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Theorem bdxmet 15525

Description: The standard bounded metric is an extended metric given an extended metric and a positive extended real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.)

**Hypothesis**
Ref	Expression
stdbdmet.1	⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ^*, < ))

**Assertion**
Ref	Expression
bdxmet	⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐷 ∈ (∞Met‘𝑋))

Distinct variable groups: 𝑥,𝑦,𝐶 𝑥,𝑅,𝑦 𝑥,𝑋,𝑦 Allowed substitution hints: 𝐷(𝑥,𝑦)

Proof of Theorem bdxmet Dummy variables 𝑎 𝑏 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1028	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐶 ∈ (∞Met‘𝑋))
2		xmetcl 15376	. . . . . . 7 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐶𝑦) ∈ ℝ^*)
3		xmetge0 15389	. . . . . . 7 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝑥𝐶𝑦))
4		elxrge0 10359	. . . . . . 7 ⊢ ((𝑥𝐶𝑦) ∈ (0[,]+∞) ↔ ((𝑥𝐶𝑦) ∈ ℝ^* ∧ 0 ≤ (𝑥𝐶𝑦)))
5	2, 3, 4	sylanbrc 421	. . . . . 6 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐶𝑦) ∈ (0[,]+∞))
6	5	3expb 1235	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ∈ (0[,]+∞))
7	1, 6	sylan 283	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ∈ (0[,]+∞))
8		xmetf 15374	. . . . . . 7 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐶:(𝑋 × 𝑋)⟶ℝ^*)
9	8	3ad2ant1 1049	. . . . . 6 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐶:(𝑋 × 𝑋)⟶ℝ^*)
10	9	ffnd 5529	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐶 Fn (𝑋 × 𝑋))
11		fnovim 6187	. . . . 5 ⊢ (𝐶 Fn (𝑋 × 𝑋) → 𝐶 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐶𝑦)))
12	10, 11	syl 14	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐶 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐶𝑦)))
13		eqidd 2239	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → (𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < )) = (𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < )))
14		preq1 3784	. . . . 5 ⊢ (𝑧 = (𝑥𝐶𝑦) → {𝑧, 𝑅} = {(𝑥𝐶𝑦), 𝑅})
15	14	infeq1d 7342	. . . 4 ⊢ (𝑧 = (𝑥𝐶𝑦) → inf({𝑧, 𝑅}, ℝ^, < ) = inf({(𝑥𝐶𝑦), 𝑅}, ℝ^, < ))
16	7, 12, 13, 15	fmpoco 6442	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < )) ∘ 𝐶) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ^, < )))
17		stdbdmet.1	. . 3 ⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ^*, < ))
18	16, 17	eqtr4di 2289	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^*, < )) ∘ 𝐶) = 𝐷)
19		elxrge0 10359	. . . . . 6 ⊢ (𝑧 ∈ (0[,]+∞) ↔ (𝑧 ∈ ℝ^* ∧ 0 ≤ 𝑧))
20	19	simplbi 274	. . . . 5 ⊢ (𝑧 ∈ (0[,]+∞) → 𝑧 ∈ ℝ^*)
21		simp2 1029	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝑅 ∈ ℝ^*)
22		xrmincl 12010	. . . . 5 ⊢ ((𝑧 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({𝑧, 𝑅}, ℝ^, < ) ∈ ℝ^*)
23	20, 21, 22	syl2anr 290	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑧 ∈ (0[,]+∞)) → inf({𝑧, 𝑅}, ℝ^, < ) ∈ ℝ^)
