Theorem isxmet2d 24358

Description: It is safe to only require the triangle inequality when the values are real (so that we can use the standard addition over the reals), but in this case the nonnegativity constraint cannot be deduced and must be provided separately. (Counterexample: 𝐷(𝑥, 𝑦) = if(𝑥 = 𝑦, 0, -∞) satisfies all hypotheses except nonnegativity.) (Contributed by Mario Carneiro, 20-Aug-2015.)

Hypotheses

Ref	Expression
isxmetd.0	⊢ (𝜑 → 𝑋 ∈ 𝑉)
isxmetd.1	⊢ (𝜑 → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
isxmet2d.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 0 ≤ (𝑥𝐷𝑦))
isxmet2d.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐷𝑦) ≤ 0 ↔ 𝑥 = 𝑦))
isxmet2d.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))

Assertion

Ref	Expression
isxmet2d	⊢ (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 isxmet2d

Step	Hyp	Ref	Expression
1		isxmetd.0	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
2		isxmetd.1	. 2 ⊢ (𝜑 → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
3	2	fovcdmda 7621	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐷𝑦) ∈ ℝ*)
4		0xr 11337	. . . 4 ⊢ 0 ∈ ℝ*
5		xrletri3 13216	. . . 4 ⊢ (((𝑥𝐷𝑦) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((𝑥𝐷𝑦) = 0 ↔ ((𝑥𝐷𝑦) ≤ 0 ∧ 0 ≤ (𝑥𝐷𝑦))))
6	3, 4, 5	sylancl 585	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐷𝑦) = 0 ↔ ((𝑥𝐷𝑦) ≤ 0 ∧ 0 ≤ (𝑥𝐷𝑦))))
7		isxmet2d.2	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 0 ≤ (𝑥𝐷𝑦))
8	7	biantrud 531	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐷𝑦) ≤ 0 ↔ ((𝑥𝐷𝑦) ≤ 0 ∧ 0 ≤ (𝑥𝐷𝑦))))
9		isxmet2d.3	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐷𝑦) ≤ 0 ↔ 𝑥 = 𝑦))
10	6, 8, 9	3bitr2d 307	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
11		isxmet2d.4	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
12	11	3expa 1118	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
13		rexadd 13294	. . . . . . 7 ⊢ (((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ) → ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) = ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
14	13	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) → ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) = ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
15	12, 14	breqtrrd 5194	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
16	15	anassrs 467	. . . 4 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) ∧ (𝑧𝐷𝑦) ∈ ℝ) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
17	3	3adantr3 1171	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐷𝑦) ∈ ℝ*)
18		pnfge 13193	. . . . . . 7 ⊢ ((𝑥𝐷𝑦) ∈ ℝ* → (𝑥𝐷𝑦) ≤ +∞)
19	17, 18	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ +∞)
20	19	ad2antrr 725	. . . . 5 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) ∧ (𝑧𝐷𝑦) = +∞) → (𝑥𝐷𝑦) ≤ +∞)
21		oveq2 7456	. . . . . 6 ⊢ ((𝑧𝐷𝑦) = +∞ → ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) = ((𝑧𝐷𝑥) +𝑒 +∞))
22	2	ffnd 6748	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 Fn (𝑋 × 𝑋))
23		elxrge0 13517	. . . . . . . . . . . . 13 ⊢ ((𝑥𝐷𝑦) ∈ (0[,]+∞) ↔ ((𝑥𝐷𝑦) ∈ ℝ* ∧ 0 ≤ (𝑥𝐷𝑦)))
24	3, 7, 23	sylanbrc 582	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐷𝑦) ∈ (0[,]+∞))
25	24	ralrimivva 3208	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐷𝑦) ∈ (0[,]+∞))
26		ffnov 7576	. . . . . . . . . . 11 ⊢ (𝐷:(𝑋 × 𝑋)⟶(0[,]+∞) ↔ (𝐷 Fn (𝑋 × 𝑋) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐷𝑦) ∈ (0[,]+∞)))
27	22, 25, 26	sylanbrc 582	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
28	27	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
29		simpr3 1196	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
30		simpr1 1194	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
31	28, 29, 30	fovcdmd 7622	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧𝐷𝑥) ∈ (0[,]+∞))
32		eliccxr 13495	. . . . . . . 8 ⊢ ((𝑧𝐷𝑥) ∈ (0[,]+∞) → (𝑧𝐷𝑥) ∈ ℝ*)
33	31, 32	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧𝐷𝑥) ∈ ℝ*)
34		renemnf 11339	. . . . . . 7 ⊢ ((𝑧𝐷𝑥) ∈ ℝ → (𝑧𝐷𝑥) ≠ -∞)
35		xaddpnf1 13288	. . . . . . 7 ⊢ (((𝑧𝐷𝑥) ∈ ℝ* ∧ (𝑧𝐷𝑥) ≠ -∞) → ((𝑧𝐷𝑥) +𝑒 +∞) = +∞)
36	33, 34, 35	syl2an 595	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) → ((𝑧𝐷𝑥) +𝑒 +∞) = +∞)
37	21, 36	sylan9eqr 2802	. . . . 5 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) ∧ (𝑧𝐷𝑦) = +∞) → ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) = +∞)
