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Theorem xblss2ps 23898
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 23901 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 23884 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
51, 2, 3, 4syl3anc 1371 . . . . 5 (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
65simprbda 499 . . . 4 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥𝑋)
71adantr 481 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝐷 ∈ (PsMet‘𝑋))
8 xblss2ps.3 . . . . . . . . 9 (𝜑𝑄𝑋)
98adantr 481 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑄𝑋)
10 psmetcl 23804 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑄𝑋𝑥𝑋) → (𝑄𝐷𝑥) ∈ ℝ*)
117, 9, 6, 10syl3anc 1371 . . . . . . 7 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑄𝐷𝑥) ∈ ℝ*)
1211adantr 481 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → (𝑄𝐷𝑥) ∈ ℝ*)
13 xblss2ps.6 . . . . . . . . . 10 (𝜑 → (𝑃𝐷𝑄) ∈ ℝ)
1413adantr 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑄) ∈ ℝ)
1514rexrd 11260 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑄) ∈ ℝ*)
163adantr 481 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑅 ∈ ℝ*)
1715, 16xaddcld 13276 . . . . . . 7 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) +𝑒 𝑅) ∈ ℝ*)
1817adantr 481 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → ((𝑃𝐷𝑄) +𝑒 𝑅) ∈ ℝ*)
19 xblss2ps.5 . . . . . . 7 (𝜑𝑆 ∈ ℝ*)
2019ad2antrr 724 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → 𝑆 ∈ ℝ*)
212adantr 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑃𝑋)
22 psmetcl 23804 . . . . . . . . . 10 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑥𝑋) → (𝑃𝐷𝑥) ∈ ℝ*)
237, 21, 6, 22syl3anc 1371 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑥) ∈ ℝ*)
2415, 23xaddcld 13276 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)) ∈ ℝ*)
25 psmettri2 23806 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑃𝑋𝑄𝑋𝑥𝑋)) → (𝑄𝐷𝑥) ≤ ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)))
267, 21, 9, 6, 25syl13anc 1372 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑄𝐷𝑥) ≤ ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)))
275simplbda 500 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑥) < 𝑅)
28 xltadd2 13232 . . . . . . . . . 10 (((𝑃𝐷𝑥) ∈ ℝ*𝑅 ∈ ℝ* ∧ (𝑃𝐷𝑄) ∈ ℝ) → ((𝑃𝐷𝑥) < 𝑅 ↔ ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)) < ((𝑃𝐷𝑄) +𝑒 𝑅)))
2923, 16, 14, 28syl3anc 1371 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑥) < 𝑅 ↔ ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)) < ((𝑃𝐷𝑄) +𝑒 𝑅)))
3027, 29mpbid 231 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)) < ((𝑃𝐷𝑄) +𝑒 𝑅))
3111, 24, 17, 26, 30xrlelttrd 13135 . . . . . . 7 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑄𝐷𝑥) < ((𝑃𝐷𝑄) +𝑒 𝑅))
3231adantr 481 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → (𝑄𝐷𝑥) < ((𝑃𝐷𝑄) +𝑒 𝑅))
3319adantr 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑆 ∈ ℝ*)
3416xnegcld 13275 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → -𝑒𝑅 ∈ ℝ*)
3533, 34xaddcld 13276 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑆 +𝑒 -𝑒𝑅) ∈ ℝ*)
36 xblss2ps.7 . . . . . . . . . 10 (𝜑 → (𝑃𝐷𝑄) ≤ (𝑆 +𝑒 -𝑒𝑅))
3736adantr 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑄) ≤ (𝑆 +𝑒 -𝑒𝑅))
38 xleadd1a 13228 . . . . . . . . 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 1373 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) +𝑒 𝑅) ≤ ((𝑆 +𝑒 -𝑒𝑅) +𝑒 𝑅))
