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Theorem xblss2ps 22407
 Description: One ball is contained in another if the center-to-center distance is less than the difference of the radii. In this version of blss2 22410 for extended metrics, we have to assume the balls are a finite distance apart, or else 𝑃 will not even be in the infinity ball around 𝑄. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
Hypotheses
Ref Expression
xblss2ps.1 (𝜑𝐷 ∈ (PsMet‘𝑋))
xblss2ps.2 (𝜑𝑃𝑋)
xblss2ps.3 (𝜑𝑄𝑋)
xblss2ps.4 (𝜑𝑅 ∈ ℝ*)
xblss2ps.5 (𝜑𝑆 ∈ ℝ*)
xblss2ps.6 (𝜑 → (𝑃𝐷𝑄) ∈ ℝ)
xblss2ps.7 (𝜑 → (𝑃𝐷𝑄) ≤ (𝑆 +𝑒 -𝑒𝑅))
Assertion
Ref Expression
xblss2ps (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆))

Proof of Theorem xblss2ps
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 xblss2ps.1 . . . . . 6 (𝜑𝐷 ∈ (PsMet‘𝑋))
2 xblss2ps.2 . . . . . 6 (𝜑𝑃𝑋)
3 xblss2ps.4 . . . . . 6 (𝜑𝑅 ∈ ℝ*)
4 elblps 22393 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
51, 2, 3, 4syl3anc 1477 . . . . 5 (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
65simprbda 654 . . . 4 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥𝑋)
71adantr 472 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝐷 ∈ (PsMet‘𝑋))
8 xblss2ps.3 . . . . . . . . 9 (𝜑𝑄𝑋)
98adantr 472 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑄𝑋)
10 psmetcl 22313 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑄𝑋𝑥𝑋) → (𝑄𝐷𝑥) ∈ ℝ*)
117, 9, 6, 10syl3anc 1477 . . . . . . 7 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑄𝐷𝑥) ∈ ℝ*)
1211adantr 472 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → (𝑄𝐷𝑥) ∈ ℝ*)
13 xblss2ps.6 . . . . . . . . . 10 (𝜑 → (𝑃𝐷𝑄) ∈ ℝ)
1413adantr 472 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑄) ∈ ℝ)
1514rexrd 10281 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑄) ∈ ℝ*)
163adantr 472 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑅 ∈ ℝ*)
1715, 16xaddcld 12324 . . . . . . 7 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) +𝑒 𝑅) ∈ ℝ*)
1817adantr 472 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → ((𝑃𝐷𝑄) +𝑒 𝑅) ∈ ℝ*)
19 xblss2ps.5 . . . . . . 7 (𝜑𝑆 ∈ ℝ*)
2019ad2antrr 764 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → 𝑆 ∈ ℝ*)
212adantr 472 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑃𝑋)
22 psmetcl 22313 . . . . . . . . . 10 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑥𝑋) → (𝑃𝐷𝑥) ∈ ℝ*)
237, 21, 6, 22syl3anc 1477 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑥) ∈ ℝ*)
2415, 23xaddcld 12324 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)) ∈ ℝ*)
25 psmettri2 22315 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑃𝑋𝑄𝑋𝑥𝑋)) → (𝑄𝐷𝑥) ≤ ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)))
267, 21, 9, 6, 25syl13anc 1479 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑄𝐷𝑥) ≤ ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)))
275simplbda 655 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑥) < 𝑅)
28 xltadd2 12280 . . . . . . . . . 10 (((𝑃𝐷𝑥) ∈ ℝ*𝑅 ∈ ℝ* ∧ (𝑃𝐷𝑄) ∈ ℝ) → ((𝑃𝐷𝑥) < 𝑅 ↔ ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)) < ((𝑃𝐷𝑄) +𝑒 𝑅)))
2923, 16, 14, 28syl3anc 1477 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑥) < 𝑅 ↔ ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)) < ((𝑃𝐷𝑄) +𝑒 𝑅)))
3027, 29mpbid 222 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)) < ((𝑃𝐷𝑄) +𝑒 𝑅))
3111, 24, 17, 26, 30xrlelttrd 12184 . . . . . . 7 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑄𝐷𝑥) < ((𝑃𝐷𝑄) +𝑒 𝑅))
3231adantr 472 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → (𝑄𝐷𝑥) < ((𝑃𝐷𝑄) +𝑒 𝑅))
3319adantr 472 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑆 ∈ ℝ*)
3416xnegcld 12323 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → -𝑒𝑅 ∈ ℝ*)
