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Theorem bdxmet 13295

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 992	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐶 ∈ (∞Met‘𝑋))
2		xmetcl 13146	. . . . . . 7 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐶𝑦) ∈ ℝ^*)
3		xmetge0 13159	. . . . . . 7 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝑥𝐶𝑦))
4		elxrge0 9935	. . . . . . 7 ⊢ ((𝑥𝐶𝑦) ∈ (0[,]+∞) ↔ ((𝑥𝐶𝑦) ∈ ℝ^* ∧ 0 ≤ (𝑥𝐶𝑦)))
5	2, 3, 4	sylanbrc 415	. . . . . 6 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐶𝑦) ∈ (0[,]+∞))
6	5	3expb 1199	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ∈ (0[,]+∞))
7	1, 6	sylan 281	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ∈ (0[,]+∞))
8		xmetf 13144	. . . . . . 7 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐶:(𝑋 × 𝑋)⟶ℝ^*)
9	8	3ad2ant1 1013	. . . . . 6 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐶:(𝑋 × 𝑋)⟶ℝ^*)
10	9	ffnd 5348	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐶 Fn (𝑋 × 𝑋))
11		fnovim 5961	. . . . 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 2171	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → (𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < )) = (𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < )))
14		preq1 3660	. . . . 5 ⊢ (𝑧 = (𝑥𝐶𝑦) → {𝑧, 𝑅} = {(𝑥𝐶𝑦), 𝑅})
15	14	infeq1d 6989	. . . 4 ⊢ (𝑧 = (𝑥𝐶𝑦) → inf({𝑧, 𝑅}, ℝ^, < ) = inf({(𝑥𝐶𝑦), 𝑅}, ℝ^, < ))
16	7, 12, 13, 15	fmpoco 6195	. . 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 2221	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^*, < )) ∘ 𝐶) = 𝐷)
19		elxrge0 9935	. . . . . 6 ⊢ (𝑧 ∈ (0[,]+∞) ↔ (𝑧 ∈ ℝ^* ∧ 0 ≤ 𝑧))
20	19	simplbi 272	. . . . 5 ⊢ (𝑧 ∈ (0[,]+∞) → 𝑧 ∈ ℝ^*)
21		simp2 993	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝑅 ∈ ℝ^*)
22		xrmincl 11229	. . . . 5 ⊢ ((𝑧 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({𝑧, 𝑅}, ℝ^, < ) ∈ ℝ^*)
23	20, 21, 22	syl2anr 288	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑧 ∈ (0[,]+∞)) → inf({𝑧, 𝑅}, ℝ^, < ) ∈ ℝ^)
24	23	fmpttd 5651	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → (𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < )):(0[,]+∞)⟶ℝ^)
25		eqid 2170	. . . . . 6 ⊢ (𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < )) = (𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < ))
26		preq1 3660	. . . . . . 7 ⊢ (𝑧 = 𝑎 → {𝑧, 𝑅} = {𝑎, 𝑅})
27	26	infeq1d 6989	. . . . . 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 9935	. . . . . . . 8 ⊢ (𝑎 ∈ (0[,]+∞) ↔ (𝑎 ∈ ℝ^* ∧ 0 ≤ 𝑎))
30	29	simplbi 272	. . . . . . 7 ⊢ (𝑎 ∈ (0[,]+∞) → 𝑎 ∈ ℝ^*)
31		xrmincl 11229	. . . . . . 7 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({𝑎, 𝑅}, ℝ^, < ) ∈ ℝ^*)
32	30, 21, 31	syl2anr 288	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → inf({𝑎, 𝑅}, ℝ^, < ) ∈ ℝ^)
33	25, 27, 28, 32	fvmptd3 5589	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^, < ))‘𝑎) = inf({𝑎, 𝑅}, ℝ^, < ))
34	33	eqeq1d 2179	. . . 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 7966	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
36	35	a1i 9	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 0 ∈ ℝ^*)
37	30	adantl 275	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 𝑎 ∈ ℝ^*)
38	21	adantr 274	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 𝑅 ∈ ℝ^*)