4039adantr 481 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → ((𝑃𝐷𝑄) +𝑒 𝑅) ≤ ((𝑆 +𝑒 -𝑒𝑅) +𝑒 𝑅))
41 xnpcan 13227 . . . . . . . 8 ((𝑆 ∈ ℝ*𝑅 ∈ ℝ) → ((𝑆 +𝑒 -𝑒𝑅) +𝑒 𝑅) = 𝑆)
4233, 41sylan 580 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → ((𝑆 +𝑒 -𝑒𝑅) +𝑒 𝑅) = 𝑆)
4340, 42breqtrd 5173 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → ((𝑃𝐷𝑄) +𝑒 𝑅) ≤ 𝑆)
4412, 18, 20, 32, 43xrltletrd 13136 . . . . 5 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → (𝑄𝐷𝑥) < 𝑆)
4511adantr 481 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑄𝐷𝑥) ∈ ℝ*)
4613ad2antrr 724 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑃𝐷𝑄) ∈ ℝ)
47 simpll 765 . . . . . . . . . 10 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝜑)
48 simplr 767 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑥 ∈ (𝑃(ball‘𝐷)𝑅))
49 simpr 485 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑅 = +∞)
5049oveq2d 7421 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑃(ball‘𝐷)𝑅) = (𝑃(ball‘𝐷)+∞))
5148, 50eleqtrd 2835 . . . . . . . . . 10 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑥 ∈ (𝑃(ball‘𝐷)+∞))
52 xblpnfps 23892 . . . . . . . . . . . 12 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋) → (𝑥 ∈ (𝑃(ball‘𝐷)+∞) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) ∈ ℝ)))
531, 2, 52syl2anc 584 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)+∞) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) ∈ ℝ)))
5453simplbda 500 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)+∞)) → (𝑃𝐷𝑥) ∈ ℝ)
5547, 51, 54syl2anc 584 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑃𝐷𝑥) ∈ ℝ)
5646, 55readdcld 11239 . . . . . . . 8 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑃𝐷𝑄) + (𝑃𝐷𝑥)) ∈ ℝ)
5756rexrd 11260 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑃𝐷𝑄) + (𝑃𝐷𝑥)) ∈ ℝ*)
58 pnfxr 11264 . . . . . . . 8 +∞ ∈ ℝ*
5958a1i 11 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → +∞ ∈ ℝ*)
601ad2antrr 724 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝐷 ∈ (PsMet‘𝑋))
612ad2antrr 724 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑃𝑋)
628ad2antrr 724 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑄𝑋)
636adantr 481 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑥𝑋)
6460, 61, 62, 63, 25syl13anc 1372 . . . . . . . 8 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑄𝐷𝑥) ≤ ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)))
6546, 55rexaddd 13209 . . . . . . . 8 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)) = ((𝑃𝐷𝑄) + (𝑃𝐷𝑥)))
6664, 65breqtrd 5173 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑄𝐷𝑥) ≤ ((𝑃𝐷𝑄) + (𝑃𝐷𝑥)))
6756ltpnfd 13097 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑃𝐷𝑄) + (𝑃𝐷𝑥)) < +∞)
6845, 57, 59, 66, 67xrlelttrd 13135 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑄𝐷𝑥) < +∞)
69 0xr 11257 . . . . . . . . . . 11 0 ∈ ℝ*
7069a1i 11 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ∈ ℝ*)
71 psmetge0 23809 . . . . . . . . . . 11 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑄𝑋) → 0 ≤ (𝑃𝐷𝑄))
727, 21, 9, 71syl3anc 1371 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ (𝑃𝐷𝑄))
7370, 15, 35, 72, 37xrletrd 13137 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ (𝑆 +𝑒 -𝑒𝑅))
74 ge0nemnf 13148 . . . . . . . . 9 (((𝑆 +𝑒 -𝑒𝑅) ∈ ℝ* ∧ 0 ≤ (𝑆 +𝑒 -𝑒𝑅)) → (𝑆 +𝑒 -𝑒𝑅) ≠ -∞)
