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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 = (𝑥𝑋, 𝑦𝑋 ↦ 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.
StepHypRef Expression
1 simp1 949 . . . . 5 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → 𝐶 ∈ (∞Met‘𝑋))
2 xmetcl 12280 . . . . . . 7 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋𝑦𝑋) → (𝑥𝐶𝑦) ∈ ℝ*)
3 xmetge0 12293 . . . . . . 7 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋𝑦𝑋) → 0 ≤ (𝑥𝐶𝑦))
4 elxrge0 9602 . . . . . . 7 ((𝑥𝐶𝑦) ∈ (0[,]+∞) ↔ ((𝑥𝐶𝑦) ∈ ℝ* ∧ 0 ≤ (𝑥𝐶𝑦)))
52, 3, 4sylanbrc 411 . . . . . 6 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋𝑦𝑋) → (𝑥𝐶𝑦) ∈ (0[,]+∞))
653expb 1150 . . . . 5 ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐶𝑦) ∈ (0[,]+∞))
71, 6sylan 279 . . . 4 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐶𝑦) ∈ (0[,]+∞))
8 xmetf 12278 . . . . . . 7 (𝐶 ∈ (∞Met‘𝑋) → 𝐶:(𝑋 × 𝑋)⟶ℝ*)
983ad2ant1 970 . . . . . 6 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → 𝐶:(𝑋 × 𝑋)⟶ℝ*)
109ffnd 5209 . . . . 5 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → 𝐶 Fn (𝑋 × 𝑋))
11 fnovim 5811 . . . . 5 (𝐶 Fn (𝑋 × 𝑋) → 𝐶 = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐶𝑦)))
1210, 11syl 14 . . . 4 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → 𝐶 = (𝑥𝑋, 𝑦𝑋 ↦ (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})
1514infeq1d 6814 . . . 4 (𝑧 = (𝑥𝐶𝑦) → inf({𝑧, 𝑅}, ℝ*, < ) = inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ))
167, 12, 13, 15fmpoco 6043 . . 3 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ*, < )) ∘ 𝐶) = (𝑥𝑋, 𝑦𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < )))
17 stdbdmet.1 . . 3 𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ))
1816, 17syl6eqr 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))
2019simplbi 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*)
2320, 21, 22syl2anr 286 . . . 4 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ 𝑧 ∈ (0[,]+∞)) → inf({𝑧, 𝑅}, ℝ*, < ) ∈ ℝ*)
2423fmpttd 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})
2726infeq1d 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))
3029simplbi 270 . . . . . . 7 (𝑎 ∈ (0[,]+∞) → 𝑎 ∈ ℝ*)
31 xrmincl 10874 . . . . . . 7 ((𝑎 ∈ ℝ*𝑅 ∈ ℝ*) → inf({𝑎, 𝑅}, ℝ*, < ) ∈ ℝ*)
3230, 21, 31syl2anr 286 . . . . . 6 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → inf({𝑎, 𝑅}, ℝ*, < ) ∈ ℝ*)
3325, 27, 28, 32fvmptd3 5446 . . . . 5 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ*, < ))‘𝑎) = inf({𝑎, 𝑅}, ℝ*, < ))
3433eqeq1d 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*
3635a1i 9 . . . . . . . 8 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 0 ∈ ℝ*)
3730adantl 273 . . . . . . . 8 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 𝑎 ∈ ℝ*)
3821adantr 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)))
4036, 37, 38, 39syl3anc 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)
4241adantr 272 . . . . . . . 8 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → 0 < 𝑅)
4342biantrud 300 . . . . . . 7 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (0 < 𝑎 ↔ (0 < 𝑎 ∧ 0 < 𝑅)))
4440, 43bitr4d 190 . . . . . 6 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (0 < inf({𝑎, 𝑅}, ℝ*, < ) ↔ 0 < 𝑎))
4544notbid 633 . . . . 5 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (¬ 0 < inf({𝑎, 𝑅}, ℝ*, < ) ↔ ¬ 0 < 𝑎))
4628, 29sylib 121 . . . . . . . . . 10 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (𝑎 ∈ ℝ* ∧ 0 ≤ 𝑎))
4746simprd 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))
4935, 21, 48sylancr 408 . . . . . . . . . . 11 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → (0 < 𝑅 → 0 ≤ 𝑅))
5041, 49mpd 13 . . . . . . . . . 10 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → 0 ≤ 𝑅)
5150adantr 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)))
5336, 37, 38, 52syl3anc 1184 . . . . . . . . 9 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (0 ≤ inf({𝑎, 𝑅}, ℝ*, < ) ↔ (0 ≤ 𝑎 ∧ 0 ≤ 𝑅)))
5447, 51, 53mpbir2and 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))
5635, 32, 55sylancr 408 . . . . . . . 8 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (0 ≤ inf({𝑎, 𝑅}, ℝ*, < ) ↔ ¬ inf({𝑎, 𝑅}, ℝ*, < ) < 0))
5754, 56mpbid 146 . . . . . . 7 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → ¬ inf({𝑎, 𝑅}, ℝ*, < ) < 0)
5857biantrurd 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*, < ))))
6032, 36, 59syl2anc 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*, < ))))
