Theorem isxmet2d 15205

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

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 6197	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐷𝑦) ∈ ℝ*)
4		0xr 8319	. . . 4 ⊢ 0 ∈ ℝ*
5		xrletri3 10136	. . . 4 ⊢ (((𝑥𝐷𝑦) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((𝑥𝐷𝑦) = 0 ↔ ((𝑥𝐷𝑦) ≤ 0 ∧ 0 ≤ (𝑥𝐷𝑦))))
6	3, 4, 5	sylancl 413	. . 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 304	. . 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 216	. 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 1230	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
13		rexadd 10184	. . . . . . 7 ⊢ (((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ) → ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) = ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
14	13	adantl 277	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) → ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) = ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
15	12, 14	breqtrrd 4136	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
16	15	anassrs 400	. . . 4 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) ∧ (𝑧𝐷𝑦) ∈ ℝ) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
17	3	3adantr3 1185	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐷𝑦) ∈ ℝ*)
18		pnfge 10121	. . . . . . 7 ⊢ ((𝑥𝐷𝑦) ∈ ℝ* → (𝑥𝐷𝑦) ≤ +∞)
19	17, 18	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ +∞)
20	19	ad2antrr 488	. . . . 5 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) ∧ (𝑧𝐷𝑦) = +∞) → (𝑥𝐷𝑦) ≤ +∞)
21		oveq2 6057	. . . . . 6 ⊢ ((𝑧𝐷𝑦) = +∞ → ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) = ((𝑧𝐷𝑥) +𝑒 +∞))
22	2	ffnd 5508	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 Fn (𝑋 × 𝑋))
23		elxrge0 10310	. . . . . . . . . . . . 13 ⊢ ((𝑥𝐷𝑦) ∈ (0[,]+∞) ↔ ((𝑥𝐷𝑦) ∈ ℝ* ∧ 0 ≤ (𝑥𝐷𝑦)))
24	3, 7, 23	sylanbrc 417	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐷𝑦) ∈ (0[,]+∞))
25	24	ralrimivva 2624	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐷𝑦) ∈ (0[,]+∞))
26		ffnov 6156	. . . . . . . . . . 11 ⊢ (𝐷:(𝑋 × 𝑋)⟶(0[,]+∞) ↔ (𝐷 Fn (𝑋 × 𝑋) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐷𝑦) ∈ (0[,]+∞)))
27	22, 25, 26	sylanbrc 417	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
28	27	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
29		simpr3 1032	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
30		simpr1 1030	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
31	28, 29, 30	fovcdmd 6198	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧𝐷𝑥) ∈ (0[,]+∞))
32		elxrge0 10310	. . . . . . . . 9 ⊢ ((𝑧𝐷𝑥) ∈ (0[,]+∞) ↔ ((𝑧𝐷𝑥) ∈ ℝ* ∧ 0 ≤ (𝑧𝐷𝑥)))
33	32	simplbi 274	. . . . . . . 8 ⊢ ((𝑧𝐷𝑥) ∈ (0[,]+∞) → (𝑧𝐷𝑥) ∈ ℝ*)
34	31, 33	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧𝐷𝑥) ∈ ℝ*)
35		renemnf 8321	. . . . . . 7 ⊢ ((𝑧𝐷𝑥) ∈ ℝ → (𝑧𝐷𝑥) ≠ -∞)
36		xaddpnf1 10178	. . . . . . 7 ⊢ (((𝑧𝐷𝑥) ∈ ℝ* ∧ (𝑧𝐷𝑥) ≠ -∞) → ((𝑧𝐷𝑥) +𝑒 +∞) = +∞)
37	34, 35, 36	syl2an 289	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) → ((𝑧𝐷𝑥) +𝑒 +∞) = +∞)
38	21, 37	sylan9eqr 2287	. . . . 5 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) ∧ (𝑧𝐷𝑦) = +∞) → ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) = +∞)
39	20, 38	breqtrrd 4136	. . . 4 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) ∧ (𝑧𝐷𝑦) = +∞) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
40		simpr2 1031	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
41	28, 29, 40	fovcdmd 6198	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧𝐷𝑦) ∈ (0[,]+∞))
42		elxrge0 10310	. . . . . . . . . . 11 ⊢ ((𝑧𝐷𝑦) ∈ (0[,]+∞) ↔ ((𝑧𝐷𝑦) ∈ ℝ* ∧ 0 ≤ (𝑧𝐷𝑦)))
43	42	simplbi 274	. . . . . . . . . 10 ⊢ ((𝑧𝐷𝑦) ∈ (0[,]+∞) → (𝑧𝐷𝑦) ∈ ℝ*)
