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Theorem bdxmet 12429

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 949	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐶 ∈ (∞Met‘𝑋))
2		xmetcl 12280	. . . . . . 7 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐶𝑦) ∈ ℝ^*)
3		xmetge0 12293	. . . . . . 7 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝑥𝐶𝑦))
4		elxrge0 9602	. . . . . . 7 ⊢ ((𝑥𝐶𝑦) ∈ (0[,]+∞) ↔ ((𝑥𝐶𝑦) ∈ ℝ^* ∧ 0 ≤ (𝑥𝐶𝑦)))
5	2, 3, 4	sylanbrc 411	. . . . . 6 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐶𝑦) ∈ (0[,]+∞))
6	5	3expb 1150	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ∈ (0[,]+∞))
7	1, 6	sylan 279	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ∈ (0[,]+∞))
8		xmetf 12278	. . . . . . 7 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐶:(𝑋 × 𝑋)⟶ℝ^*)
9	8	3ad2ant1 970	. . . . . 6 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐶:(𝑋 × 𝑋)⟶ℝ^*)
10	9	ffnd 5209	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐶 Fn (𝑋 × 𝑋))
11		fnovim 5811	. . . . 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 2101	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → (𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < )) = (𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < )))
14		preq1 3547	. . . . 5 ⊢ (𝑧 = (𝑥𝐶𝑦) → {𝑧, 𝑅} = {(𝑥𝐶𝑦), 𝑅})
15	14	infeq1d 6814	. . . 4 ⊢ (𝑧 = (𝑥𝐶𝑦) → inf({𝑧, 𝑅}, ℝ^, < ) = inf({(𝑥𝐶𝑦), 𝑅}, ℝ^, < ))
16	7, 12, 13, 15	fmpoco 6043	. . 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	syl6eqr 2150	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^*, < )) ∘ 𝐶) = 𝐷)
19		elxrge0 9602	. . . . . 6 ⊢ (𝑧 ∈ (0[,]+∞) ↔ (𝑧 ∈ ℝ^* ∧ 0 ≤ 𝑧))
20	19	simplbi 270	. . . . 5 ⊢ (𝑧 ∈ (0[,]+∞) → 𝑧 ∈ ℝ^*)
21		simp2 950	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝑅 ∈ ℝ^*)
22		xrmincl 10874	. . . . 5 ⊢ ((𝑧 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({𝑧, 𝑅}, ℝ^, < ) ∈ ℝ^*)
23	20, 21, 22	syl2anr 286	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑧 ∈ (0[,]+∞)) → inf({𝑧, 𝑅}, ℝ^, < ) ∈ ℝ^)
24	23	fmpttd 5507	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → (𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < )):(0[,]+∞)⟶ℝ^)
25		eqid 2100	. . . . . 6 ⊢ (𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < )) = (𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < ))
26		preq1 3547	. . . . . . 7 ⊢ (𝑧 = 𝑎 → {𝑧, 𝑅} = {𝑎, 𝑅})
27	26	infeq1d 6814	. . . . . 6 ⊢ (𝑧 = 𝑎 → inf({𝑧, 𝑅}, ℝ^, < ) = inf({𝑎, 𝑅}, ℝ^, < ))
28		simpr 109	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 𝑎 ∈ (0[,]+∞))
29		elxrge0 9602	. . . . . . . 8 ⊢ (𝑎 ∈ (0[,]+∞) ↔ (𝑎 ∈ ℝ^* ∧ 0 ≤ 𝑎))
30	29	simplbi 270	. . . . . . 7 ⊢ (𝑎 ∈ (0[,]+∞) → 𝑎 ∈ ℝ^*)
31		xrmincl 10874	. . . . . . 7 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({𝑎, 𝑅}, ℝ^, < ) ∈ ℝ^*)
32	30, 21, 31	syl2anr 286	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → inf({𝑎, 𝑅}, ℝ^, < ) ∈ ℝ^)
33	25, 27, 28, 32	fvmptd3 5446	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < ))‘𝑎) = inf({𝑎, 𝑅}, ℝ^, < ))
34	33	eqeq1d 2108	. . . 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 7684	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
36	35	a1i 9	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 0 ∈ ℝ^*)
37	30	adantl 273	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 𝑎 ∈ ℝ^*)
38	21	adantr 272	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 𝑅 ∈ ℝ^*)
39		xrltmininf 10878	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ 𝑎 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → (0 < inf({𝑎, 𝑅}, ℝ^, < ) ↔ (0 < 𝑎 ∧ 0 < 𝑅)))
40	36, 37, 38, 39	syl3anc 1184	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (0 < inf({𝑎, 𝑅}, ℝ^*, < ) ↔ (0 < 𝑎 ∧ 0 < 𝑅)))
41		simp3 951	. . . . . . . . 9 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 0 < 𝑅)
42	41	adantr 272	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 0 < 𝑅)
43	42	biantrud 300	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (0 < 𝑎 ↔ (0 < 𝑎 ∧ 0 < 𝑅)))
44	40, 43	bitr4d 190	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (0 < inf({𝑎, 𝑅}, ℝ^*, < ) ↔ 0 < 𝑎))