7535, 73, 74syl2anc 584 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑆 +𝑒 -𝑒𝑅) ≠ -∞)
7675adantr 481 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 +𝑒 -𝑒𝑅) ≠ -∞)
7719ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑆 ∈ ℝ*)
78 xaddmnf1 13203 . . . . . . . . . . 11 ((𝑆 ∈ ℝ*𝑆 ≠ +∞) → (𝑆 +𝑒 -∞) = -∞)
7978ex 413 . . . . . . . . . 10 (𝑆 ∈ ℝ* → (𝑆 ≠ +∞ → (𝑆 +𝑒 -∞) = -∞))
8077, 79syl 17 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 ≠ +∞ → (𝑆 +𝑒 -∞) = -∞))
81 xnegeq 13182 . . . . . . . . . . . . 13 (𝑅 = +∞ → -𝑒𝑅 = -𝑒+∞)
8249, 81syl 17 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → -𝑒𝑅 = -𝑒+∞)
83 xnegpnf 13184 . . . . . . . . . . . 12 -𝑒+∞ = -∞
8482, 83eqtrdi 2788 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → -𝑒𝑅 = -∞)
8584oveq2d 7421 . . . . . . . . . 10 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 +𝑒 -𝑒𝑅) = (𝑆 +𝑒 -∞))
8685eqeq1d 2734 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑆 +𝑒 -𝑒𝑅) = -∞ ↔ (𝑆 +𝑒 -∞) = -∞))
8780, 86sylibrd 258 . . . . . . . 8 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 ≠ +∞ → (𝑆 +𝑒 -𝑒𝑅) = -∞))
8887necon1d 2962 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑆 +𝑒 -𝑒𝑅) ≠ -∞ → 𝑆 = +∞))
8976, 88mpd 15 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑆 = +∞)
9068, 89breqtrrd 5175 . . . . 5 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑄𝐷𝑥) < 𝑆)
91 psmetge0 23809 . . . . . . . . . . 11 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑥𝑋) → 0 ≤ (𝑃𝐷𝑥))
927, 21, 6, 91syl3anc 1371 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ (𝑃𝐷𝑥))
9370, 23, 16, 92, 27xrlelttrd 13135 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 < 𝑅)
9470, 16, 93xrltled 13125 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ 𝑅)
95 ge0nemnf 13148 . . . . . . . 8 ((𝑅 ∈ ℝ* ∧ 0 ≤ 𝑅) → 𝑅 ≠ -∞)
9616, 94, 95syl2anc 584 . . . . . . 7 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑅 ≠ -∞)
9716, 96jca 512 . . . . . 6 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑅 ∈ ℝ*𝑅 ≠ -∞))
98 xrnemnf 13093 . . . . . 6 ((𝑅 ∈ ℝ*𝑅 ≠ -∞) ↔ (𝑅 ∈ ℝ ∨ 𝑅 = +∞))
9997, 98sylib 217 . . . . 5 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑅 ∈ ℝ ∨ 𝑅 = +∞))
10044, 90, 99mpjaodan 957 . . . 4 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑄𝐷𝑥) < 𝑆)
101 elblps 23884 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑄𝑋𝑆 ∈ ℝ*) → (𝑥 ∈ (𝑄(ball‘𝐷)𝑆) ↔ (𝑥𝑋 ∧ (𝑄𝐷𝑥) < 𝑆)))
1027, 9, 33, 101syl3anc 1371 . . . 4 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥 ∈ (𝑄(ball‘𝐷)𝑆) ↔ (𝑥𝑋 ∧ (𝑄𝐷𝑥) < 𝑆)))
1036, 100, 102mpbir2and 711 . . 3 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ (𝑄(ball‘𝐷)𝑆))
104103ex 413 . 2 (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) → 𝑥 ∈ (𝑄(ball‘𝐷)𝑆)))
105104ssrdv 3987 1 (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  wne 2940  wss 3947   class class class wbr 5147  cfv 6540  (class class class)co 7405  cr 11105  0cc0 11106   + caddc 11109  +∞cpnf 11241  -∞cmnf 11242  *cxr 11243   < clt 11244  cle 11245  -𝑒cxne 13085   +𝑒 cxad 13086  PsMetcpsmet 20920  ballcbl 20923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-po 5587  df-so 5588  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-2 12271  df-rp 12971  df-xneg 13088  df-xadd 13089  df-xmul 13090  df-psmet 20928  df-bl 20931
This theorem is referenced by:  blss2ps  23900  ssblps  23919
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