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Theorem xmeteq0 22952
 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 6677 . . . . . 6 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
2 isxmet 22938 . . . . . 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 270 . . . 4 (𝐷 ∈ (∞Met‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
5 simpl 486 . . . . 5 ((((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
652ralimi 3129 . . . 4 (∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))) → ∀𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
74, 6simpl2im 507 . . 3 (𝐷 ∈ (∞Met‘𝑋) → ∀𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
8 oveq1 7142 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐷𝑦) = (𝐴𝐷𝑦))
98eqeq1d 2800 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝐷𝑦) = 0 ↔ (𝐴𝐷𝑦) = 0))
10 eqeq1 2802 . . . . 5 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
119, 10bibi12d 349 . . . 4 (𝑥 = 𝐴 → (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ↔ ((𝐴𝐷𝑦) = 0 ↔ 𝐴 = 𝑦)))
12 oveq2 7143 . . . . . 6 (𝑦 = 𝐵 → (𝐴𝐷𝑦) = (𝐴𝐷𝐵))
1312eqeq1d 2800 . . . . 5 (𝑦 = 𝐵 → ((𝐴𝐷𝑦) = 0 ↔ (𝐴𝐷𝐵) = 0))
14 eqeq2 2810 . . . . 5 (𝑦 = 𝐵 → (𝐴 = 𝑦𝐴 = 𝐵))
1513, 14bibi12d 349 . . . 4 (𝑦 = 𝐵 → (((𝐴𝐷𝑦) = 0 ↔ 𝐴 = 𝑦) ↔ ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵)))
1611, 15rspc2v 3581 . . 3 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵)))
177, 16syl5com 31 . 2 (𝐷 ∈ (∞Met‘𝑋) → ((𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵)))
18173impib 1113 1 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106   class class class wbr 5030   × cxp 5517  dom cdm 5519  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  0cc0 10528  ℝ*cxr 10665   ≤ cle 10667   +𝑒 cxad 12495  ∞Metcxmet 20079 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-cnex 10584  ax-resscn 10585 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8393  df-xr 10670  df-xmet 20087 This theorem is referenced by:  meteq0  22953  xmet0  22956  xmetgt0  22972  xmetres2  22975  prdsxmetlem  22982  imasf1oxmet  22989  xblss2  23016  xmseq0  23078  comet  23127
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