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Theorem metidv 31245
Description: 𝐴 and 𝐵 identify by the metric 𝐷 if their distance is zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)
Assertion
Ref Expression
metidv ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴(~Met𝐷)𝐵 ↔ (𝐴𝐷𝐵) = 0))

Proof of Theorem metidv
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2877 . . . . . 6 (𝑎 = 𝐴 → (𝑎𝑋𝐴𝑋))
2 eleq1 2877 . . . . . 6 (𝑏 = 𝐵 → (𝑏𝑋𝐵𝑋))
31, 2bi2anan9 638 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑎𝑋𝑏𝑋) ↔ (𝐴𝑋𝐵𝑋)))
4 oveq12 7144 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎𝐷𝑏) = (𝐴𝐷𝐵))
54eqeq1d 2800 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑎𝐷𝑏) = 0 ↔ (𝐴𝐷𝐵) = 0))
63, 5anbi12d 633 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → (((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0) ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) = 0)))
7 eqid 2798 . . . 4 {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)}
86, 7brabga 5386 . . 3 ((𝐴𝑋𝐵𝑋) → (𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵 ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) = 0)))
98adantl 485 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵 ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) = 0)))
10 metidval 31243 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)})
1110adantr 484 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (~Met𝐷) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)})
1211breqd 5041 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴(~Met𝐷)𝐵𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵))
13 ibar 532 . . 3 ((𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵) = 0 ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) = 0)))
1413adantl 485 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → ((𝐴𝐷𝐵) = 0 ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) = 0)))
159, 12, 143bitr4d 314 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴(~Met𝐷)𝐵 ↔ (𝐴𝐷𝐵) = 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2111   class class class wbr 5030  {copab 5092  cfv 6324  (class class class)co 7135  0cc0 10526  PsMetcpsmet 20075  ~Metcmetid 31239
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583
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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8391  df-xr 10668  df-psmet 20083  df-metid 31241
This theorem is referenced by:  metideq  31246  metider  31247  pstmfval  31249  pstmxmet  31250
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