45	44	notbid 633	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (¬ 0 < inf({𝑎, 𝑅}, ℝ^*, < ) ↔ ¬ 0 < 𝑎))
46	28, 29	sylib 121	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (𝑎 ∈ ℝ^* ∧ 0 ≤ 𝑎))
47	46	simprd 113	. . . . . . . . 9 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 0 ≤ 𝑎)
48		xrltle 9425	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^*) → (0 < 𝑅 → 0 ≤ 𝑅))
49	35, 21, 48	sylancr 408	. . . . . . . . . . 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 272	. . . . . . . . 9 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 0 ≤ 𝑅)
52		xrlemininf 10879	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ 𝑎 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → (0 ≤ inf({𝑎, 𝑅}, ℝ^, < ) ↔ (0 ≤ 𝑎 ∧ 0 ≤ 𝑅)))
53	36, 37, 38, 52	syl3anc 1184	. . . . . . . . 9 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (0 ≤ inf({𝑎, 𝑅}, ℝ^*, < ) ↔ (0 ≤ 𝑎 ∧ 0 ≤ 𝑅)))
54	47, 51, 53	mpbir2and 896	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 0 ≤ inf({𝑎, 𝑅}, ℝ^*, < ))
55		xrlenlt 7701	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ inf({𝑎, 𝑅}, ℝ^, < ) ∈ ℝ^) → (0 ≤ inf({𝑎, 𝑅}, ℝ^, < ) ↔ ¬ inf({𝑎, 𝑅}, ℝ^, < ) < 0))
56	35, 32, 55	sylancr 408	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (0 ≤ inf({𝑎, 𝑅}, ℝ^, < ) ↔ ¬ inf({𝑎, 𝑅}, ℝ^, < ) < 0))
57	54, 56	mpbid 146	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → ¬ inf({𝑎, 𝑅}, ℝ^*, < ) < 0)
58	57	biantrurd 301	. . . . . 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 9424	. . . . . . 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 406	. . . . . 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 190	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (¬ 0 < inf({𝑎, 𝑅}, ℝ^, < ) ↔ inf({𝑎, 𝑅}, ℝ^, < ) = 0))
62		xrlenlt 7701	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ 𝑎 ∈ ℝ^*) → (0 ≤ 𝑎 ↔ ¬ 𝑎 < 0))
63	35, 37, 62	sylancr 408	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (0 ≤ 𝑎 ↔ ¬ 𝑎 < 0))
64	47, 63	mpbid 146	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → ¬ 𝑎 < 0)
65	64	biantrurd 301	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (¬ 0 < 𝑎 ↔ (¬ 𝑎 < 0 ∧ ¬ 0 < 𝑎)))
66		xrlttri3 9424	. . . . . . 7 ⊢ ((𝑎 ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (𝑎 = 0 ↔ (¬ 𝑎 < 0 ∧ ¬ 0 < 𝑎)))
67	37, 36, 66	syl2anc 406	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (𝑎 = 0 ↔ (¬ 𝑎 < 0 ∧ ¬ 0 < 𝑎)))
68	65, 67	bitr4d 190	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (¬ 0 < 𝑎 ↔ 𝑎 = 0))
69	45, 61, 68	3bitr3d 217	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (inf({𝑎, 𝑅}, ℝ^*, < ) = 0 ↔ 𝑎 = 0))
70	34, 69	bitrd 187	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^*, < ))‘𝑎) = 0 ↔ 𝑎 = 0))
71	30	ad2antrl 477	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 𝑎 ∈ ℝ^*)
72	21	adantr 272	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 𝑅 ∈ ℝ^*)
73		xrmin1inf 10875	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({𝑎, 𝑅}, ℝ^, < ) ≤ 𝑎)
74	71, 72, 73	syl2anc 406	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → inf({𝑎, 𝑅}, ℝ^*, < ) ≤ 𝑎)
75	71, 72, 31	syl2anc 406	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → inf({𝑎, 𝑅}, ℝ^, < ) ∈ ℝ^)
76		elxrge0 9602	. . . . . . . . . 10 ⊢ (𝑏 ∈ (0[,]+∞) ↔ (𝑏 ∈ ℝ^* ∧ 0 ≤ 𝑏))
77	76	simplbi 270	. . . . . . . . 9 ⊢ (𝑏 ∈ (0[,]+∞) → 𝑏 ∈ ℝ^*)
78	77	ad2antll 478	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 𝑏 ∈ ℝ^*)
79		xrletr 9432	. . . . . . . 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 1184	. . . . . . 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 423	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎 ≤ 𝑏 → inf({𝑎, 𝑅}, ℝ^*, < ) ≤ 𝑏))
82		xrmin2inf 10876	. . . . . . 7 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({𝑎, 𝑅}, ℝ^, < ) ≤ 𝑅)
83	71, 72, 82	syl2anc 406	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → inf({𝑎, 𝑅}, ℝ^*, < ) ≤ 𝑅)
