Theorem isxmet2d 24317

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 7534	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐷𝑦) ∈ ℝ*)
4		0xr 11190	. . . 4 ⊢ 0 ∈ ℝ*
5		xrletri3 13103	. . . 4 ⊢ (((𝑥𝐷𝑦) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((𝑥𝐷𝑦) = 0 ↔ ((𝑥𝐷𝑦) ≤ 0 ∧ 0 ≤ (𝑥𝐷𝑦))))
6	3, 4, 5	sylancl 592	. . 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 536	. . 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 308	. 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 1124	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
13		rexadd 13182	. . . . . . 7 ⊢ (((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ) → ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) = ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
14	13	adantl 482	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) → ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) = ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
15	12, 14	breqtrrd 5107	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
16	15	anassrs 468	. . . 4 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) ∧ (𝑧𝐷𝑦) ∈ ℝ) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
17	3	3adantr3 1178	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐷𝑦) ∈ ℝ*)
18		pnfge 13079	. . . . . . 7 ⊢ ((𝑥𝐷𝑦) ∈ ℝ* → (𝑥𝐷𝑦) ≤ +∞)
19	17, 18	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ +∞)
20	19	ad2antrr 732	. . . . 5 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) ∧ (𝑧𝐷𝑦) = +∞) → (𝑥𝐷𝑦) ≤ +∞)
21		oveq2 7371	. . . . . 6 ⊢ ((𝑧𝐷𝑦) = +∞ → ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) = ((𝑧𝐷𝑥) +𝑒 +∞))
22	2	ffnd 6663	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 Fn (𝑋 × 𝑋))
23		elxrge0 13408	. . . . . . . . . . . . 13 ⊢ ((𝑥𝐷𝑦) ∈ (0[,]+∞) ↔ ((𝑥𝐷𝑦) ∈ ℝ* ∧ 0 ≤ (𝑥𝐷𝑦)))
24	3, 7, 23	sylanbrc 589	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐷𝑦) ∈ (0[,]+∞))
25	24	ralrimivva 3183	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐷𝑦) ∈ (0[,]+∞))
26		ffnov 7489	. . . . . . . . . . 11 ⊢ (𝐷:(𝑋 × 𝑋)⟶(0[,]+∞) ↔ (𝐷 Fn (𝑋 × 𝑋) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐷𝑦) ∈ (0[,]+∞)))
27	22, 25, 26	sylanbrc 589	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
28	27	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
29		simpr3 1203	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
30		simpr1 1201	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
31	28, 29, 30	fovcdmd 7535	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧𝐷𝑥) ∈ (0[,]+∞))
32		eliccxr 13386	. . . . . . . 8 ⊢ ((𝑧𝐷𝑥) ∈ (0[,]+∞) → (𝑧𝐷𝑥) ∈ ℝ*)
33	31, 32	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧𝐷𝑥) ∈ ℝ*)
34		renemnf 11192	. . . . . . 7 ⊢ ((𝑧𝐷𝑥) ∈ ℝ → (𝑧𝐷𝑥) ≠ -∞)
35		xaddpnf1 13176	. . . . . . 7 ⊢ (((𝑧𝐷𝑥) ∈ ℝ* ∧ (𝑧𝐷𝑥) ≠ -∞) → ((𝑧𝐷𝑥) +𝑒 +∞) = +∞)
36	33, 34, 35	syl2an 602	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) → ((𝑧𝐷𝑥) +𝑒 +∞) = +∞)
37	21, 36	sylan9eqr 2797	. . . . 5 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) ∧ (𝑧𝐷𝑦) = +∞) → ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) = +∞)
38	20, 37	breqtrrd 5107	. . . 4 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) ∧ (𝑧𝐷𝑦) = +∞) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
39		simpr2 1202	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
40	28, 29, 39	fovcdmd 7535	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧𝐷𝑦) ∈ (0[,]+∞))
41		eliccxr 13386	. . . . . . . . . 10 ⊢ ((𝑧𝐷𝑦) ∈ (0[,]+∞) → (𝑧𝐷𝑦) ∈ ℝ*)
42	40, 41	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧𝐷𝑦) ∈ ℝ*)
