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Theorem metidv 30269
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 2837 . . . . . 6 (𝑎 = 𝐴 → (𝑎𝑋𝐴𝑋))
2 eleq1 2837 . . . . . 6 (𝑏 = 𝐵 → (𝑏𝑋𝐵𝑋))
31, 2bi2anan9 612 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑎𝑋𝑏𝑋) ↔ (𝐴𝑋𝐵𝑋)))
4 oveq12 6801 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎𝐷𝑏) = (𝐴𝐷𝐵))
54eqeq1d 2772 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑎𝐷𝑏) = 0 ↔ (𝐴𝐷𝐵) = 0))
63, 5anbi12d 608 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → (((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0) ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) = 0)))
7 eqid 2770 . . . 4 {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)}
86, 7brabga 5122 . . 3 ((𝐴𝑋𝐵𝑋) → (𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵 ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) = 0)))
98adantl 467 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵 ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) = 0)))
10 metidval 30267 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)})
1110adantr 466 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (~Met𝐷) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)})
1211breqd 4795 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴(~Met𝐷)𝐵𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵))
13 ibar 512 . . 3 ((𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵) = 0 ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) = 0)))
1413adantl 467 . 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 382   = wceq 1630  wcel 2144   class class class wbr 4784  {copab 4844  cfv 6031  (class class class)co 6792  0cc0 10137  PsMetcpsmet 19944  ~Metcmetid 30263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-map 8010  df-xr 10279  df-psmet 19952  df-metid 30265
This theorem is referenced by:  metideq  30270  metider  30271  pstmfval  30273  pstmxmet  30274
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