3533, 34xaddcld 12324 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑆 +𝑒 -𝑒𝑅) ∈ ℝ*)
36 xblss2ps.7 . . . . . . . . . 10 (𝜑 → (𝑃𝐷𝑄) ≤ (𝑆 +𝑒 -𝑒𝑅))
3736adantr 472 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑄) ≤ (𝑆 +𝑒 -𝑒𝑅))
38 xleadd1a 12276 . . . . . . . . 9 ((((𝑃𝐷𝑄) ∈ ℝ* ∧ (𝑆 +𝑒 -𝑒𝑅) ∈ ℝ*𝑅 ∈ ℝ*) ∧ (𝑃𝐷𝑄) ≤ (𝑆 +𝑒 -𝑒𝑅)) → ((𝑃𝐷𝑄) +𝑒 𝑅) ≤ ((𝑆 +𝑒 -𝑒𝑅) +𝑒 𝑅))
3915, 35, 16, 37, 38syl31anc 1480 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) +𝑒 𝑅) ≤ ((𝑆 +𝑒 -𝑒𝑅) +𝑒 𝑅))
4039adantr 472 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → ((𝑃𝐷𝑄) +𝑒 𝑅) ≤ ((𝑆 +𝑒 -𝑒𝑅) +𝑒 𝑅))
41 xnpcan 12275 . . . . . . . 8 ((𝑆 ∈ ℝ*𝑅 ∈ ℝ) → ((𝑆 +𝑒 -𝑒𝑅) +𝑒 𝑅) = 𝑆)
4233, 41sylan 489 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → ((𝑆 +𝑒 -𝑒𝑅) +𝑒 𝑅) = 𝑆)
4340, 42breqtrd 4830 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → ((𝑃𝐷𝑄) +𝑒 𝑅) ≤ 𝑆)
4412, 18, 20, 32, 43xrltletrd 12185 . . . . 5 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → (𝑄𝐷𝑥) < 𝑆)
4511adantr 472 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑄𝐷𝑥) ∈ ℝ*)
4613ad2antrr 764 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑃𝐷𝑄) ∈ ℝ)
47 simpll 807 . . . . . . . . . 10 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝜑)
48 simplr 809 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑥 ∈ (𝑃(ball‘𝐷)𝑅))
49 simpr 479 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑅 = +∞)
5049oveq2d 6829 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑃(ball‘𝐷)𝑅) = (𝑃(ball‘𝐷)+∞))
5148, 50eleqtrd 2841 . . . . . . . . . 10 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑥 ∈ (𝑃(ball‘𝐷)+∞))
52 xblpnfps 22401 . . . . . . . . . . . 12 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋) → (𝑥 ∈ (𝑃(ball‘𝐷)+∞) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) ∈ ℝ)))
531, 2, 52syl2anc 696 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)+∞) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) ∈ ℝ)))
5453simplbda 655 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)+∞)) → (𝑃𝐷𝑥) ∈ ℝ)
5547, 51, 54syl2anc 696 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑃𝐷𝑥) ∈ ℝ)
5646, 55readdcld 10261 . . . . . . . 8 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑃𝐷𝑄) + (𝑃𝐷𝑥)) ∈ ℝ)
5756rexrd 10281 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑃𝐷𝑄) + (𝑃𝐷𝑥)) ∈ ℝ*)
58 pnfxr 10284 . . . . . . . 8 +∞ ∈ ℝ*
5958a1i 11 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → +∞ ∈ ℝ*)
601ad2antrr 764 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝐷 ∈ (PsMet‘𝑋))
612ad2antrr 764 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑃𝑋)
628ad2antrr 764 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑄𝑋)
636adantr 472 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑥𝑋)
6460, 61, 62, 63, 25syl13anc 1479 . . . . . . . 8 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑄𝐷𝑥) ≤ ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)))
65 rexadd 12256 . . . . . . . . 9 (((𝑃𝐷𝑄) ∈ ℝ ∧ (𝑃𝐷𝑥) ∈ ℝ) → ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)) = ((𝑃𝐷𝑄) + (𝑃𝐷𝑥)))
6646, 55, 65syl2anc 696 . . . . . . . 8 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)) = ((𝑃𝐷𝑄) + (𝑃𝐷𝑥)))
6764, 66breqtrd 4830 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑄𝐷𝑥) ≤ ((𝑃𝐷𝑄) + (𝑃𝐷𝑥)))
68 ltpnf 12147 . . . . . . . 8 (((𝑃𝐷𝑄) + (𝑃𝐷𝑥)) ∈ ℝ → ((𝑃𝐷𝑄) + (𝑃𝐷𝑥)) < +∞)
6956, 68syl 17 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑃𝐷𝑄) + (𝑃𝐷𝑥)) < +∞)
7045, 57, 59, 67, 69xrlelttrd 12184 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑄𝐷𝑥) < +∞)
71 0xr 10278 . . . . . . . . . . 11 0 ∈ ℝ*
7271a1i 11 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ∈ ℝ*)
73 psmetge0 22318 . . . . . . . . . . 11 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑄𝑋) → 0 ≤ (𝑃𝐷𝑄))
747, 21, 9, 73syl3anc 1477 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ (𝑃𝐷𝑄))
