Theorem ismet 14580

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 2774	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ V)
2		fnmap 6714	. . . . . . . 8 ⊢ ↑𝑚 Fn (V × V)
3		reex 8013	. . . . . . . 8 ⊢ ℝ ∈ V
4		sqxpexg 4779	. . . . . . . 8 ⊢ (𝑋 ∈ V → (𝑋 × 𝑋) ∈ V)
5		fnovex 5955	. . . . . . . 8 ⊢ (( ↑𝑚 Fn (V × V) ∧ ℝ ∈ V ∧ (𝑋 × 𝑋) ∈ V) → (ℝ ↑𝑚 (𝑋 × 𝑋)) ∈ V)
6	2, 3, 4, 5	mp3an12i 1352	. . . . . . 7 ⊢ (𝑋 ∈ V → (ℝ ↑𝑚 (𝑋 × 𝑋)) ∈ V)
7		rabexg 4176	. . . . . . 7 ⊢ ((ℝ ↑𝑚 (𝑋 × 𝑋)) ∈ V → {𝑑 ∈ (ℝ ↑𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))} ∈ V)
8	6, 7	syl 14	. . . . . 6 ⊢ (𝑋 ∈ V → {𝑑 ∈ (ℝ ↑𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))} ∈ V)
9		xpeq12 4682	. . . . . . . . . 10 ⊢ ((𝑡 = 𝑋 ∧ 𝑡 = 𝑋) → (𝑡 × 𝑡) = (𝑋 × 𝑋))
10	9	anidms 397	. . . . . . . . 9 ⊢ (𝑡 = 𝑋 → (𝑡 × 𝑡) = (𝑋 × 𝑋))
11	10	oveq2d 5938	. . . . . . . 8 ⊢ (𝑡 = 𝑋 → (ℝ ↑𝑚 (𝑡 × 𝑡)) = (ℝ ↑𝑚 (𝑋 × 𝑋)))
12		raleq 2693	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑋 → (∀𝑧 ∈ 𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)) ↔ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦))))
13	12	anbi2d 464	. . . . . . . . . 10 ⊢ (𝑡 = 𝑋 → ((((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦))) ↔ (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))))
14	13	raleqbi1dv 2705	. . . . . . . . 9 ⊢ (𝑡 = 𝑋 → (∀𝑦 ∈ 𝑡 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦))) ↔ ∀𝑦 ∈ 𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))))
15	14	raleqbi1dv 2705	. . . . . . . 8 ⊢ (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)))))
16	11, 15	rabeqbidv 2758	. . . . . . 7 ⊢ (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)))})
17		df-met 14101	. . . . . . 7 ⊢ Met = (𝑡 ∈ V ↦ {𝑑 ∈ (ℝ ↑𝑚 (𝑡 × 𝑡)) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))})
18	16, 17	fvmptg 5637	. . . . . 6 ⊢ ((𝑋 ∈ V ∧ {𝑑 ∈ (ℝ ↑𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))} ∈ V) → (Met‘𝑋) = {𝑑 ∈ (ℝ ↑𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))})
19	8, 18	mpdan 421	. . . . 5 ⊢ (𝑋 ∈ V → (Met‘𝑋) = {𝑑 ∈ (ℝ ↑𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))})
20	1, 19	syl 14	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (Met‘𝑋) = {𝑑 ∈ (ℝ ↑𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))})
21	20	eleq2d 2266	. . 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)))}))
22		oveq 5928	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
23	22	eqeq1d 2205	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑥𝑑𝑦) = 0 ↔ (𝑥𝐷𝑦) = 0))
24	23	bibi1d 233	. . . . . 6 ⊢ (𝑑 = 𝐷 → (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ↔ ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦)))
25		oveq 5928	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (𝑧𝑑𝑥) = (𝑧𝐷𝑥))
26		oveq 5928	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (𝑧𝑑𝑦) = (𝑧𝐷𝑦))
27	25, 26	oveq12d 5940	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)) = ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
28	22, 27	breq12d 4046	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)) ↔ (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))
29	28	ralbidv 2497	. . . . . 6 ⊢ (𝑑 = 𝐷 → (∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)) ↔ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))
30	24, 29	anbi12d 473	. . . . 5 ⊢ (𝑑 = 𝐷 → ((((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦))) ↔ (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))))
31	30	2ralbidv 2521	. . . 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)))))
32	31	elrab 2920	. . 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)))))
33	21, 32	bitrdi 196	. 2 ⊢ (𝑋 ∈ 𝐴 → (𝐷 ∈ (Met‘𝑋) ↔ (𝐷 ∈ (ℝ ↑𝑚 (𝑋 × 𝑋)) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))))
34		sqxpexg 4779	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (𝑋 × 𝑋) ∈ V)
35		elmapg 6720	. . . 4 ⊢ ((ℝ ∈ V ∧ (𝑋 × 𝑋) ∈ V) → (𝐷 ∈ (ℝ ↑𝑚 (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ))
36	3, 34, 35	sylancr 414	. . 3 ⊢ (𝑋 ∈ 𝐴 → (𝐷 ∈ (ℝ ↑𝑚 (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ))
37	36	anbi1d 465	. 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))))))
38	33, 37	bitrd 188	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   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  ∀wral 2475  {crab 2479  Vcvv 2763   class class class wbr 4033   × cxp 4661   Fn wfn 5253  ⟶wf 5254  ‘cfv 5258  (class class class)co 5922   ↑_𝑚 cmap 6707  ℝcr 7878  0cc0 7879   + caddc 7882   ≤ cle 8062  Metcmet 14093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-map 6709  df-met 14101

This theorem is referenced by:  ismeti  14582  metflem  14585  ismet2  14590