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Theorem metidv 30260
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 2873 . . . . . 6 (𝑎 = 𝐴 → (𝑎𝑋𝐴𝑋))
2 eleq1 2873 . . . . . 6 (𝑏 = 𝐵 → (𝑏𝑋𝐵𝑋))
31, 2bi2anan9 622 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑎𝑋𝑏𝑋) ↔ (𝐴𝑋𝐵𝑋)))
4 oveq12 6883 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎𝐷𝑏) = (𝐴𝐷𝐵))
54eqeq1d 2808 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑎𝐷𝑏) = 0 ↔ (𝐴𝐷𝐵) = 0))
63, 5anbi12d 618 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → (((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0) ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) = 0)))
7 eqid 2806 . . . 4 {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)}
86, 7brabga 5184 . . 3 ((𝐴𝑋𝐵𝑋) → (𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵 ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) = 0)))
98adantl 469 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵 ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) = 0)))
10 metidval 30258 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)})
1110adantr 468 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (~Met𝐷) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)})
1211breqd 4855 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴(~Met𝐷)𝐵𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵))
13 ibar 520 . . 3 ((𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵) = 0 ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) = 0)))
1413adantl 469 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → ((𝐴𝐷𝐵) = 0 ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) = 0)))
159, 12, 143bitr4d 302 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴(~Met𝐷)𝐵 ↔ (𝐴𝐷𝐵) = 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2156   class class class wbr 4844  {copab 4906  cfv 6101  (class class class)co 6874  0cc0 10221  PsMetcpsmet 19938  ~Metcmetid 30254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179  ax-cnex 10277  ax-resscn 10278
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-map 8094  df-xr 10363  df-psmet 19946  df-metid 30256
This theorem is referenced by:  metideq  30261  metider  30262  pstmfval  30264  pstmxmet  30265
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