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Theorem metidv 33838
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 2832 . . . . . 6 (𝑎 = 𝐴 → (𝑎𝑋𝐴𝑋))
2 eleq1 2832 . . . . . 6 (𝑏 = 𝐵 → (𝑏𝑋𝐵𝑋))
31, 2bi2anan9 637 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑎𝑋𝑏𝑋) ↔ (𝐴𝑋𝐵𝑋)))
4 oveq12 7457 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎𝐷𝑏) = (𝐴𝐷𝐵))
54eqeq1d 2742 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑎𝐷𝑏) = 0 ↔ (𝐴𝐷𝐵) = 0))
63, 5anbi12d 631 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → (((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0) ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) = 0)))
7 eqid 2740 . . . 4 {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)}
86, 7brabga 5553 . . 3 ((𝐴𝑋𝐵𝑋) → (𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵 ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) = 0)))
98adantl 481 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵 ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) = 0)))
10 metidval 33836 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)})
1110adantr 480 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (~Met𝐷) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)})
1211breqd 5177 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴(~Met𝐷)𝐵𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑋𝑏𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵))
13 ibar 528 . . 3 ((𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵) = 0 ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) = 0)))
1413adantl 481 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → ((𝐴𝐷𝐵) = 0 ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) = 0)))
159, 12, 143bitr4d 311 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴(~Met𝐷)𝐵 ↔ (𝐴𝐷𝐵) = 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2108   class class class wbr 5166  {copab 5228  cfv 6573  (class class class)co 7448  0cc0 11184  PsMetcpsmet 21371  ~Metcmetid 33832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-xr 11328  df-psmet 21379  df-metid 33834
This theorem is referenced by:  metideq  33839  metider  33840  pstmfval  33842  pstmxmet  33843
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