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Theorem psmet0 22332
Description: The distance function of a pseudometric space is zero if its arguments are equal. (Contributed by Thierry Arnoux, 7-Feb-2018.)
Assertion
Ref Expression
psmet0 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = 0)

Proof of Theorem psmet0
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6362 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
2 ispsmet 22328 . . . . . . . 8 (𝑋 ∈ V → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑎𝑋 ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏𝑋𝑐𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))))
31, 2syl 17 . . . . . . 7 (𝐷 ∈ (PsMet‘𝑋) → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑎𝑋 ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏𝑋𝑐𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))))
43ibi 256 . . . . . 6 (𝐷 ∈ (PsMet‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑎𝑋 ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏𝑋𝑐𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))))
54simprd 477 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎𝑋 ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏𝑋𝑐𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
65r19.21bi 3080 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) → ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏𝑋𝑐𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
76simpld 476 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) → (𝑎𝐷𝑎) = 0)
87ralrimiva 3114 . 2 (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎𝑋 (𝑎𝐷𝑎) = 0)
9 id 22 . . . . 5 (𝑎 = 𝐴𝑎 = 𝐴)
109, 9oveq12d 6810 . . . 4 (𝑎 = 𝐴 → (𝑎𝐷𝑎) = (𝐴𝐷𝐴))
1110eqeq1d 2772 . . 3 (𝑎 = 𝐴 → ((𝑎𝐷𝑎) = 0 ↔ (𝐴𝐷𝐴) = 0))
1211rspcv 3454 . 2 (𝐴𝑋 → (∀𝑎𝑋 (𝑎𝐷𝑎) = 0 → (𝐴𝐷𝐴) = 0))
138, 12mpan9 490 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  wral 3060  Vcvv 3349   class class class wbr 4784   × cxp 5247  wf 6027  cfv 6031  (class class class)co 6792  0cc0 10137  *cxr 10274  cle 10276   +𝑒 cxad 12148  PsMetcpsmet 19944
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-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
This theorem is referenced by:  psmetsym  22334  psmetge0  22336  psmetres2  22338  distspace  22340  xblcntrps  22434  ssblps  22446  metustid  22578  metider  30271  pstmfval  30273
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