet)
29 fveq2 6229 . . . . . 6 (𝑥 = 𝑋 → (PsMet‘𝑥) = (PsMet‘𝑋))
3029eleq2d 2716 . . . . 5 (𝑥 = 𝑋 → (𝐷 ∈ (PsMet‘𝑥) ↔ 𝐷 ∈ (PsMet‘𝑋)))
3130rspcev 3340 . . . 4 ((𝑋 ∈ dom PsMet ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥))
3228, 31mpancom 704 . . 3 (𝐷 ∈ (PsMet‘𝑋) → ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥))
33 df-psmet 19786 . . . . 5 PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑎𝑥 ((𝑎𝑑𝑎) = 0 ∧ ∀𝑏𝑥𝑐𝑥 (𝑎𝑑𝑏) ≤ ((𝑐𝑑𝑎) +𝑒 (𝑐𝑑𝑏)))})
3433funmpt2 5965 . . . 4 Fun PsMet
35 elunirn 6549 . . . 4 (Fun PsMet → (𝐷 ran PsMet ↔ ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥)))
3634, 35ax-mp 5 . . 3 (𝐷 ran PsMet ↔ ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥))
3732, 36sylibr 224 . 2 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ran PsMet)
38 elfvex 6259 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
39 qsexg 7848 . . . 4 (𝑋 ∈ V → (𝑋 / ) ∈ V)
4038, 39syl 17 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (𝑋 / ) ∈ V)
41 mpt2exga 7291 . . 3 (((𝑋 / ) ∈ V ∧ (𝑋 / ) ∈ V) → (𝑎 ∈ (𝑋 / ), 𝑏 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)}) ∈ V)
4240, 40, 41syl2anc 694 . 2 (𝐷 ∈ (PsMet‘𝑋) → (𝑎 ∈ (𝑋 / ), 𝑏 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)}) ∈ V)
432, 27, 37, 42fvmptd 6327 1 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑎 ∈ (𝑋 / ), 𝑏 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  {cab 2637  wral 2941  wrex 2942  {crab 2945  Vcvv 3231   cuni 4468   class class class wbr 4685  cmpt 4762   × cxp 5141  dom cdm 5143  ran crn 5144  Fun wfun 5920  cfv 5926  (class class class)co 6690  cmpt2 6692   / cqs 7786  𝑚 cmap 7899  0cc0 9974  *cxr 10111  cle 10113   +𝑒 cxad 11982  PsMetcpsmet 19778  ~Metcmetid 30057  pstoMetcpstm 30058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-ec 7789  df-qs 7793  df-map 7901  df-xr 10116  df-psmet 19786  df-pstm 30060
This theorem is referenced by:  pstmfval  30067  pstmxmet  30068
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