Theorem ismet 23089

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 3418	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ V)
2		xpeq12 5560	. . . . . . . . 9 ⊢ ((𝑡 = 𝑋 ∧ 𝑡 = 𝑋) → (𝑡 × 𝑡) = (𝑋 × 𝑋))
3	2	anidms 570	. . . . . . . 8 ⊢ (𝑡 = 𝑋 → (𝑡 × 𝑡) = (𝑋 × 𝑋))
4	3	oveq2d 7199	. . . . . . 7 ⊢ (𝑡 = 𝑋 → (ℝ ↑m (𝑡 × 𝑡)) = (ℝ ↑m (𝑋 × 𝑋)))
5		raleq 3311	. . . . . . . . . 10 ⊢ (𝑡 = 𝑋 → (∀𝑧 ∈ 𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)) ↔ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦))))
6	5	anbi2d 632	. . . . . . . . 9 ⊢ (𝑡 = 𝑋 → ((((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦))) ↔ (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))))
7	6	raleqbi1dv 3309	. . . . . . . 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 3309	. . . . . . 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 3388	. . . . . 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 20224	. . . . . 6 ⊢ Met = (𝑡 ∈ V ↦ {𝑑 ∈ (ℝ ↑m (𝑡 × 𝑡)) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))})
11		ovex 7216	. . . . . . 7 ⊢ (ℝ ↑m (𝑋 × 𝑋)) ∈ V
12	11	rabex 5210	. . . . . 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 6788	. . . . 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 2819	. . 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 7189	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
17	16	eqeq1d 2741	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑥𝑑𝑦) = 0 ↔ (𝑥𝐷𝑦) = 0))
18	17	bibi1d 347	. . . . . 6 ⊢ (𝑑 = 𝐷 → (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ↔ ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦)))
19		oveq 7189	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (𝑧𝑑𝑥) = (𝑧𝐷𝑥))
20		oveq 7189	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (𝑧𝑑𝑦) = (𝑧𝐷𝑦))
21	19, 20	oveq12d 7201	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)) = ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
22	16, 21	breq12d 5053	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)) ↔ (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))
23	22	ralbidv 3110	. . . . . 6 ⊢ (𝑑 = 𝐷 → (∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)) ↔ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))
24	18, 23	anbi12d 634	. . . . 5 ⊢ (𝑑 = 𝐷 → ((((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦))) ↔ (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))))
25	24	2ralbidv 3112	. . . 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 3593	. . 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 290	. 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 10719	. . . 4 ⊢ ℝ ∈ V
29		sqxpexg 7509	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (𝑋 × 𝑋) ∈ V)
30		elmapg 8463	. . . 4 ⊢ ((ℝ ∈ V ∧ (𝑋 × 𝑋) ∈ V) → (𝐷 ∈ (ℝ ↑m (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ))
31	28, 29, 30	sylancr 590	. . 3 ⊢ (𝑋 ∈ 𝐴 → (𝐷 ∈ (ℝ ↑m (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ))
32	31	anbi1d 633	. 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 282	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 209   ∧ wa 399   = wceq 1542   ∈ wcel 2114  ∀wral 3054  {crab 3058  Vcvv 3400   class class class wbr 5040   × cxp 5533  ⟶wf 6346  ‘cfv 6350  (class class class)co 7183   ↑_m cmap 8450  ℝcr 10627  0cc0 10628   + caddc 10631   ≤ cle 10767  Metcmet 20216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7492  ax-cnex 10684  ax-resscn 10685

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-iota 6308  df-fun 6352  df-fn 6353  df-f 6354  df-fv 6358  df-ov 7186  df-oprab 7187  df-mpo 7188  df-map 8452  df-met 20224

This theorem is referenced by:  ismeti  23091  metflem  23094  ismet2  23099  dscmet  23338  nrmmetd  23340  rrxmet  24173  metf1o  35569  rrnmet  35643