6158, 60bitr4d 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))
6335, 37, 62sylancr 408 . . . . . . . 8 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (0 ≤ 𝑎 ↔ ¬ 𝑎 < 0))
6447, 63mpbid 146 . . . . . . 7 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → ¬ 𝑎 < 0)
6564biantrurd 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)))
6737, 36, 66syl2anc 406 . . . . . 6 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (𝑎 = 0 ↔ (¬ 𝑎 < 0 ∧ ¬ 0 < 𝑎)))
6865, 67bitr4d 190 . . . . 5 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (¬ 0 < 𝑎𝑎 = 0))
6945, 61, 683bitr3d 217 . . . 4 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (inf({𝑎, 𝑅}, ℝ*, < ) = 0 ↔ 𝑎 = 0))
7034, 69bitrd 187 . . 3 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ 𝑎 ∈ (0[,]+∞)) → (((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ*, < ))‘𝑎) = 0 ↔ 𝑎 = 0))
7130ad2antrl 477 . . . . . . . 8 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 𝑎 ∈ ℝ*)
7221adantr 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)
7471, 72, 73syl2anc 406 . . . . . . 7 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → inf({𝑎, 𝑅}, ℝ*, < ) ≤ 𝑎)
7571, 72, 31syl2anc 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))
7776simplbi 270 . . . . . . . . 9 (𝑏 ∈ (0[,]+∞) → 𝑏 ∈ ℝ*)
7877ad2antll 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*, < ) ≤ 𝑎𝑎𝑏) → inf({𝑎, 𝑅}, ℝ*, < ) ≤ 𝑏))
8075, 71, 78, 79syl3anc 1184 . . . . . . 7 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → ((inf({𝑎, 𝑅}, ℝ*, < ) ≤ 𝑎𝑎𝑏) → inf({𝑎, 𝑅}, ℝ*, < ) ≤ 𝑏))
8174, 80mpand 423 . . . . . 6 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎𝑏 → inf({𝑎, 𝑅}, ℝ*, < ) ≤ 𝑏))
82 xrmin2inf 10876 . . . . . . 7 ((𝑎 ∈ ℝ*𝑅 ∈ ℝ*) → inf({𝑎, 𝑅}, ℝ*, < ) ≤ 𝑅)
8371, 72, 82syl2anc 406 . . . . . 6 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → inf({𝑎, 𝑅}, ℝ*, < ) ≤ 𝑅)
8481, 83jctird 313 . . . . 5 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎𝑏 → (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)))
8675, 78, 72, 85syl3anc 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)))
8784, 86sylibrd 168 . . . 4 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎𝑏 → inf({𝑎, 𝑅}, ℝ*, < ) ≤ inf({𝑏, 𝑅}, ℝ*, < )))
8833adantrr 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})
9089infeq1d 6814 . . . . . 6 (𝑧 = 𝑏 → inf({𝑧, 𝑅}, ℝ*, < ) = inf({𝑏, 𝑅}, ℝ*, < ))
91 simpr 109 . . . . . . 7 ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → 𝑏 ∈ (0[,]+∞))
9291adantl 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*)
9478, 72, 93syl2anc 406 . . . . . 6 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → inf({𝑏, 𝑅}, ℝ*, < ) ∈ ℝ*)
9525, 90, 92, 94fvmptd3 5446 . . . . 5 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ*, < ))‘𝑏) = inf({𝑏, 𝑅}, ℝ*, < ))
9688, 95breq12d 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*, < )))
9787, 96sylibrd 168 . . 3 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎𝑏 → ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ*, < ))‘𝑎) ≤ ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ*, < ))‘𝑏)))
9829simprbi 271 . . . . . 6 (𝑎 ∈ (0[,]+∞) → 0 ≤ 𝑎)
9998ad2antrl 477 . . . . 5 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 0 ≤ 𝑎)
10076simprbi 271 . . . . . 6 (𝑏 ∈ (0[,]+∞) → 0 ≤ 𝑏)
101100ad2antll 478 . . . . 5 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → 0 ≤ 𝑏)
10241adantr 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*, < )))
10471, 99, 78, 101, 72, 102, 103syl222anc 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})
106105infeq1d 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[,]+∞))
108107adantl 273 . . . . 5 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → (𝑎 +𝑒 𝑏) ∈ (0[,]+∞))
10971, 78xaddcld 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*)
111109, 72, 110syl2anc 406 . . . . 5 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞))) → inf({(𝑎 +𝑒 𝑏), 𝑅}, ℝ*, < ) ∈ ℝ*)
11225, 106, 108, 111fvmptd3 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*, < ))
11388, 95oveq12d 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*, < )))
114104, 112, 1133brtr4d 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)))
1151, 24, 70, 97, 114comet 12427 . 2 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → ((𝑧 ∈ (0[,]+∞) ↦ inf({𝑧, 𝑅}, ℝ*, < )) ∘ 𝐶) ∈ (∞Met‘𝑋))
11618, 115eqeltrrd 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
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