Theorem ismet 24222

Description: Express the predicate "𝐷 is a metric." (Contributed by NM, 25-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
ismet	⊢ (𝑋 ∈ 𝐴 → (𝐷 ∈ (Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐷   𝑥,𝑋,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)

Proof of Theorem ismet

Dummy variables 𝑑 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3488	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ V)
2		xpeq12 5697	. . . . . . . . 9 ⊢ ((𝑡 = 𝑋 ∧ 𝑡 = 𝑋) → (𝑡 × 𝑡) = (𝑋 × 𝑋))
3	2	anidms 566	. . . . . . . 8 ⊢ (𝑡 = 𝑋 → (𝑡 × 𝑡) = (𝑋 × 𝑋))
4	3	oveq2d 7430	. . . . . . 7 ⊢ (𝑡 = 𝑋 → (ℝ ↑m (𝑡 × 𝑡)) = (ℝ ↑m (𝑋 × 𝑋)))
5		raleq 3317	. . . . . . . . . 10 ⊢ (𝑡 = 𝑋 → (∀𝑧 ∈ 𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)) ↔ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦))))
6	5	anbi2d 628	. . . . . . . . 9 ⊢ (𝑡 = 𝑋 → ((((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦))) ↔ (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))))
7	6	raleqbi1dv 3328	. . . . . . . 8 ⊢ (𝑡 = 𝑋 → (∀𝑦 ∈ 𝑡 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦))) ↔ ∀𝑦 ∈ 𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))))
8	7	raleqbi1dv 3328	. . . . . . 7 ⊢ (𝑡 = 𝑋 → (∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦))) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))))
9	4, 8	rabeqbidv 3444	. . . . . 6 ⊢ (𝑡 = 𝑋 → {𝑑 ∈ (ℝ ↑m (𝑡 × 𝑡)) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))} = {𝑑 ∈ (ℝ ↑m (𝑋 × 𝑋)) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))})
10		df-met 21266	. . . . . 6 ⊢ Met = (𝑡 ∈ V ↦ {𝑑 ∈ (ℝ ↑m (𝑡 × 𝑡)) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))})
11		ovex 7447	. . . . . . 7 ⊢ (ℝ ↑m (𝑋 × 𝑋)) ∈ V
12	11	rabex 5328	. . . . . 6 ⊢ {𝑑 ∈ (ℝ ↑m (𝑋 × 𝑋)) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))} ∈ V
13	9, 10, 12	fvmpt 6999	. . . . 5 ⊢ (𝑋 ∈ V → (Met‘𝑋) = {𝑑 ∈ (ℝ ↑m (𝑋 × 𝑋)) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))})
14	1, 13	syl 17	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (Met‘𝑋) = {𝑑 ∈ (ℝ ↑m (𝑋 × 𝑋)) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))})
15	14	eleq2d 2814	. . 3 ⊢ (𝑋 ∈ 𝐴 → (𝐷 ∈ (Met‘𝑋) ↔ 𝐷 ∈ {𝑑 ∈ (ℝ ↑m (𝑋 × 𝑋)) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))}))
16		oveq 7420	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
17	16	eqeq1d 2729	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑥𝑑𝑦) = 0 ↔ (𝑥𝐷𝑦) = 0))
18	17	bibi1d 343	. . . . . 6 ⊢ (𝑑 = 𝐷 → (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ↔ ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦)))
19		oveq 7420	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (𝑧𝑑𝑥) = (𝑧𝐷𝑥))
20		oveq 7420	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (𝑧𝑑𝑦) = (𝑧𝐷𝑦))
21	19, 20	oveq12d 7432	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)) = ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
22	16, 21	breq12d 5155	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)) ↔ (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))
23	22	ralbidv 3172	. . . . . 6 ⊢ (𝑑 = 𝐷 → (∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)) ↔ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))
24	18, 23	anbi12d 630	. . . . 5 ⊢ (𝑑 = 𝐷 → ((((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦))) ↔ (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))))
25	24	2ralbidv 3213	. . . 4 ⊢ (𝑑 = 𝐷 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦))) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))))
26	25	elrab 3680	. . 3 ⊢ (𝐷 ∈ {𝑑 ∈ (ℝ ↑m (𝑋 × 𝑋)) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))} ↔ (𝐷 ∈ (ℝ ↑m (𝑋 × 𝑋)) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))))
27	15, 26	bitrdi 287	. 2 ⊢ (𝑋 ∈ 𝐴 → (𝐷 ∈ (Met‘𝑋) ↔ (𝐷 ∈ (ℝ ↑m (𝑋 × 𝑋)) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))))
28		reex 11223	. . . 4 ⊢ ℝ ∈ V
29		sqxpexg 7751	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (𝑋 × 𝑋) ∈ V)
30		elmapg 8851	. . . 4 ⊢ ((ℝ ∈ V ∧ (𝑋 × 𝑋) ∈ V) → (𝐷 ∈ (ℝ ↑m (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ))
31	28, 29, 30	sylancr 586	. . 3 ⊢ (𝑋 ∈ 𝐴 → (𝐷 ∈ (ℝ ↑m (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ))
32	31	anbi1d 629	. 2 ⊢ (𝑋 ∈ 𝐴 → ((𝐷 ∈ (ℝ ↑m (𝑋 × 𝑋)) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))))
33	27, 32	bitrd 279	1 ⊢ (𝑋 ∈ 𝐴 → (𝐷 ∈ (Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3056  {crab 3427  Vcvv 3469   class class class wbr 5142   × cxp 5670  ⟶wf 6538  ‘cfv 6542  (class class class)co 7414   ↑_m cmap 8838  ℝcr 11131  0cc0 11132   + caddc 11135   ≤ cle 11273  Metcmet 21258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11188  ax-resscn 11189

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8840  df-met 21266

This theorem is referenced by:  ismeti  24224  metflem  24227  ismet2  24232  dscmet  24474  nrmmetd  24476  rrxmet  25329  metf1o  37217  rrnmet  37291