38	20, 37	breqtrrd 5194	. . . 4 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) ∧ (𝑧𝐷𝑦) = +∞) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
39		simpr2 1195	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
40	28, 29, 39	fovcdmd 7622	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧𝐷𝑦) ∈ (0[,]+∞))
41		eliccxr 13495	. . . . . . . . . 10 ⊢ ((𝑧𝐷𝑦) ∈ (0[,]+∞) → (𝑧𝐷𝑦) ∈ ℝ*)
42	40, 41	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧𝐷𝑦) ∈ ℝ*)
43		elxrge0 13517	. . . . . . . . . . 11 ⊢ ((𝑧𝐷𝑦) ∈ (0[,]+∞) ↔ ((𝑧𝐷𝑦) ∈ ℝ* ∧ 0 ≤ (𝑧𝐷𝑦)))
44	43	simprbi 496	. . . . . . . . . 10 ⊢ ((𝑧𝐷𝑦) ∈ (0[,]+∞) → 0 ≤ (𝑧𝐷𝑦))
45	40, 44	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 0 ≤ (𝑧𝐷𝑦))
46		ge0nemnf 13235	. . . . . . . . 9 ⊢ (((𝑧𝐷𝑦) ∈ ℝ* ∧ 0 ≤ (𝑧𝐷𝑦)) → (𝑧𝐷𝑦) ≠ -∞)
47	42, 45, 46	syl2anc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧𝐷𝑦) ≠ -∞)
48	47	a1d 25	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (¬ (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) → (𝑧𝐷𝑦) ≠ -∞))
49	48	necon4bd 2966	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑧𝐷𝑦) = -∞ → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
50	49	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) → ((𝑧𝐷𝑦) = -∞ → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
51	50	imp 406	. . . 4 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) ∧ (𝑧𝐷𝑦) = -∞) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
52	42	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) → (𝑧𝐷𝑦) ∈ ℝ*)
53		elxr 13179	. . . . 5 ⊢ ((𝑧𝐷𝑦) ∈ ℝ* ↔ ((𝑧𝐷𝑦) ∈ ℝ ∨ (𝑧𝐷𝑦) = +∞ ∨ (𝑧𝐷𝑦) = -∞))
54	52, 53	sylib 218	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) → ((𝑧𝐷𝑦) ∈ ℝ ∨ (𝑧𝐷𝑦) = +∞ ∨ (𝑧𝐷𝑦) = -∞))
55	16, 38, 51, 54	mpjao3dan 1432	. . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
56	19	adantr 480	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) = +∞) → (𝑥𝐷𝑦) ≤ +∞)
57		oveq1 7455	. . . . 5 ⊢ ((𝑧𝐷𝑥) = +∞ → ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) = (+∞ +𝑒 (𝑧𝐷𝑦)))
58		xaddpnf2 13289	. . . . . 6 ⊢ (((𝑧𝐷𝑦) ∈ ℝ* ∧ (𝑧𝐷𝑦) ≠ -∞) → (+∞ +𝑒 (𝑧𝐷𝑦)) = +∞)
59	42, 47, 58	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (+∞ +𝑒 (𝑧𝐷𝑦)) = +∞)
60	57, 59	sylan9eqr 2802	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) = +∞) → ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) = +∞)
61	56, 60	breqtrrd 5194	. . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) = +∞) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
62		elxrge0 13517	. . . . . . . . 9 ⊢ ((𝑧𝐷𝑥) ∈ (0[,]+∞) ↔ ((𝑧𝐷𝑥) ∈ ℝ* ∧ 0 ≤ (𝑧𝐷𝑥)))
63	62	simprbi 496	. . . . . . . 8 ⊢ ((𝑧𝐷𝑥) ∈ (0[,]+∞) → 0 ≤ (𝑧𝐷𝑥))
64	31, 63	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 0 ≤ (𝑧𝐷𝑥))
65		ge0nemnf 13235	. . . . . . 7 ⊢ (((𝑧𝐷𝑥) ∈ ℝ* ∧ 0 ≤ (𝑧𝐷𝑥)) → (𝑧𝐷𝑥) ≠ -∞)
66	33, 64, 65	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧𝐷𝑥) ≠ -∞)
67	66	a1d 25	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (¬ (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) → (𝑧𝐷𝑥) ≠ -∞))
68	67	necon4bd 2966	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑧𝐷𝑥) = -∞ → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
69	68	imp 406	. . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) = -∞) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
70		elxr 13179	. . . 4 ⊢ ((𝑧𝐷𝑥) ∈ ℝ* ↔ ((𝑧𝐷𝑥) ∈ ℝ ∨ (𝑧𝐷𝑥) = +∞ ∨ (𝑧𝐷𝑥) = -∞))
71	33, 70	sylib 218	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑧𝐷𝑥) ∈ ℝ ∨ (𝑧𝐷𝑥) = +∞ ∨ (𝑧𝐷𝑥) = -∞))
72	55, 61, 69, 71	mpjao3dan 1432	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
73	1, 2, 10, 72	isxmetd 24357	1 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ w3o 1086   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067   class class class wbr 5166   × cxp 5698   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  ℝcr 11183  0cc0 11184   + caddc 11187  +∞cpnf 11321  -∞cmnf 11322  ℝ^*cxr 11323   ≤ cle 11325   +_𝑒 cxad 13173  [,]cicc 13410  ∞Metcxmet 21372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-i2m1 11252  ax-rnegex 11255  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-po 5607  df-so 5608  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-xadd 13176  df-icc 13414  df-xmet 21380

This theorem is referenced by:  prdsxmetlem  24399  xrsxmet  24850