Theorem xmeteq0 24364

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) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵))

Proof of Theorem xmeteq0

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6944	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
2		isxmet 24350	. . . . . 6 ⊢ (𝑋 ∈ dom ∞Met → (𝐷 ∈ (∞Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
3	1, 2	syl 17	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ∈ (∞Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
4	3	ibi 267	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
5		simpl 482	. . . . 5 ⊢ ((((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
6	5	2ralimi 3121	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
7	4, 6	simpl2im 503	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
8		oveq1 7438	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐷𝑦) = (𝐴𝐷𝑦))
9	8	eqeq1d 2737	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝐷𝑦) = 0 ↔ (𝐴𝐷𝑦) = 0))
10		eqeq1 2739	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝐴 = 𝑦))
11	9, 10	bibi12d 345	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ↔ ((𝐴𝐷𝑦) = 0 ↔ 𝐴 = 𝑦)))
12		oveq2 7439	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴𝐷𝑦) = (𝐴𝐷𝐵))
13	12	eqeq1d 2737	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴𝐷𝑦) = 0 ↔ (𝐴𝐷𝐵) = 0))
14		eqeq2 2747	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐵))
15	13, 14	bibi12d 345	. . . 4 ⊢ (𝑦 = 𝐵 → (((𝐴𝐷𝑦) = 0 ↔ 𝐴 = 𝑦) ↔ ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵)))
16	11, 15	rspc2v 3633	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵)))
17	7, 16	syl5com 31	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵)))
18	17	3impib 1115	1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059   class class class wbr 5148   × cxp 5687  dom cdm 5689  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  0cc0 11153  ℝ^*cxr 11292   ≤ cle 11294   +_𝑒 cxad 13150  ∞Metcxmet 21367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-xr 11297  df-xmet 21375

This theorem is referenced by:  meteq0  24365  xmet0  24368  xmetgt0  24384  xmetres2  24387  prdsxmetlem  24394  imasf1oxmet  24401  xblss2  24428  xmseq0  24490  comet  24542