24	23	fmpttd 5854	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → (𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < )):(0[,]+∞)⟶ℝ^)
25		eqid 2238	. . . . . 6 ⊢ (𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < )) = (𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < ))
26		preq1 3784	. . . . . . 7 ⊢ (𝑧 = 𝑎 → {𝑧, 𝑅} = {𝑎, 𝑅})
27	26	infeq1d 7342	. . . . . 6 ⊢ (𝑧 = 𝑎 → inf({𝑧, 𝑅}, ℝ^, < ) = inf({𝑎, 𝑅}, ℝ^, < ))
28		simpr 110	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 𝑎 ∈ (0[,]+∞))
29		elxrge0 10359	. . . . . . . 8 ⊢ (𝑎 ∈ (0[,]+∞) ↔ (𝑎 ∈ ℝ^* ∧ 0 ≤ 𝑎))
30	29	simplbi 274	. . . . . . 7 ⊢ (𝑎 ∈ (0[,]+∞) → 𝑎 ∈ ℝ^*)
31		xrmincl 12010	. . . . . . 7 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({𝑎, 𝑅}, ℝ^, < ) ∈ ℝ^*)
32	30, 21, 31	syl2anr 290	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → inf({𝑎, 𝑅}, ℝ^, < ) ∈ ℝ^)
33	25, 27, 28, 32	fvmptd3 5793	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < ))‘𝑎) = inf({𝑎, 𝑅}, ℝ^, < ))
34	33	eqeq1d 2247	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < ))‘𝑎) = 0 ↔ inf({𝑎, 𝑅}, ℝ^, < ) = 0))
35		0xr 8362	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
36	35	a1i 9	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 0 ∈ ℝ^*)
37	30	adantl 277	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 𝑎 ∈ ℝ^*)
38	21	adantr 276	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 𝑅 ∈ ℝ^*)
39		xrltmininf 12014	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ 𝑎 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → (0 < inf({𝑎, 𝑅}, ℝ^, < ) ↔ (0 < 𝑎 ∧ 0 < 𝑅)))
40	36, 37, 38, 39	syl3anc 1278	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (0 < inf({𝑎, 𝑅}, ℝ^*, < ) ↔ (0 < 𝑎 ∧ 0 < 𝑅)))
41		simp3 1030	. . . . . . . . 9 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 0 < 𝑅)
42	41	adantr 276	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 0 < 𝑅)
43	42	biantrud 304	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (0 < 𝑎 ↔ (0 < 𝑎 ∧ 0 < 𝑅)))
44	40, 43	bitr4d 191	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (0 < inf({𝑎, 𝑅}, ℝ^*, < ) ↔ 0 < 𝑎))
45	44	notbid 677	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (¬ 0 < inf({𝑎, 𝑅}, ℝ^*, < ) ↔ ¬ 0 < 𝑎))
46	28, 29	sylib 122	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (𝑎 ∈ ℝ^* ∧ 0 ≤ 𝑎))
47	46	simprd 114	. . . . . . . . 9 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 0 ≤ 𝑎)
48		xrltle 10179	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^*) → (0 < 𝑅 → 0 ≤ 𝑅))
49	35, 21, 48	sylancr 418	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → (0 < 𝑅 → 0 ≤ 𝑅))
50	41, 49	mpd 13	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 0 ≤ 𝑅)
51	50	adantr 276	. . . . . . . . 9 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 0 ≤ 𝑅)
52		xrlemininf 12015	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ 𝑎 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → (0 ≤ inf({𝑎, 𝑅}, ℝ^, < ) ↔ (0 ≤ 𝑎 ∧ 0 ≤ 𝑅)))
53	36, 37, 38, 52	syl3anc 1278	. . . . . . . . 9 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (0 ≤ inf({𝑎, 𝑅}, ℝ^*, < ) ↔ (0 ≤ 𝑎 ∧ 0 ≤ 𝑅)))
