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Theorem xblss2ps 22428
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 22431 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 22414 . . . . . 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 22334 . . . . . . . 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 10302 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑄) ∈ ℝ*)
163adantr 472 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑅 ∈ ℝ*)
1715, 16xaddcld 12345 . . . . . . 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 22334 . . . . . . . . . 10 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑥𝑋) → (𝑃𝐷𝑥) ∈ ℝ*)
237, 21, 6, 22syl3anc 1477 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑥) ∈ ℝ*)
2415, 23xaddcld 12345 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)) ∈ ℝ*)
25 psmettri2 22336 . . . . . . . . 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 12301 . . . . . . . . . 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 12205 . . . . . . 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 12344 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → -𝑒𝑅 ∈ ℝ*)
3533, 34xaddcld 12345 . . . . . . . . 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 12297 . . . . . . . . 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 12296 . . . . . . . 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 4831 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → ((𝑃𝐷𝑄) +𝑒 𝑅) ≤ 𝑆)
4412, 18, 20, 32, 43xrltletrd 12206 . . . . 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 6831 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑃(ball‘𝐷)𝑅) = (𝑃(ball‘𝐷)+∞))
5148, 50eleqtrd 2842 . . . . . . . . . 10 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑥 ∈ (𝑃(ball‘𝐷)+∞))
52 xblpnfps 22422 . . . . . . . . . . . 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 10282 . . . . . . . 8 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑃𝐷𝑄) + (𝑃𝐷𝑥)) ∈ ℝ)
5756rexrd 10302 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑃𝐷𝑄) + (𝑃𝐷𝑥)) ∈ ℝ*)
58 pnfxr 10305 . . . . . . . 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 12277 . . . . . . . . 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 4831 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑄𝐷𝑥) ≤ ((𝑃𝐷𝑄) + (𝑃𝐷𝑥)))
68 ltpnf 12168 . . . . . . . 8 (((𝑃𝐷𝑄) + (𝑃𝐷𝑥)) ∈ ℝ → ((𝑃𝐷𝑄) + (𝑃𝐷𝑥)) < +∞)
6956, 68syl 17 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑃𝐷𝑄) + (𝑃𝐷𝑥)) < +∞)
7045, 57, 59, 67, 69xrlelttrd 12205 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑄𝐷𝑥) < +∞)
71 0xr 10299 . . . . . . . . . . 11 0 ∈ ℝ*
7271a1i 11 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ∈ ℝ*)
73 psmetge0 22339 . . . . . . . . . . 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 12207 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ (𝑆 +𝑒 -𝑒𝑅))
76 ge0nemnf 12218 . . . . . . . . 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 12273 . . . . . . . . . . 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 12252 . . . . . . . . . . . . 13 (𝑅 = +∞ → -𝑒𝑅 = -𝑒+∞)
8449, 83syl 17 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → -𝑒𝑅 = -𝑒+∞)
85 xnegpnf 12254 . . . . . . . . . . . 12 -𝑒+∞ = -∞
8684, 85syl6eq 2811 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → -𝑒𝑅 = -∞)
8786oveq2d 6831 . . . . . . . . . 10 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 +𝑒 -𝑒𝑅) = (𝑆 +𝑒 -∞))
8887eqeq1d 2763 . . . . . . . . 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 2955 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑆 +𝑒 -𝑒𝑅) ≠ -∞ → 𝑆 = +∞))
9178, 90mpd 15 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑆 = +∞)
9270, 91breqtrrd 4833 . . . . 5 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑄𝐷𝑥) < 𝑆)
93 psmetge0 22339 . . . . . . . . . . 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 12205 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 < 𝑅)
96 xrltle 12196 . . . . . . . . . 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 12218 . . . . . . . 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 12165 . . . . . 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 22414 . . . . 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 3751 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 2140  wne 2933  wss 3716   class class class wbr 4805  cfv 6050  (class class class)co 6815  cr 10148  0cc0 10149   + caddc 10152  +∞cpnf 10284  -∞cmnf 10285  *cxr 10286   < clt 10287  cle 10288  -𝑒cxne 12157   +𝑒 cxad 12158  PsMetcpsmet 19953  ballcbl 19956
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 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-cnex 10205  ax-resscn 10206  ax-1cn 10207  ax-icn 10208  ax-addcl 10209  ax-addrcl 10210  ax-mulcl 10211  ax-mulrcl 10212  ax-mulcom 10213  ax-addass 10214  ax-mulass 10215  ax-distr 10216  ax-i2m1 10217  ax-1ne0 10218  ax-1rid 10219  ax-rnegex 10220  ax-rrecex 10221  ax-cnre 10222  ax-pre-lttri 10223  ax-pre-lttrn 10224  ax-pre-ltadd 10225  ax-pre-mulgt0 10226
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 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-po 5188  df-so 5189  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-1st 7335  df-2nd 7336  df-er 7914  df-map 8028  df-en 8125  df-dom 8126  df-sdom 8127  df-pnf 10289  df-mnf 10290  df-xr 10291  df-ltxr 10292  df-le 10293  df-sub 10481  df-neg 10482  df-div 10898  df-2 11292  df-rp 12047  df-xneg 12160  df-xadd 12161  df-xmul 12162  df-psmet 19961  df-bl 19964
This theorem is referenced by:  blss2ps  22430  ssblps  22449
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