7572, 15, 35, 74, 37xrletrd 12186 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ (𝑆 +𝑒 -𝑒𝑅))
76 ge0nemnf 12197 . . . . . . . . 9 (((𝑆 +𝑒 -𝑒𝑅) ∈ ℝ* ∧ 0 ≤ (𝑆 +𝑒 -𝑒𝑅)) → (𝑆 +𝑒 -𝑒𝑅) ≠ -∞)
7735, 75, 76syl2anc 696 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑆 +𝑒 -𝑒𝑅) ≠ -∞)
7877adantr 472 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 +𝑒 -𝑒𝑅) ≠ -∞)
7919ad2antrr 764 . . . . . . . . . 10 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑆 ∈ ℝ*)
80 xaddmnf1 12252 . . . . . . . . . . 11 ((𝑆 ∈ ℝ*𝑆 ≠ +∞) → (𝑆 +𝑒 -∞) = -∞)
8180ex 449 . . . . . . . . . 10 (𝑆 ∈ ℝ* → (𝑆 ≠ +∞ → (𝑆 +𝑒 -∞) = -∞))
8279, 81syl 17 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 ≠ +∞ → (𝑆 +𝑒 -∞) = -∞))
83 xnegeq 12231 . . . . . . . . . . . . 13 (𝑅 = +∞ → -𝑒𝑅 = -𝑒+∞)
8449, 83syl 17 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → -𝑒𝑅 = -𝑒+∞)
85 xnegpnf 12233 . . . . . . . . . . . 12 -𝑒+∞ = -∞
8684, 85syl6eq 2810 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → -𝑒𝑅 = -∞)
8786oveq2d 6829 . . . . . . . . . 10 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 +𝑒 -𝑒𝑅) = (𝑆 +𝑒 -∞))
8887eqeq1d 2762 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑆 +𝑒 -𝑒𝑅) = -∞ ↔ (𝑆 +𝑒 -∞) = -∞))
8982, 88sylibrd 249 . . . . . . . 8 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 ≠ +∞ → (𝑆 +𝑒 -𝑒𝑅) = -∞))
9089necon1d 2954 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑆 +𝑒 -𝑒𝑅) ≠ -∞ → 𝑆 = +∞))
9178, 90mpd 15 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑆 = +∞)
9270, 91breqtrrd 4832 . . . . 5 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑄𝐷𝑥) < 𝑆)
93 psmetge0 22318 . . . . . . . . . . 11 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑥𝑋) → 0 ≤ (𝑃𝐷𝑥))
947, 21, 6, 93syl3anc 1477 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ (𝑃𝐷𝑥))
9572, 23, 16, 94, 27xrlelttrd 12184 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 < 𝑅)
96 xrltle 12175 . . . . . . . . . 10 ((0 ∈ ℝ*𝑅 ∈ ℝ*) → (0 < 𝑅 → 0 ≤ 𝑅))
9771, 16, 96sylancr 698 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (0 < 𝑅 → 0 ≤ 𝑅))
9895, 97mpd 15 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ 𝑅)
99 ge0nemnf 12197 . . . . . . . 8 ((𝑅 ∈ ℝ* ∧ 0 ≤ 𝑅) → 𝑅 ≠ -∞)
10016, 98, 99syl2anc 696 . . . . . . 7 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑅 ≠ -∞)
10116, 100jca 555 . . . . . 6 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑅 ∈ ℝ*𝑅 ≠ -∞))
102 xrnemnf 12144 . . . . . 6 ((𝑅 ∈ ℝ*𝑅 ≠ -∞) ↔ (𝑅 ∈ ℝ ∨ 𝑅 = +∞))
103101, 102sylib 208 . . . . 5 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑅 ∈ ℝ ∨ 𝑅 = +∞))
10444, 92, 103mpjaodan 862 . . . 4 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑄𝐷𝑥) < 𝑆)
105 elblps 22393 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑄𝑋𝑆 ∈ ℝ*) → (𝑥 ∈ (𝑄(ball‘𝐷)𝑆) ↔ (𝑥𝑋 ∧ (𝑄𝐷𝑥) < 𝑆)))
1067, 9, 33, 105syl3anc 1477 . . . 4 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥 ∈ (𝑄(ball‘𝐷)𝑆) ↔ (𝑥𝑋 ∧ (𝑄𝐷𝑥) < 𝑆)))
1076, 104, 106mpbir2and 995 . . 3 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ (𝑄(ball‘𝐷)𝑆))
108107ex 449 . 2 (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) → 𝑥 ∈ (𝑄(ball‘𝐷)𝑆)))
109108ssrdv 3750 1 (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   = wceq 1632   ∈ wcel 2139   ≠ wne 2932   ⊆ wss 3715   class class class wbr 4804  ‘cfv 6049  (class class class)co 6813  ℝcr 10127  0cc0 10128   + caddc 10131  +∞cpnf 10263  -∞cmnf 10264  ℝ*cxr 10265   < clt 10266   ≤ cle 10267  -𝑒cxne 12136   +𝑒 cxad 12137  PsMetcpsmet 19932  ballcbl 19935 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-2 11271  df-rp 12026  df-xneg 12139  df-xadd 12140  df-xmul 12141  df-psmet 19940  df-bl 19943 This theorem is referenced by:  blss2ps  22409  ssblps  22428
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