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Theorem xmeteq0 24349
Description: The value of an extended metric is zero iff its arguments are equal. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xmeteq0 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵))

Proof of Theorem xmeteq0
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 6942 . . . . . 6 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
2 isxmet 24335 . . . . . 6 (𝑋 ∈ dom ∞Met → (𝐷 ∈ (∞Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
31, 2syl 17 . . . . 5 (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ∈ (∞Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
43ibi 267 . . . 4 (𝐷 ∈ (∞Met‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
5 simpl 482 . . . . 5 ((((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
652ralimi 3122 . . . 4 (∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))) → ∀𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
74, 6simpl2im 503 . . 3 (𝐷 ∈ (∞Met‘𝑋) → ∀𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
8 oveq1 7439 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐷𝑦) = (𝐴𝐷𝑦))
98eqeq1d 2738 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝐷𝑦) = 0 ↔ (𝐴𝐷𝑦) = 0))
10 eqeq1 2740 . . . . 5 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
119, 10bibi12d 345 . . . 4 (𝑥 = 𝐴 → (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ↔ ((𝐴𝐷𝑦) = 0 ↔ 𝐴 = 𝑦)))
12 oveq2 7440 . . . . . 6 (𝑦 = 𝐵 → (𝐴𝐷𝑦) = (𝐴𝐷𝐵))
1312eqeq1d 2738 . . . . 5 (𝑦 = 𝐵 → ((𝐴𝐷𝑦) = 0 ↔ (𝐴𝐷𝐵) = 0))
14 eqeq2 2748 . . . . 5 (𝑦 = 𝐵 → (𝐴 = 𝑦𝐴 = 𝐵))
1513, 14bibi12d 345 . . . 4 (𝑦 = 𝐵 → (((𝐴𝐷𝑦) = 0 ↔ 𝐴 = 𝑦) ↔ ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵)))
1611, 15rspc2v 3632 . . 3 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵)))
177, 16syl5com 31 . 2 (𝐷 ∈ (∞Met‘𝑋) → ((𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵)))
18173impib 1116 1 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060   class class class wbr 5142   × cxp 5682  dom cdm 5684  wf 6556  cfv 6560  (class class class)co 7432  0cc0 11156  *cxr 11295  cle 11297   +𝑒 cxad 13153  ∞Metcxmet 21350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-map 8869  df-xr 11300  df-xmet 21358
This theorem is referenced by:  meteq0  24350  xmet0  24353  xmetgt0  24369  xmetres2  24372  prdsxmetlem  24379  imasf1oxmet  24386  xblss2  24413  xmseq0  24475  comet  24527
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