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Theorem pstmval 30247
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 30241 . . 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 22311 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
43adantr 472 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑋 = dom dom 𝐷)
5 dmeq 5479 . . . . . . . . 9 (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
65dmeqd 5481 . . . . . . . 8 (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
76adantl 473 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = dom dom 𝐷)
84, 7eqtr4d 2797 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑋 = dom dom 𝑑)
9 qseq1 7963 . . . . . 6 (𝑋 = dom dom 𝑑 → (𝑋 / ) = (dom dom 𝑑 / ))
108, 9syl 17 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑋 / ) = (dom dom 𝑑 / ))
11 fveq2 6352 . . . . . . . 8 (𝑑 = 𝐷 → (~Met𝑑) = (~Met𝐷))
12 pstmval.1 . . . . . . . 8 = (~Met𝐷)
1311, 12syl6reqr 2813 . . . . . . 7 (𝑑 = 𝐷 = (~Met𝑑))
14 qseq2 7964 . . . . . . 7 ( = (~Met𝑑) → (dom dom 𝑑 / ) = (dom dom 𝑑 / (~Met𝑑)))
1513, 14syl 17 . . . . . 6 (𝑑 = 𝐷 → (dom dom 𝑑 / ) = (dom dom 𝑑 / (~Met𝑑)))
1615adantl 473 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (dom dom 𝑑 / ) = (dom dom 𝑑 / (~Met𝑑)))
1710, 16eqtr2d 2795 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (dom dom 𝑑 / (~Met𝑑)) = (𝑋 / ))
18 mpt2eq12 6880 . . . 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 696 . . 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 1241 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ) ∧ 𝑏 ∈ (𝑋 / )) → 𝑑 = 𝐷)
2120oveqd 6830 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ) ∧ 𝑏 ∈ (𝑋 / )) → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
2221eqeq2d 2770 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ) ∧ 𝑏 ∈ (𝑋 / )) → (𝑧 = (𝑥𝑑𝑦) ↔ 𝑧 = (𝑥𝐷𝑦)))
23222rexbidv 3195 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ) ∧ 𝑏 ∈ (𝑋 / )) → (∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦) ↔ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)))
2423abbidv 2879 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ) ∧ 𝑏 ∈ (𝑋 / )) → {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦)} = {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)})
2524unieqd 4598 . . . 4 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ) ∧ 𝑏 ∈ (𝑋 / )) → {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦)} = {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)})
2625mpt2eq3dva 6884 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑎 ∈ (𝑋 / ), 𝑏 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦)}) = (𝑎 ∈ (𝑋 / ), 𝑏 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)}))
2719, 26eqtrd 2794 . 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 6381 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ dom PsMet)
29 fveq2 6352 . . . . . 6 (𝑥 = 𝑋 → (PsMet‘𝑥) = (PsMet‘𝑋))
3029eleq2d 2825 . . . . 5 (𝑥 = 𝑋 → (𝐷 ∈ (PsMet‘𝑥) ↔ 𝐷 ∈ (PsMet‘𝑋)))
3130rspcev 3449 . . . 4 ((𝑋 ∈ dom PsMet ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥))
3228, 31mpancom 706 . . 3 (𝐷 ∈ (PsMet‘𝑋) → ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥))
33 df-psmet 19940 . . . . 5 PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑎𝑥 ((𝑎𝑑𝑎) = 0 ∧ ∀𝑏𝑥𝑐𝑥 (𝑎𝑑𝑏) ≤ ((𝑐𝑑𝑎) +𝑒 (𝑐𝑑𝑏)))})
3433funmpt2 6088 . . . 4 Fun PsMet
35 elunirn 6672 . . . 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 6382 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
39 qsexg 7972 . . . 4 (𝑋 ∈ V → (𝑋 / ) ∈ V)
4038, 39syl 17 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (𝑋 / ) ∈ V)
41 mpt2exga 7414 . . 3 (((𝑋 / ) ∈ V ∧ (𝑋 / ) ∈ V) → (𝑎 ∈ (𝑋 / ), 𝑏 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)}) ∈ V)
4240, 40, 41syl2anc 696 . 2 (𝐷 ∈ (PsMet‘𝑋) → (𝑎 ∈ (𝑋 / ), 𝑏 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)}) ∈ V)
432, 27, 37, 42fvmptd 6450 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 1072   = wceq 1632  wcel 2139  {cab 2746  wral 3050  wrex 3051  {crab 3054  Vcvv 3340   cuni 4588   class class class wbr 4804  cmpt 4881   × cxp 5264  dom cdm 5266  ran crn 5267  Fun wfun 6043  cfv 6049  (class class class)co 6813  cmpt2 6815   / cqs 7910  𝑚 cmap 8023  0cc0 10128  *cxr 10265  cle 10267   +𝑒 cxad 12137  PsMetcpsmet 19932  ~Metcmetid 30238  pstoMetcpstm 30239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-ec 7913  df-qs 7917  df-map 8025  df-xr 10270  df-psmet 19940  df-pstm 30241
This theorem is referenced by:  pstmfval  30248  pstmxmet  30249
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