54	47, 51, 53	mpbir2and 957	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 0 ≤ inf({𝑎, 𝑅}, ℝ^*, < ))
55		xrlenlt 8380	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ inf({𝑎, 𝑅}, ℝ^, < ) ∈ ℝ^) → (0 ≤ inf({𝑎, 𝑅}, ℝ^, < ) ↔ ¬ inf({𝑎, 𝑅}, ℝ^, < ) < 0))
56	35, 32, 55	sylancr 418	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (0 ≤ inf({𝑎, 𝑅}, ℝ^, < ) ↔ ¬ inf({𝑎, 𝑅}, ℝ^, < ) < 0))
57	54, 56	mpbid 147	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → ¬ inf({𝑎, 𝑅}, ℝ^*, < ) < 0)
58	57	biantrurd 305	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (¬ 0 < inf({𝑎, 𝑅}, ℝ^, < ) ↔ (¬ inf({𝑎, 𝑅}, ℝ^, < ) < 0 ∧ ¬ 0 < inf({𝑎, 𝑅}, ℝ^*, < ))))
59		xrlttri3 10178	. . . . . . 7 ⊢ ((inf({𝑎, 𝑅}, ℝ^, < ) ∈ ℝ^ ∧ 0 ∈ ℝ^) → (inf({𝑎, 𝑅}, ℝ^, < ) = 0 ↔ (¬ inf({𝑎, 𝑅}, ℝ^, < ) < 0 ∧ ¬ 0 < inf({𝑎, 𝑅}, ℝ^, < ))))
60	32, 36, 59	syl2anc 415	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (inf({𝑎, 𝑅}, ℝ^, < ) = 0 ↔ (¬ inf({𝑎, 𝑅}, ℝ^, < ) < 0 ∧ ¬ 0 < inf({𝑎, 𝑅}, ℝ^*, < ))))
61	58, 60	bitr4d 191	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (¬ 0 < inf({𝑎, 𝑅}, ℝ^, < ) ↔ inf({𝑎, 𝑅}, ℝ^, < ) = 0))
62		xrlenlt 8380	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ 𝑎 ∈ ℝ^*) → (0 ≤ 𝑎 ↔ ¬ 𝑎 < 0))
63	35, 37, 62	sylancr 418	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (0 ≤ 𝑎 ↔ ¬ 𝑎 < 0))
64	47, 63	mpbid 147	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → ¬ 𝑎 < 0)
65	64	biantrurd 305	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (¬ 0 < 𝑎 ↔ (¬ 𝑎 < 0 ∧ ¬ 0 < 𝑎)))
66		xrlttri3 10178	. . . . . . 7 ⊢ ((𝑎 ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (𝑎 = 0 ↔ (¬ 𝑎 < 0 ∧ ¬ 0 < 𝑎)))
67	37, 36, 66	syl2anc 415	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (𝑎 = 0 ↔ (¬ 𝑎 < 0 ∧ ¬ 0 < 𝑎)))
68	65, 67	bitr4d 191	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (¬ 0 < 𝑎 ↔ 𝑎 = 0))
69	45, 61, 68	3bitr3d 218	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (inf({𝑎, 𝑅}, ℝ^*, < ) = 0 ↔ 𝑎 = 0))
70	34, 69	bitrd 188	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^*, < ))‘𝑎) = 0 ↔ 𝑎 = 0))
71	30	ad2antrl 494	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 𝑎 ∈ ℝ^*)
72	21	adantr 276	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 𝑅 ∈ ℝ^*)
73		xrmin1inf 12011	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({𝑎, 𝑅}, ℝ^, < ) ≤ 𝑎)
74	71, 72, 73	syl2anc 415	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → inf({𝑎, 𝑅}, ℝ^*, < ) ≤ 𝑎)
75	71, 72, 31	syl2anc 415	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → inf({𝑎, 𝑅}, ℝ^, < ) ∈ ℝ^)
76		elxrge0 10359	. . . . . . . . . 10 ⊢ (𝑏 ∈ (0[,]+∞) ↔ (𝑏 ∈ ℝ^* ∧ 0 ≤ 𝑏))
77	76	simplbi 274	. . . . . . . . 9 ⊢ (𝑏 ∈ (0[,]+∞) → 𝑏 ∈ ℝ^*)
78	77	ad2antll 495	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 𝑏 ∈ ℝ^*)
79		xrletr 10189	. . . . . . . 8 ⊢ ((inf({𝑎, 𝑅}, ℝ^, < ) ∈ ℝ^ ∧ 𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) → ((inf({𝑎, 𝑅}, ℝ^, < ) ≤ 𝑎 ∧ 𝑎 ≤ 𝑏) → inf({𝑎, 𝑅}, ℝ^*, < ) ≤ 𝑏))
