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Theorem bdxmet 13668

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 997	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐶 ∈ (∞Met‘𝑋))
2		xmetcl 13519	. . . . . . 7 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐶𝑦) ∈ ℝ^*)
3		xmetge0 13532	. . . . . . 7 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝑥𝐶𝑦))
4		elxrge0 9965	. . . . . . 7 ⊢ ((𝑥𝐶𝑦) ∈ (0[,]+∞) ↔ ((𝑥𝐶𝑦) ∈ ℝ^* ∧ 0 ≤ (𝑥𝐶𝑦)))
5	2, 3, 4	sylanbrc 417	. . . . . 6 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐶𝑦) ∈ (0[,]+∞))
6	5	3expb 1204	. . . . 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 13517	. . . . . . 7 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐶:(𝑋 × 𝑋)⟶ℝ^*)
9	8	3ad2ant1 1018	. . . . . 6 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐶:(𝑋 × 𝑋)⟶ℝ^*)
10	9	ffnd 5362	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐶 Fn (𝑋 × 𝑋))
11		fnovim 5977	. . . . 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 2178	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → (𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < )) = (𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < )))
14		preq1 3668	. . . . 5 ⊢ (𝑧 = (𝑥𝐶𝑦) → {𝑧, 𝑅} = {(𝑥𝐶𝑦), 𝑅})
15	14	infeq1d 7005	. . . 4 ⊢ (𝑧 = (𝑥𝐶𝑦) → inf({𝑧, 𝑅}, ℝ^, < ) = inf({(𝑥𝐶𝑦), 𝑅}, ℝ^, < ))
16	7, 12, 13, 15	fmpoco 6211	. . 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 2228	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^*, < )) ∘ 𝐶) = 𝐷)
19		elxrge0 9965	. . . . . 6 ⊢ (𝑧 ∈ (0[,]+∞) ↔ (𝑧 ∈ ℝ^* ∧ 0 ≤ 𝑧))
20	19	simplbi 274	. . . . 5 ⊢ (𝑧 ∈ (0[,]+∞) → 𝑧 ∈ ℝ^*)
21		simp2 998	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝑅 ∈ ℝ^*)
22		xrmincl 11258	. . . . 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 5667	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → (𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < )):(0[,]+∞)⟶ℝ^)
25		eqid 2177	. . . . . 6 ⊢ (𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < )) = (𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < ))
26		preq1 3668	. . . . . . 7 ⊢ (𝑧 = 𝑎 → {𝑧, 𝑅} = {𝑎, 𝑅})
27	26	infeq1d 7005	. . . . . 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 9965	. . . . . . . 8 ⊢ (𝑎 ∈ (0[,]+∞) ↔ (𝑎 ∈ ℝ^* ∧ 0 ≤ 𝑎))
30	29	simplbi 274	. . . . . . 7 ⊢ (𝑎 ∈ (0[,]+∞) → 𝑎 ∈ ℝ^*)
31		xrmincl 11258	. . . . . . 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 5605	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < ))‘𝑎) = inf({𝑎, 𝑅}, ℝ^, < ))
34	33	eqeq1d 2186	. . . 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 7994	. . . . . . . . 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 11262	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ 𝑎 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → (0 < inf({𝑎, 𝑅}, ℝ^, < ) ↔ (0 < 𝑎 ∧ 0 < 𝑅)))
40	36, 37, 38, 39	syl3anc 1238	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (0 < inf({𝑎, 𝑅}, ℝ^*, < ) ↔ (0 < 𝑎 ∧ 0 < 𝑅)))
41		simp3 999	. . . . . . . . 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 667	. . . . 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 9785	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^*) → (0 < 𝑅 → 0 ≤ 𝑅))
49	35, 21, 48	sylancr 414	. . . . . . . . . . 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 11263	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ 𝑎 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → (0 ≤ inf({𝑎, 𝑅}, ℝ^, < ) ↔ (0 ≤ 𝑎 ∧ 0 ≤ 𝑅)))
53	36, 37, 38, 52	syl3anc 1238	. . . . . . . . 9 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (0 ≤ inf({𝑎, 𝑅}, ℝ^*, < ) ↔ (0 ≤ 𝑎 ∧ 0 ≤ 𝑅)))
54	47, 51, 53	mpbir2and 944	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 0 ≤ inf({𝑎, 𝑅}, ℝ^*, < ))
55		xrlenlt 8012	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ inf({𝑎, 𝑅}, ℝ^, < ) ∈ ℝ^) → (0 ≤ inf({𝑎, 𝑅}, ℝ^, < ) ↔ ¬ inf({𝑎, 𝑅}, ℝ^, < ) < 0))
56	35, 32, 55	sylancr 414	. . . . . . . 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 9784	. . . . . . 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 411	. . . . . 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 8012	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ 𝑎 ∈ ℝ^*) → (0 ≤ 𝑎 ↔ ¬ 𝑎 < 0))
