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Theorem metidv 29909
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 2687 . . . . . 6 (𝑎 = 𝐴 → (𝑎𝑋𝐴𝑋))
2 eleq1 2687 . . . . . 6 (𝑏 = 𝐵 → (𝑏𝑋𝐵𝑋))
31, 2bi2anan9 916 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑎𝑋𝑏𝑋) ↔ (𝐴𝑋𝐵𝑋)))
4 oveq12 6644 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎𝐷𝑏) = (𝐴𝐷𝐵))
54eqeq1d 2622 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑎𝐷𝑏) = 0 ↔ (𝐴𝐷𝐵) = 0))
63, 5anbi12d 746 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → (((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0) ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) = 0)))
7 eqid 2620 . . . 4 {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)}
86, 7brabga 4979 . . 3 ((𝐴𝑋𝐵𝑋) → (𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵 ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) = 0)))
98adantl 482 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵 ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) = 0)))
10 metidval 29907 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)})
1110adantr 481 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (~Met𝐷) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)})
1211breqd 4655 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴(~Met𝐷)𝐵𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵))
13 ibar 525 . . 3 ((𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵) = 0 ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) = 0)))
1413adantl 482 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → ((𝐴𝐷𝐵) = 0 ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) = 0)))
159, 12, 143bitr4d 300 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴(~Met𝐷)𝐵 ↔ (𝐴𝐷𝐵) = 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988   class class class wbr 4644  {copab 4703  cfv 5876  (class class class)co 6635  0cc0 9921  PsMetcpsmet 19711  ~Metcmetid 29903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-map 7844  df-xr 10063  df-psmet 19719  df-metid 29905
This theorem is referenced by:  metideq  29910  metider  29911  pstmfval  29913  pstmxmet  29914
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