Theorem psmet0 14869

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) ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 0)

Proof of Theorem psmet0

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-psmet 14375	. . . . . . . . 9 ⊢ PsMet = (𝑑 ∈ V ↦ {𝑒 ∈ (ℝ* ↑𝑚 (𝑑 × 𝑑)) ∣ ∀𝑎 ∈ 𝑑 ((𝑎𝑒𝑎) = 0 ∧ ∀𝑏 ∈ 𝑑 ∀𝑐 ∈ 𝑑 (𝑎𝑒𝑏) ≤ ((𝑐𝑒𝑎) +𝑒 (𝑐𝑒𝑏)))})
2	1	mptrcl 5674	. . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
3		ispsmet 14865	. . . . . . . 8 ⊢ (𝑋 ∈ V → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑎 ∈ 𝑋 ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))))
4	2, 3	syl 14	. . . . . . 7 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑎 ∈ 𝑋 ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))))
5	4	ibi 176	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑎 ∈ 𝑋 ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))))
6	5	simprd 114	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎 ∈ 𝑋 ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
7	6	r19.21bi 2595	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) → ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
8	7	simpld 112	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) → (𝑎𝐷𝑎) = 0)
9	8	ralrimiva 2580	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎 ∈ 𝑋 (𝑎𝐷𝑎) = 0)
10		id 19	. . . . 5 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
11	10, 10	oveq12d 5974	. . . 4 ⊢ (𝑎 = 𝐴 → (𝑎𝐷𝑎) = (𝐴𝐷𝐴))
12	11	eqeq1d 2215	. . 3 ⊢ (𝑎 = 𝐴 → ((𝑎𝐷𝑎) = 0 ↔ (𝐴𝐷𝐴) = 0))
13	12	rspcv 2877	. 2 ⊢ (𝐴 ∈ 𝑋 → (∀𝑎 ∈ 𝑋 (𝑎𝐷𝑎) = 0 → (𝐴𝐷𝐴) = 0))
14	9, 13	mpan9 281	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 0)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373   ∈ wcel 2177  ∀wral 2485  {crab 2489  Vcvv 2773   class class class wbr 4050   × cxp 4680  ⟶wf 5275  ‘cfv 5279  (class class class)co 5956   ↑_𝑚 cmap 6747  0cc0 7940  ℝ^*cxr 8121   ≤ cle 8123   +_𝑒 cxad 9907  PsMetcpsmet 14367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4169  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-setind 4592  ax-cnex 8031  ax-resscn 8032

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-br 4051  df-opab 4113  df-mpt 4114  df-id 4347  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-fv 5287  df-ov 5959  df-oprab 5960  df-mpo 5961  df-map 6749  df-pnf 8124  df-mnf 8125  df-xr 8126  df-psmet 14375

This theorem is referenced by:  psmetsym  14871  psmetge0  14873  psmetres2  14875  distspace  14877  xblcntrps  14955  ssblps  14967