Theorem isxmet2d 15451

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 6233	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐷𝑦) ∈ ℝ*)
4		0xr 8372	. . . 4 ⊢ 0 ∈ ℝ*
5		xrletri3 10208	. . . 4 ⊢ (((𝑥𝐷𝑦) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((𝑥𝐷𝑦) = 0 ↔ ((𝑥𝐷𝑦) ≤ 0 ∧ 0 ≤ (𝑥𝐷𝑦))))
6	3, 4, 5	sylancl 417	. . 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 1234	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
13		rexadd 10256	. . . . . . 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 4158	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
16	15	anassrs 404	. . . 4 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) ∧ (𝑧𝐷𝑦) ∈ ℝ) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
17	3	3adantr3 1189	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐷𝑦) ∈ ℝ*)
18		pnfge 10193	. . . . . . 7 ⊢ ((𝑥𝐷𝑦) ∈ ℝ* → (𝑥𝐷𝑦) ≤ +∞)
19	17, 18	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ +∞)
20	19	ad2antrr 492	. . . . 5 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) ∧ (𝑧𝐷𝑦) = +∞) → (𝑥𝐷𝑦) ≤ +∞)
21		oveq2 6093	. . . . . 6 ⊢ ((𝑧𝐷𝑦) = +∞ → ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) = ((𝑧𝐷𝑥) +𝑒 +∞))
22	2	ffnd 5534	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 Fn (𝑋 × 𝑋))
23		elxrge0 10382	. . . . . . . . . . . . 13 ⊢ ((𝑥𝐷𝑦) ∈ (0[,]+∞) ↔ ((𝑥𝐷𝑦) ∈ ℝ* ∧ 0 ≤ (𝑥𝐷𝑦)))
24	3, 7, 23	sylanbrc 421	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐷𝑦) ∈ (0[,]+∞))
25	24	ralrimivva 2632	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐷𝑦) ∈ (0[,]+∞))
26		ffnov 6192	. . . . . . . . . . 11 ⊢ (𝐷:(𝑋 × 𝑋)⟶(0[,]+∞) ↔ (𝐷 Fn (𝑋 × 𝑋) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐷𝑦) ∈ (0[,]+∞)))
27	22, 25, 26	sylanbrc 421	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
28	27	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
29		simpr3 1036	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
30		simpr1 1034	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
31	28, 29, 30	fovcdmd 6234	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧𝐷𝑥) ∈ (0[,]+∞))
32		elxrge0 10382	. . . . . . . . 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 8374	. . . . . . 7 ⊢ ((𝑧𝐷𝑥) ∈ ℝ → (𝑧𝐷𝑥) ≠ -∞)
36		xaddpnf1 10250	. . . . . . 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 2293	. . . . 5 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) ∧ (𝑧𝐷𝑦) = +∞) → ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) = +∞)
39	20, 38	breqtrrd 4158	. . . 4 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) ∈ ℝ) ∧ (𝑧𝐷𝑦) = +∞) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
40		simpr2 1035	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
41	28, 29, 40	fovcdmd 6234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧𝐷𝑦) ∈ (0[,]+∞))
42		elxrge0 10382	. . . . . . . . . . 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 10228	. . . . . . . . 9 ⊢ (((𝑧𝐷𝑦) ∈ ℝ* ∧ 0 ≤ (𝑧𝐷𝑦)) → (𝑧𝐷𝑦) ≠ -∞)
48	44, 46, 47	syl2anc 415	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧𝐷𝑦) ≠ -∞)
49	48	neneqd 2441	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ¬ (𝑧𝐷𝑦) = -∞)
50	49	pm2.21d 628	. . . . . 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 10180	. . . . 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 1348	. . 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 6092	. . . . 5 ⊢ ((𝑧𝐷𝑥) = +∞ → ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) = (+∞ +𝑒 (𝑧𝐷𝑦)))
59		xaddpnf2 10251	. . . . . 6 ⊢ (((𝑧𝐷𝑦) ∈ ℝ* ∧ (𝑧𝐷𝑦) ≠ -∞) → (+∞ +𝑒 (𝑧𝐷𝑦)) = +∞)
60	44, 48, 59	syl2anc 415	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (+∞ +𝑒 (𝑧𝐷𝑦)) = +∞)
61	58, 60	sylan9eqr 2293	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (𝑧𝐷𝑥) = +∞) → ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) = +∞)
62	57, 61	breqtrrd 4158	. . 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 10228	. . . . . . 7 ⊢ (((𝑧𝐷𝑥) ∈ ℝ* ∧ 0 ≤ (𝑧𝐷𝑥)) → (𝑧𝐷𝑥) ≠ -∞)
66	34, 64, 65	syl2anc 415	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧𝐷𝑥) ≠ -∞)
67	66	neneqd 2441	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ¬ (𝑧𝐷𝑥) = -∞)
68	67	pm2.21d 628	. . . 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 10180	. . . 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 1348	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
73	1, 2, 10, 72	isxmetd 15450	1 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ w3o 1008   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209   ≠ wne 2420  ∀wral 2528  Vcvv 2821   class class class wbr 4130   × cxp 4772   Fn wfn 5372  ⟶wf 5373  ‘cfv 5377  (class class class)co 6085  ℝcr 8178  0cc0 8179   + caddc 8182  +∞cpnf 8357  -∞cmnf 8358  ℝ^*cxr 8359   ≤ cle 8361   +_𝑒 cxad 10174  [,]cicc 10295  ∞Metcxmet 14875

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-map 6924  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-xadd 10177  df-icc 10299  df-xmet 14883

This theorem is used by: (None)