Theorem xmeteq0 23835

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 6925	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
2		isxmet 23821	. . . . . 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 266	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
5		simpl 483	. . . . 5 ⊢ ((((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
6	5	2ralimi 3123	. . . 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 504	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
8		oveq1 7412	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐷𝑦) = (𝐴𝐷𝑦))
9	8	eqeq1d 2734	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝐷𝑦) = 0 ↔ (𝐴𝐷𝑦) = 0))
10		eqeq1 2736	. . . . 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 7413	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴𝐷𝑦) = (𝐴𝐷𝐵))
13	12	eqeq1d 2734	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴𝐷𝑦) = 0 ↔ (𝐴𝐷𝐵) = 0))
14		eqeq2 2744	. . . . 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 3621	. . 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 1116	1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061   class class class wbr 5147   × cxp 5673  dom cdm 5675  ⟶wf 6536  ‘cfv 6540  (class class class)co 7405  0cc0 11106  ℝ^*cxr 11243   ≤ cle 11245   +_𝑒 cxad 13086  ∞Metcxmet 20921

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8818  df-xr 11248  df-xmet 20929

This theorem is referenced by:  meteq0  23836  xmet0  23839  xmetgt0  23855  xmetres2  23858  prdsxmetlem  23865  imasf1oxmet  23872  xblss2  23899  xmseq0  23961  comet  24013