80	75, 71, 78, 79	syl3anc 1278	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → ((inf({𝑎, 𝑅}, ℝ^, < ) ≤ 𝑎 ∧ 𝑎 ≤ 𝑏) → inf({𝑎, 𝑅}, ℝ^, < ) ≤ 𝑏))
81	74, 80	mpand 433	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎 ≤ 𝑏 → inf({𝑎, 𝑅}, ℝ^*, < ) ≤ 𝑏))
82		xrmin2inf 12012	. . . . . . 7 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({𝑎, 𝑅}, ℝ^, < ) ≤ 𝑅)
83	71, 72, 82	syl2anc 415	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → inf({𝑎, 𝑅}, ℝ^*, < ) ≤ 𝑅)
84	81, 83	jctird 317	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎 ≤ 𝑏 → (inf({𝑎, 𝑅}, ℝ^, < ) ≤ 𝑏 ∧ inf({𝑎, 𝑅}, ℝ^, < ) ≤ 𝑅)))
85		xrlemininf 12015	. . . . . 6 ⊢ ((inf({𝑎, 𝑅}, ℝ^, < ) ∈ ℝ^ ∧ 𝑏 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → (inf({𝑎, 𝑅}, ℝ^, < ) ≤ inf({𝑏, 𝑅}, ℝ^, < ) ↔ (inf({𝑎, 𝑅}, ℝ^, < ) ≤ 𝑏 ∧ inf({𝑎, 𝑅}, ℝ^*, < ) ≤ 𝑅)))
86	75, 78, 72, 85	syl3anc 1278	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (inf({𝑎, 𝑅}, ℝ^, < ) ≤ inf({𝑏, 𝑅}, ℝ^, < ) ↔ (inf({𝑎, 𝑅}, ℝ^, < ) ≤ 𝑏 ∧ inf({𝑎, 𝑅}, ℝ^, < ) ≤ 𝑅)))
87	84, 86	sylibrd 169	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎 ≤ 𝑏 → inf({𝑎, 𝑅}, ℝ^, < ) ≤ inf({𝑏, 𝑅}, ℝ^, < )))
88	33	adantrr 483	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < ))‘𝑎) = inf({𝑎, 𝑅}, ℝ^, < ))
89		preq1 3784	. . . . . . 7 ⊢ (𝑧 = 𝑏 → {𝑧, 𝑅} = {𝑏, 𝑅})
90	89	infeq1d 7342	. . . . . 6 ⊢ (𝑧 = 𝑏 → inf({𝑧, 𝑅}, ℝ^, < ) = inf({𝑏, 𝑅}, ℝ^, < ))
91		simpr 110	. . . . . . 7 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → 𝑏 ∈ (0[,]+∞))
92	91	adantl 277	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 𝑏 ∈ (0[,]+∞))
93		xrmincl 12010	. . . . . . 7 ⊢ ((𝑏 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({𝑏, 𝑅}, ℝ^, < ) ∈ ℝ^*)
94	78, 72, 93	syl2anc 415	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → inf({𝑏, 𝑅}, ℝ^, < ) ∈ ℝ^)
95	25, 90, 92, 94	fvmptd3 5793	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < ))‘𝑏) = inf({𝑏, 𝑅}, ℝ^, < ))
96	88, 95	breq12d 4138	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < ))‘𝑎) ≤ ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < ))‘𝑏) ↔ inf({𝑎, 𝑅}, ℝ^, < ) ≤ inf({𝑏, 𝑅}, ℝ^, < )))
97	87, 96	sylibrd 169	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎 ≤ 𝑏 → ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < ))‘𝑎) ≤ ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < ))‘𝑏)))
98	29	simprbi 275	. . . . . 6 ⊢ (𝑎 ∈ (0[,]+∞) → 0 ≤ 𝑎)
99	98	ad2antrl 494	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 0 ≤ 𝑎)
100	76	simprbi 275	. . . . . 6 ⊢ (𝑏 ∈ (0[,]+∞) → 0 ≤ 𝑏)
101	100	ad2antll 495	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 0 ≤ 𝑏)
102	41	adantr 276	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 0 < 𝑅)
103		xrbdtri 12020	. . . . 5 ⊢ (((𝑎 ∈ ℝ^* ∧ 0 ≤ 𝑎) ∧ (𝑏 ∈ ℝ^* ∧ 0 ≤ 𝑏) ∧ (𝑅 ∈ ℝ^* ∧ 0 < 𝑅)) → inf({(𝑎 +_𝑒 𝑏), 𝑅}, ℝ^, < ) ≤ (inf({𝑎, 𝑅}, ℝ^, < ) +_𝑒 inf({𝑏, 𝑅}, ℝ^*, < )))