84	81, 83	jctird 313	. . . . 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 10879	. . . . . 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 1184	. . . . 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 168	. . . 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 466	. . . . 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 3547	. . . . . . 7 ⊢ (𝑧 = 𝑏 → {𝑧, 𝑅} = {𝑏, 𝑅})
90	89	infeq1d 6814	. . . . . 6 ⊢ (𝑧 = 𝑏 → inf({𝑧, 𝑅}, ℝ^, < ) = inf({𝑏, 𝑅}, ℝ^, < ))
91		simpr 109	. . . . . . 7 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → 𝑏 ∈ (0[,]+∞))
92	91	adantl 273	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 𝑏 ∈ (0[,]+∞))
93		xrmincl 10874	. . . . . . 7 ⊢ ((𝑏 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({𝑏, 𝑅}, ℝ^, < ) ∈ ℝ^*)
94	78, 72, 93	syl2anc 406	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → inf({𝑏, 𝑅}, ℝ^, < ) ∈ ℝ^)
95	25, 90, 92, 94	fvmptd3 5446	. . . . 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 3888	. . . 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 168	. . 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 271	. . . . . 6 ⊢ (𝑎 ∈ (0[,]+∞) → 0 ≤ 𝑎)
99	98	ad2antrl 477	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 0 ≤ 𝑎)
100	76	simprbi 271	. . . . . 6 ⊢ (𝑏 ∈ (0[,]+∞) → 0 ≤ 𝑏)
101	100	ad2antll 478	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 0 ≤ 𝑏)
102	41	adantr 272	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 0 < 𝑅)
103		xrbdtri 10884	. . . . 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 1200	. . . 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 3547	. . . . . 6 ⊢ (𝑧 = (𝑎 +_𝑒 𝑏) → {𝑧, 𝑅} = {(𝑎 +_𝑒 𝑏), 𝑅})
106	105	infeq1d 6814	. . . . 5 ⊢ (𝑧 = (𝑎 +_𝑒 𝑏) → inf({𝑧, 𝑅}, ℝ^, < ) = inf({(𝑎 +_𝑒 𝑏), 𝑅}, ℝ^, < ))
107		ge0xaddcl 9607	. . . . . 6 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → (𝑎 +_𝑒 𝑏) ∈ (0[,]+∞))
108	107	adantl 273	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎 +_𝑒 𝑏) ∈ (0[,]+∞))
109	71, 78	xaddcld 9508	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎 +_𝑒 𝑏) ∈ ℝ^*)
110		xrmincl 10874	. . . . . 6 ⊢ (((𝑎 +_𝑒 𝑏) ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({(𝑎 +_𝑒 𝑏), 𝑅}, ℝ^, < ) ∈ ℝ^*)
111	109, 72, 110	syl2anc 406	. . . . 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 5446	. . . 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 5724	. . . 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 3905	. . 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 12427	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^*, < )) ∘ 𝐶) ∈ (∞Met‘𝑋))
116	18, 115	eqeltrrd 2177	1 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐷 ∈ (∞Met‘𝑋))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 930 = wceq 1299 ∈ wcel 1448 {cpr 3475 class class class wbr 3875 ↦ cmpt 3929 × cxp 4475 ∘ ccom 4481 Fn wfn 5054 ⟶wf 5055 ‘cfv 5059 (class class class)co 5706 ∈ cmpo 5708 infcinf 6785 0cc0 7500 +∞cpnf 7669 ℝ^*cxr 7671 < clt 7672 ≤ cle 7673 +_𝑒 cxad 9398 [,]cicc 9515 ∞Metcxmet 11931

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-mulrcl 7594 ax-addcom 7595 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-1rid 7602 ax-0id 7603 ax-rnegex 7604 ax-precex 7605 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-apti 7610 ax-pre-ltadd 7611 ax-pre-mulgt0 7612 ax-pre-mulext 7613 ax-arch 7614 ax-caucvg 7615

This theorem depends on definitions: df-bi 116 df-dc 787 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rmo 2383 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-if 3422 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-id 4153 df-po 4156 df-iso 4157 df-iord 4226 df-on 4228 df-ilim 4229 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-isom 5068 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-recs 6132 df-frec 6218 df-map 6474 df-sup 6786 df-inf 6787 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-reap 8203 df-ap 8210 df-div 8294 df-inn 8579 df-2 8637 df-3 8638 df-4 8639 df-n0 8830 df-z 8907 df-uz 9177 df-rp 9292 df-xneg 9400 df-xadd 9401 df-icc 9519 df-seqfrec 10060 df-exp 10134 df-cj 10455 df-re 10456 df-im 10457 df-rsqrt 10610 df-abs 10611 df-xmet 11939

This theorem is referenced by: bdmet 12430 bdbl 12431 bdmopn 12432

W3C validator