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Theorem isxmetd 24214

Description: Properties that determine an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) (Revised by AV, 9-Apr-2024.)

**Hypotheses**
Ref	Expression
isxmetd.0	⊢ (𝜑 → 𝑋 ∈ 𝑉)
isxmetd.1	⊢ (𝜑 → 𝐷:(𝑋 × 𝑋)⟶ℝ^*)
isxmetd.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
isxmetd.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))

**Assertion**
Ref	Expression
isxmetd	⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))

Distinct variable groups: 𝑥,𝑦,𝑧,𝐷 𝜑,𝑥,𝑦,𝑧 𝑥,𝑋,𝑦,𝑧 Allowed substitution hints: 𝑉(𝑥,𝑦,𝑧)

**Proof of Theorem isxmetd**
Step	Hyp	Ref	Expression
1		isxmetd.1	. 2 ⊢ (𝜑 → 𝐷:(𝑋 × 𝑋)⟶ℝ^*)
2		isxmetd.2	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
3		isxmetd.3	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))
4	3	3exp2 1355	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 → (𝑦 ∈ 𝑋 → (𝑧 ∈ 𝑋 → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))))
5	4	imp32 418	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑧 ∈ 𝑋 → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))
6	5	ralrimiv 3124	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))
7	2, 6	jca 511	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))
8	7	ralrimivva 3180	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))
9		isxmetd.0	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
10		isxmet 24212	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝐷 ∈ (∞Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ^* ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))))
11	9, 10	syl 17	. 2 ⊢ (𝜑 → (𝐷 ∈ (∞Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ^* ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))))
12	1, 8, 11	mpbir2and 713	1 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 class class class wbr 5107 × cxp 5636 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 0cc0 11068 ℝ^*cxr 11207 ≤ cle 11209 +_𝑒 cxad 13070 ∞Metcxmet 21249

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-sbc 3754 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-fv 6519 df-ov 7390 df-oprab 7391 df-mpo 7392 df-map 8801 df-xr 11212 df-xmet 21257

This theorem is referenced by: isxmet2d 24215 xmetres2 24249 comet 24401

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