63	35, 37, 62	sylancr 414	. . . . . . . 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 9784	. . . . . . 7 ⊢ ((𝑎 ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (𝑎 = 0 ↔ (¬ 𝑎 < 0 ∧ ¬ 0 < 𝑎)))
67	37, 36, 66	syl2anc 411	. . . . . 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 490	. . . . . . . 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 11259	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({𝑎, 𝑅}, ℝ^, < ) ≤ 𝑎)
74	71, 72, 73	syl2anc 411	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → inf({𝑎, 𝑅}, ℝ^*, < ) ≤ 𝑎)
75	71, 72, 31	syl2anc 411	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → inf({𝑎, 𝑅}, ℝ^, < ) ∈ ℝ^)
76		elxrge0 9965	. . . . . . . . . 10 ⊢ (𝑏 ∈ (0[,]+∞) ↔ (𝑏 ∈ ℝ^* ∧ 0 ≤ 𝑏))
77	76	simplbi 274	. . . . . . . . 9 ⊢ (𝑏 ∈ (0[,]+∞) → 𝑏 ∈ ℝ^*)
78	77	ad2antll 491	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 𝑏 ∈ ℝ^*)
79		xrletr 9795	. . . . . . . 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 1238	. . . . . . 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 429	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎 ≤ 𝑏 → inf({𝑎, 𝑅}, ℝ^*, < ) ≤ 𝑏))
82		xrmin2inf 11260	. . . . . . 7 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({𝑎, 𝑅}, ℝ^, < ) ≤ 𝑅)
83	71, 72, 82	syl2anc 411	. . . . . 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 11263	. . . . . 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 1238	. . . . 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 479	. . . . 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 3668	. . . . . . 7 ⊢ (𝑧 = 𝑏 → {𝑧, 𝑅} = {𝑏, 𝑅})
90	89	infeq1d 7005	. . . . . 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 11258	. . . . . . 7 ⊢ ((𝑏 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({𝑏, 𝑅}, ℝ^, < ) ∈ ℝ^*)
94	78, 72, 93	syl2anc 411	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → inf({𝑏, 𝑅}, ℝ^, < ) ∈ ℝ^)
95	25, 90, 92, 94	fvmptd3 5605	. . . . 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 4013	. . . 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 490	. . . . 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 491	. . . . 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 11268	. . . . 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 1254	. . . 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 3668	. . . . . 6 ⊢ (𝑧 = (𝑎 +_𝑒 𝑏) → {𝑧, 𝑅} = {(𝑎 +_𝑒 𝑏), 𝑅})
106	105	infeq1d 7005	. . . . 5 ⊢ (𝑧 = (𝑎 +_𝑒 𝑏) → inf({𝑧, 𝑅}, ℝ^, < ) = inf({(𝑎 +_𝑒 𝑏), 𝑅}, ℝ^, < ))
107		ge0xaddcl 9970	. . . . . 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 9871	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎 +_𝑒 𝑏) ∈ ℝ^*)
110		xrmincl 11258	. . . . . 6 ⊢ (((𝑎 +_𝑒 𝑏) ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({(𝑎 +_𝑒 𝑏), 𝑅}, ℝ^, < ) ∈ ℝ^*)
111	109, 72, 110	syl2anc 411	. . . . 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 5605	. . . 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 5887	. . . 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 4032	. . 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 13666	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^*, < )) ∘ 𝐶) ∈ (∞Met‘𝑋))
116	18, 115	eqeltrrd 2255	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 978 = wceq 1353 ∈ wcel 2148 {cpr 3592 class class class wbr 4000 ↦ cmpt 4061 × cxp 4621 ∘ ccom 4627 Fn wfn 5207 ⟶wf 5208 ‘cfv 5212 (class class class)co 5869 ∈ cmpo 5871 infcinf 6976 0cc0 7802 +∞cpnf 7979 ℝ^*cxr 7981 < clt 7982 ≤ cle 7983 +_𝑒 cxad 9757 [,]cicc 9878 ∞Metcxmet 13147

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921 ax-caucvg 7922

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-isom 5221 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-frec 6386 df-map 6644 df-sup 6977 df-inf 6978 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-n0 9166 df-z 9243 df-uz 9518 df-rp 9641 df-xneg 9759 df-xadd 9760 df-icc 9882 df-seqfrec 10432 df-exp 10506 df-cj 10835 df-re 10836 df-im 10837 df-rsqrt 10991 df-abs 10992 df-xmet 13155

This theorem is referenced by: bdmet 13669 bdbl 13670 bdmopn 13671

W3C validator