Theorem xmeteq0 24669

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 6927	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
2		isxmet 24655	. . . . . 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 18	. . . . 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 270	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +e (𝑧𝐷𝑦)))))
5		simpl 488	. . . . 5 ⊢ ((((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +e (𝑧𝐷𝑦))) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
6	5	2ralimi 3135	. . . 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 513	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
8		oveq1 7434	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐷𝑦) = (𝐴𝐷𝑦))
9	8	eqeq1d 2765	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝐷𝑦) = 0 ↔ (𝐴𝐷𝑦) = 0))
10		eqeq1 2767	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝐴 = 𝑦))
11	9, 10	bibi12d 348	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ↔ ((𝐴𝐷𝑦) = 0 ↔ 𝐴 = 𝑦)))
12		oveq2 7435	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴𝐷𝑦) = (𝐴𝐷𝐵))
13	12	eqeq1d 2765	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴𝐷𝑦) = 0 ↔ (𝐴𝐷𝐵) = 0))
14		eqeq2 2775	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐵))
15	13, 14	bibi12d 348	. . . 4 ⊢ (𝑦 = 𝐵 → (((𝐴𝐷𝑦) = 0 ↔ 𝐴 = 𝑦) ↔ ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵)))
16	11, 15	rspc2v 3589	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵)))
17	7, 16	syl5com 32	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵)))
18	17	3impib 1134	1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  ∀wral 3079   class class class wbr 5105   × cxp 5652  dom cdm 5654  ⟶wf 6542  ‘cfv 6546  (class class class)co 7427  0cc0 11210  ℝ^*cxr 11352   ≤ cle 11354   +_e cxad 13249  ∞Metcxmet 21675

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758  ax-cnex 11266  ax-resscn 11267

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-sbc 3742  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-map 8860  df-xr 11357  df-xmet 21683

This theorem is used by:  meteq0  24670  xmet0  24673  xmetgt0  24689  xmetres2  24692  prdsxmetlem  24699  imasf1oxmet  24706  xblss2  24733  xmseq0  24795  comet  24844