39		xrltmininf 11233	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ 𝑎 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → (0 < inf({𝑎, 𝑅}, ℝ^, < ) ↔ (0 < 𝑎 ∧ 0 < 𝑅)))
40	36, 37, 38, 39	syl3anc 1233	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (0 < inf({𝑎, 𝑅}, ℝ^*, < ) ↔ (0 < 𝑎 ∧ 0 < 𝑅)))
41		simp3 994	. . . . . . . . 9 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 0 < 𝑅)
42	41	adantr 274	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 0 < 𝑅)
43	42	biantrud 302	. . . . . . 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 662	. . . . 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 9755	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^*) → (0 < 𝑅 → 0 ≤ 𝑅))
49	35, 21, 48	sylancr 412	. . . . . . . . . . 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 274	. . . . . . . . 9 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 0 ≤ 𝑅)
52		xrlemininf 11234	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ 𝑎 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → (0 ≤ inf({𝑎, 𝑅}, ℝ^, < ) ↔ (0 ≤ 𝑎 ∧ 0 ≤ 𝑅)))
53	36, 37, 38, 52	syl3anc 1233	. . . . . . . . 9 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (0 ≤ inf({𝑎, 𝑅}, ℝ^*, < ) ↔ (0 ≤ 𝑎 ∧ 0 ≤ 𝑅)))
54	47, 51, 53	mpbir2and 939	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 0 ≤ inf({𝑎, 𝑅}, ℝ^*, < ))
55		xrlenlt 7984	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ inf({𝑎, 𝑅}, ℝ^, < ) ∈ ℝ^) → (0 ≤ inf({𝑎, 𝑅}, ℝ^, < ) ↔ ¬ inf({𝑎, 𝑅}, ℝ^, < ) < 0))
56	35, 32, 55	sylancr 412	. . . . . . . 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 303	. . . . . 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 9754	. . . . . . 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 409	. . . . . 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 7984	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ 𝑎 ∈ ℝ^*) → (0 ≤ 𝑎 ↔ ¬ 𝑎 < 0))
63	35, 37, 62	sylancr 412	. . . . . . . 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 303	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (¬ 0 < 𝑎 ↔ (¬ 𝑎 < 0 ∧ ¬ 0 < 𝑎)))
66		xrlttri3 9754	. . . . . . 7 ⊢ ((𝑎 ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (𝑎 = 0 ↔ (¬ 𝑎 < 0 ∧ ¬ 0 < 𝑎)))
67	37, 36, 66	syl2anc 409	. . . . . 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 487	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 𝑎 ∈ ℝ^*)
72	21	adantr 274	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 𝑅 ∈ ℝ^*)
73		xrmin1inf 11230	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({𝑎, 𝑅}, ℝ^, < ) ≤ 𝑎)
74	71, 72, 73	syl2anc 409	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → inf({𝑎, 𝑅}, ℝ^*, < ) ≤ 𝑎)
75	71, 72, 31	syl2anc 409	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → inf({𝑎, 𝑅}, ℝ^, < ) ∈ ℝ^)
76		elxrge0 9935	. . . . . . . . . 10 ⊢ (𝑏 ∈ (0[,]+∞) ↔ (𝑏 ∈ ℝ^* ∧ 0 ≤ 𝑏))
77	76	simplbi 272	. . . . . . . . 9 ⊢ (𝑏 ∈ (0[,]+∞) → 𝑏 ∈ ℝ^*)
78	77	ad2antll 488	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 𝑏 ∈ ℝ^*)
79		xrletr 9765	. . . . . . . 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 1233	. . . . . . 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 427	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎 ≤ 𝑏 → inf({𝑎, 𝑅}, ℝ^*, < ) ≤ 𝑏))
82		xrmin2inf 11231	. . . . . . 7 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({𝑎, 𝑅}, ℝ^, < ) ≤ 𝑅)