43		elxrge0 13408	. . . . . . . . . . 11 ⊢ ((𝑧𝐷𝑦) ∈ (0[,]+∞) ↔ ((𝑧𝐷𝑦) ∈ ℝ* ∧ 0 ≤ (𝑧𝐷𝑦)))
44	43	simprbi 498	. . . . . . . . . 10 ⊢ ((𝑧𝐷𝑦) ∈ (0[,]+∞) → 0 ≤ (𝑧𝐷𝑦))
45	40, 44	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 0 ≤ (𝑧𝐷𝑦))
46		ge0nemnf 13123	. . . . . . . . 9 ⊢ (((𝑧𝐷𝑦) ∈ ℝ* ∧ 0 ≤ (𝑧𝐷𝑦)) → (𝑧𝐷𝑦) ≠ -∞)
47	42, 45, 46	syl2anc 590	. . . . . . . 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 2955	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑧𝐷𝑦) = -∞ → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
50	49	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) → ((𝑧𝐷𝑦) = -∞ → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
51	50	imp 407	. . . 4 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) ∧ (𝑧𝐷𝑦) = -∞) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
52	42	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) → (𝑧𝐷𝑦) ∈ ℝ*)
53		elxr 13065	. . . . 5 ⊢ ((𝑧𝐷𝑦) ∈ ℝ* ↔ ((𝑧𝐷𝑦) ∈ ℝ ∨ (𝑧𝐷𝑦) = +∞ ∨ (𝑧𝐷𝑦) = -∞))
54	52, 53	sylib 219	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) → ((𝑧𝐷𝑦) ∈ ℝ ∨ (𝑧𝐷𝑦) = +∞ ∨ (𝑧𝐷𝑦) = -∞))
55	16, 38, 51, 54	mpjao3dan 1440	. . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
56	19	adantr 481	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) = +∞) → (𝑥𝐷𝑦) ≤ +∞)
57		oveq1 7370	. . . . 5 ⊢ ((𝑧𝐷𝑥) = +∞ → ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) = (+∞ +𝑒 (𝑧𝐷𝑦)))
58		xaddpnf2 13177	. . . . . 6 ⊢ (((𝑧𝐷𝑦) ∈ ℝ* ∧ (𝑧𝐷𝑦) ≠ -∞) → (+∞ +𝑒 (𝑧𝐷𝑦)) = +∞)
59	42, 47, 58	syl2anc 590	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (+∞ +𝑒 (𝑧𝐷𝑦)) = +∞)
60	57, 59	sylan9eqr 2797	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) = +∞) → ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) = +∞)
61	56, 60	breqtrrd 5107	. . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) = +∞) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
62		elxrge0 13408	. . . . . . . . 9 ⊢ ((𝑧𝐷𝑥) ∈ (0[,]+∞) ↔ ((𝑧𝐷𝑥) ∈ ℝ* ∧ 0 ≤ (𝑧𝐷𝑥)))
63	62	simprbi 498	. . . . . . . 8 ⊢ ((𝑧𝐷𝑥) ∈ (0[,]+∞) → 0 ≤ (𝑧𝐷𝑥))
64	31, 63	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 0 ≤ (𝑧𝐷𝑥))
65		ge0nemnf 13123	. . . . . . 7 ⊢ (((𝑧𝐷𝑥) ∈ ℝ* ∧ 0 ≤ (𝑧𝐷𝑥)) → (𝑧𝐷𝑥) ≠ -∞)
66	33, 64, 65	syl2anc 590	. . . . . 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 2955	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑧𝐷𝑥) = -∞ → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
69	68	imp 407	. . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) = -∞) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
70		elxr 13065	. . . 4 ⊢ ((𝑧𝐷𝑥) ∈ ℝ* ↔ ((𝑧𝐷𝑥) ∈ ℝ ∨ (𝑧𝐷𝑥) = +∞ ∨ (𝑧𝐷𝑥) = -∞))
71	33, 70	sylib 219	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑧𝐷𝑥) ∈ ℝ ∨ (𝑧𝐷𝑥) = +∞ ∨ (𝑧𝐷𝑥) = -∞))
72	55, 61, 69, 71	mpjao3dan 1440	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
73	1, 2, 10, 72	isxmetd 24316	1 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ w3o 1091   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054   class class class wbr 5079   × cxp 5623   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363  ℝcr 11035  0cc0 11036   + caddc 11039  +∞cpnf 11174  -∞cmnf 11175  ℝ^*cxr 11176   ≤ cle 11178   +_𝑒 cxad 13059  [,]cicc 13299  ∞Metcxmet 21339

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-i2m1 11104  ax-rnegex 11107  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-po 5533  df-so 5534  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-er 8640  df-map 8772  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-xadd 13062  df-icc 13303  df-xmet 21347

This theorem is referenced by:  prdsxmetlem  24358  xrsxmet  24800