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Theorem psmet0 24216
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 6852 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
2 ispsmet 24212 . . . . . . . 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 267 . . . . . 6 (𝐷 ∈ (PsMet‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑎𝑋 ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏𝑋𝑐𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))))
54simprd 495 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎𝑋 ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏𝑋𝑐𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
65r19.21bi 3222 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) → ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏𝑋𝑐𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
76simpld 494 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) → (𝑎𝐷𝑎) = 0)
87ralrimiva 3122 . 2 (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎𝑋 (𝑎𝐷𝑎) = 0)
9 id 22 . . . . 5 (𝑎 = 𝐴𝑎 = 𝐴)
109, 9oveq12d 7359 . . . 4 (𝑎 = 𝐴 → (𝑎𝐷𝑎) = (𝐴𝐷𝐴))
1110eqeq1d 2732 . . 3 (𝑎 = 𝐴 → ((𝑎𝐷𝑎) = 0 ↔ (𝐴𝐷𝐴) = 0))
1211rspcv 3571 . 2 (𝐴𝑋 → (∀𝑎𝑋 (𝑎𝐷𝑎) = 0 → (𝐴𝐷𝐴) = 0))
138, 12mpan9 506 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1541  wcel 2110  wral 3045  Vcvv 3434   class class class wbr 5089   × cxp 5612  wf 6473  cfv 6477  (class class class)co 7341  0cc0 10998  *cxr 11137  cle 11139   +𝑒 cxad 13001  PsMetcpsmet 21268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-cnex 11054  ax-resscn 11055
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-sbc 3740  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-fv 6485  df-ov 7344  df-oprab 7345  df-mpo 7346  df-map 8747  df-xr 11142  df-psmet 21276
This theorem is referenced by:  psmetsym  24218  psmetge0  24220  psmetres2  24222  distspace  24224  xblcntrps  24318  ssblps  24330  metustid  24462  metider  33897  pstmfval  33899
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