83	71, 72, 82	syl2anc 409	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → inf({𝑎, 𝑅}, ℝ^*, < ) ≤ 𝑅)
84	81, 83	jctird 315	. . . . 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 11234	. . . . . 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 1233	. . . . 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 476	. . . . 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 3660	. . . . . . 7 ⊢ (𝑧 = 𝑏 → {𝑧, 𝑅} = {𝑏, 𝑅})
90	89	infeq1d 6989	. . . . . 6 ⊢ (𝑧 = 𝑏 → inf({𝑧, 𝑅}, ℝ^, < ) = inf({𝑏, 𝑅}, ℝ^, < ))
91		simpr 109	. . . . . . 7 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → 𝑏 ∈ (0[,]+∞))
92	91	adantl 275	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 𝑏 ∈ (0[,]+∞))
93		xrmincl 11229	. . . . . . 7 ⊢ ((𝑏 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({𝑏, 𝑅}, ℝ^, < ) ∈ ℝ^*)
94	78, 72, 93	syl2anc 409	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → inf({𝑏, 𝑅}, ℝ^, < ) ∈ ℝ^)
95	25, 90, 92, 94	fvmptd3 5589	. . . . 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 4002	. . . 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 273	. . . . . 6 ⊢ (𝑎 ∈ (0[,]+∞) → 0 ≤ 𝑎)
99	98	ad2antrl 487	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 0 ≤ 𝑎)
100	76	simprbi 273	. . . . . 6 ⊢ (𝑏 ∈ (0[,]+∞) → 0 ≤ 𝑏)
101	100	ad2antll 488	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 0 ≤ 𝑏)
102	41	adantr 274	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 0 < 𝑅)
103		xrbdtri 11239	. . . . 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 1249	. . . 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 3660	. . . . . 6 ⊢ (𝑧 = (𝑎 +_𝑒 𝑏) → {𝑧, 𝑅} = {(𝑎 +_𝑒 𝑏), 𝑅})
106	105	infeq1d 6989	. . . . 5 ⊢ (𝑧 = (𝑎 +_𝑒 𝑏) → inf({𝑧, 𝑅}, ℝ^, < ) = inf({(𝑎 +_𝑒 𝑏), 𝑅}, ℝ^, < ))
107		ge0xaddcl 9940	. . . . . 6 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → (𝑎 +_𝑒 𝑏) ∈ (0[,]+∞))
108	107	adantl 275	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎 +_𝑒 𝑏) ∈ (0[,]+∞))
109	71, 78	xaddcld 9841	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎 +_𝑒 𝑏) ∈ ℝ^*)
110		xrmincl 11229	. . . . . 6 ⊢ (((𝑎 +_𝑒 𝑏) ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({(𝑎 +_𝑒 𝑏), 𝑅}, ℝ^, < ) ∈ ℝ^*)
111	109, 72, 110	syl2anc 409	. . . . 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 5589	. . . 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 5871	. . . 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 4021	. . 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 13293	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ^*, < )) ∘ 𝐶) ∈ (∞Met‘𝑋))
116	18, 115	eqeltrrd 2248	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 973 = wceq 1348 ∈ wcel 2141 {cpr 3584 class class class wbr 3989 ↦ cmpt 4050 × cxp 4609 ∘ ccom 4615 Fn wfn 5193 ⟶wf 5194 ‘cfv 5198 (class class class)co 5853 ∈ cmpo 5855 infcinf 6960 0cc0 7774 +∞cpnf 7951 ℝ^*cxr 7953 < clt 7954 ≤ cle 7955 +_𝑒 cxad 9727 [,]cicc 9848 ∞Metcxmet 12774

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-map 6628 df-sup 6961 df-inf 6962 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-rp 9611 df-xneg 9729 df-xadd 9730 df-icc 9852 df-seqfrec 10402 df-exp 10476 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-xmet 12782

This theorem is referenced by: bdmet 13296 bdbl 13297 bdmopn 13298

W3C validator