104	71, 99, 78, 101, 72, 102, 103	syl222anc 1294	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → inf({(𝑎 +_𝑒 𝑏), 𝑅}, ℝ^, < ) ≤ (inf({𝑎, 𝑅}, ℝ^, < ) +_𝑒 inf({𝑏, 𝑅}, ℝ^*, < )))
105		preq1 3784	. . . . . 6 ⊢ (𝑧 = (𝑎 +_𝑒 𝑏) → {𝑧, 𝑅} = {(𝑎 +_𝑒 𝑏), 𝑅})
106	105	infeq1d 7342	. . . . 5 ⊢ (𝑧 = (𝑎 +_𝑒 𝑏) → inf({𝑧, 𝑅}, ℝ^, < ) = inf({(𝑎 +_𝑒 𝑏), 𝑅}, ℝ^, < ))
107		ge0xaddcl 10364	. . . . . 6 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → (𝑎 +_𝑒 𝑏) ∈ (0[,]+∞))
108	107	adantl 277	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎 +_𝑒 𝑏) ∈ (0[,]+∞))
109	71, 78	xaddcld 10265	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎 +_𝑒 𝑏) ∈ ℝ^*)
110		xrmincl 12010	. . . . . 6 ⊢ (((𝑎 +_𝑒 𝑏) ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({(𝑎 +_𝑒 𝑏), 𝑅}, ℝ^, < ) ∈ ℝ^*)
111	109, 72, 110	syl2anc 415	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → inf({(𝑎 +_𝑒 𝑏), 𝑅}, ℝ^, < ) ∈ ℝ^)
112	25, 106, 108, 111	fvmptd3 5793	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < ))‘(𝑎 +_𝑒 𝑏)) = inf({(𝑎 +_𝑒 𝑏), 𝑅}, ℝ^, < ))
113	88, 95	oveq12d 6093	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < ))‘𝑎) +_𝑒 ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < ))‘𝑏)) = (inf({𝑎, 𝑅}, ℝ^, < ) +_𝑒 inf({𝑏, 𝑅}, ℝ^, < )))
114	104, 112, 113	3brtr4d 4157	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < ))‘(𝑎 +_𝑒 𝑏)) ≤ (((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < ))‘𝑎) +_𝑒 ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^*, < ))‘𝑏)))
115	1, 24, 70, 97, 114	comet 15523	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^*, < )) ∘ 𝐶) ∈ (∞Met‘𝑋))
116	18, 115	eqeltrrd 2316	1 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐷 ∈ (∞Met‘𝑋))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 {cpr 3706 class class class wbr 4125 ↦ cmpt 4187 × cxp 4767 ∘ ccom 4773 Fn wfn 5367 ⟶wf 5368 ‘cfv 5372 (class class class)co 6075 ∈ cmpo 6077 infcinf 7313 0cc0 8169 +∞cpnf 8347 ℝ^*cxr 8349 < clt 8350 ≤ cle 8351 +_𝑒 cxad 10151 [,]cicc 10272 ∞Metcxmet 14845

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 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-coll 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287 ax-arch 8288 ax-caucvg 8289

This theorem 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-reu 2535 df-rmo 2536 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-nul 3521 df-if 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-po 4436 df-iso 4437 df-iord 4506 df-on 4508 df-ilim 4509 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-isom 5381 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-frec 6652 df-map 6914 df-sup 7314 df-inf 7315 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-inn 9284 df-2 9342 df-3 9343 df-4 9344 df-n0 9543 df-z 9624 df-uz 9901 df-rp 10034 df-xneg 10153 df-xadd 10154 df-icc 10276 df-seqfrec 10863 df-exp 10954 df-cj 11585 df-re 11586 df-im 11587 df-rsqrt 11742 df-abs 11743 df-xmet 14853

This theorem is referenced by: bdmet 15526 bdbl 15527 bdmopn 15528

W3C validator