44	41, 43	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧𝐷𝑦) ∈ ℝ*)
45	42	simprbi 275	. . . . . . . . . 10 ⊢ ((𝑧𝐷𝑦) ∈ (0[,]+∞) → 0 ≤ (𝑧𝐷𝑦))
46	41, 45	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 0 ≤ (𝑧𝐷𝑦))
47		ge0nemnf 10156	. . . . . . . . 9 ⊢ (((𝑧𝐷𝑦) ∈ ℝ* ∧ 0 ≤ (𝑧𝐷𝑦)) → (𝑧𝐷𝑦) ≠ -∞)
48	44, 46, 47	syl2anc 411	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧𝐷𝑦) ≠ -∞)
49	48	neneqd 2433	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ¬ (𝑧𝐷𝑦) = -∞)
50	49	pm2.21d 624	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑧𝐷𝑦) = -∞ → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
51	50	adantr 276	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) → ((𝑧𝐷𝑦) = -∞ → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
52	51	imp 124	. . . 4 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) ∧ (𝑧𝐷𝑦) = -∞) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
53	44	adantr 276	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) → (𝑧𝐷𝑦) ∈ ℝ*)
54		elxr 10108	. . . . 5 ⊢ ((𝑧𝐷𝑦) ∈ ℝ* ↔ ((𝑧𝐷𝑦) ∈ ℝ ∨ (𝑧𝐷𝑦) = +∞ ∨ (𝑧𝐷𝑦) = -∞))
55	53, 54	sylib 122	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) → ((𝑧𝐷𝑦) ∈ ℝ ∨ (𝑧𝐷𝑦) = +∞ ∨ (𝑧𝐷𝑦) = -∞))
56	16, 39, 52, 55	mpjao3dan 1344	. . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
57	19	adantr 276	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) = +∞) → (𝑥𝐷𝑦) ≤ +∞)
58		oveq1 6056	. . . . 5 ⊢ ((𝑧𝐷𝑥) = +∞ → ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) = (+∞ +𝑒 (𝑧𝐷𝑦)))
59		xaddpnf2 10179	. . . . . 6 ⊢ (((𝑧𝐷𝑦) ∈ ℝ* ∧ (𝑧𝐷𝑦) ≠ -∞) → (+∞ +𝑒 (𝑧𝐷𝑦)) = +∞)
60	44, 48, 59	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (+∞ +𝑒 (𝑧𝐷𝑦)) = +∞)
61	58, 60	sylan9eqr 2287	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) = +∞) → ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) = +∞)
62	57, 61	breqtrrd 4136	. . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) = +∞) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
63	32	simprbi 275	. . . . . . . 8 ⊢ ((𝑧𝐷𝑥) ∈ (0[,]+∞) → 0 ≤ (𝑧𝐷𝑥))
64	31, 63	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 0 ≤ (𝑧𝐷𝑥))
65		ge0nemnf 10156	. . . . . . 7 ⊢ (((𝑧𝐷𝑥) ∈ ℝ* ∧ 0 ≤ (𝑧𝐷𝑥)) → (𝑧𝐷𝑥) ≠ -∞)
66	34, 64, 65	syl2anc 411	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧𝐷𝑥) ≠ -∞)
67	66	neneqd 2433	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ¬ (𝑧𝐷𝑥) = -∞)
68	67	pm2.21d 624	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑧𝐷𝑥) = -∞ → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
69	68	imp 124	. . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) = -∞) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
70		elxr 10108	. . . 4 ⊢ ((𝑧𝐷𝑥) ∈ ℝ* ↔ ((𝑧𝐷𝑥) ∈ ℝ ∨ (𝑧𝐷𝑥) = +∞ ∨ (𝑧𝐷𝑥) = -∞))
71	34, 70	sylib 122	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑧𝐷𝑥) ∈ ℝ ∨ (𝑧𝐷𝑥) = +∞ ∨ (𝑧𝐷𝑥) = -∞))
72	56, 62, 69, 71	mpjao3dan 1344	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
73	1, 2, 10, 72	isxmetd 15204	1 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ w3o 1004   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203   ≠ wne 2412  ∀wral 2520  Vcvv 2812   class class class wbr 4108   × cxp 4746   Fn wfn 5346  ⟶wf 5347  ‘cfv 5351  (class class class)co 6049  ℝcr 8125  0cc0 8126   + caddc 8129  +∞cpnf 8304  -∞cmnf 8305  ℝ^*cxr 8306   ≤ cle 8308   +_𝑒 cxad 10102  [,]cicc 10223  ∞Metcxmet 14676

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1re 8220  ax-addrcl 8223  ax-rnegex 8235  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-po 4416  df-iso 4417  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-map 6883  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-xadd 10105  df-icc 10227  df-xmet 14684

This theorem is referenced by: (None)