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Theorem xblss2 12613
 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 12615 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.)
Hypotheses
Ref Expression
xblss2.1 (𝜑𝐷 ∈ (∞Met‘𝑋))
xblss2.2 (𝜑𝑃𝑋)
xblss2.3 (𝜑𝑄𝑋)
xblss2.4 (𝜑𝑅 ∈ ℝ*)
xblss2.5 (𝜑𝑆 ∈ ℝ*)
xblss2.6 (𝜑 → (𝑃𝐷𝑄) ∈ ℝ)
xblss2.7 (𝜑 → (𝑃𝐷𝑄) ≤ (𝑆 +𝑒 -𝑒𝑅))
Assertion
Ref Expression
xblss2 (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆))

Proof of Theorem xblss2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 xblss2.1 . . . . . 6 (𝜑𝐷 ∈ (∞Met‘𝑋))
2 xblss2.2 . . . . . 6 (𝜑𝑃𝑋)
3 xblss2.4 . . . . . 6 (𝜑𝑅 ∈ ℝ*)
4 elbl 12599 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
51, 2, 3, 4syl3anc 1217 . . . . 5 (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
65simprbda 381 . . . 4 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥𝑋)
71adantr 274 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝐷 ∈ (∞Met‘𝑋))
8 xblss2.3 . . . . . . . . 9 (𝜑𝑄𝑋)
98adantr 274 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑄𝑋)
10 xmetcl 12560 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑄𝑋𝑥𝑋) → (𝑄𝐷𝑥) ∈ ℝ*)
117, 9, 6, 10syl3anc 1217 . . . . . . 7 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑄𝐷𝑥) ∈ ℝ*)
1211adantr 274 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → (𝑄𝐷𝑥) ∈ ℝ*)
13 xblss2.6 . . . . . . . . . 10 (𝜑 → (𝑃𝐷𝑄) ∈ ℝ)
1413adantr 274 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑄) ∈ ℝ)
1514rexrd 7839 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑄) ∈ ℝ*)
163adantr 274 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑅 ∈ ℝ*)
1715, 16xaddcld 9697 . . . . . . 7 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) +𝑒 𝑅) ∈ ℝ*)
1817adantr 274 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → ((𝑃𝐷𝑄) +𝑒 𝑅) ∈ ℝ*)
19 xblss2.5 . . . . . . 7 (𝜑𝑆 ∈ ℝ*)
2019ad2antrr 480 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → 𝑆 ∈ ℝ*)
212adantr 274 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑃𝑋)
22 xmetcl 12560 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑥𝑋) → (𝑃𝐷𝑥) ∈ ℝ*)
237, 21, 6, 22syl3anc 1217 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑥) ∈ ℝ*)
2415, 23xaddcld 9697 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)) ∈ ℝ*)
25 xmettri2 12569 . . . . . . . . 9 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑃𝑋𝑄𝑋𝑥𝑋)) → (𝑄𝐷𝑥) ≤ ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)))
267, 21, 9, 6, 25syl13anc 1219 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑄𝐷𝑥) ≤ ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)))
275simplbda 382 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑥) < 𝑅)
28 xltadd2 9690 . . . . . . . . . 10 (((𝑃𝐷𝑥) ∈ ℝ*𝑅 ∈ ℝ* ∧ (𝑃𝐷𝑄) ∈ ℝ) → ((𝑃𝐷𝑥) < 𝑅 ↔ ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)) < ((𝑃𝐷𝑄) +𝑒 𝑅)))
2923, 16, 14, 28syl3anc 1217 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑥) < 𝑅 ↔ ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)) < ((𝑃𝐷𝑄) +𝑒 𝑅)))
3027, 29mpbid 146 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)) < ((𝑃𝐷𝑄) +𝑒 𝑅))
3111, 24, 17, 26, 30xrlelttrd 9623 . . . . . . 7 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑄𝐷𝑥) < ((𝑃𝐷𝑄) +𝑒 𝑅))
3231adantr 274 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → (𝑄𝐷𝑥) < ((𝑃𝐷𝑄) +𝑒 𝑅))
3319adantr 274 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑆 ∈ ℝ*)
3416xnegcld 9668 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → -𝑒𝑅 ∈ ℝ*)
3533, 34xaddcld 9697 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑆 +𝑒 -𝑒𝑅) ∈ ℝ*)
36 xblss2.7 . . . . . . . . . 10 (𝜑 → (𝑃𝐷𝑄) ≤ (𝑆 +𝑒 -𝑒𝑅))
3736adantr 274 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑄) ≤ (𝑆 +𝑒 -𝑒𝑅))
38 xleadd1a 9686 . . . . . . . . 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 1220 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) +𝑒 𝑅) ≤ ((𝑆 +𝑒 -𝑒𝑅) +𝑒 𝑅))
4039adantr 274 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → ((𝑃𝐷𝑄) +𝑒 𝑅) ≤ ((𝑆 +𝑒 -𝑒𝑅) +𝑒 𝑅))
41 xnpcan 9685 . . . . . . . 8 ((𝑆 ∈ ℝ*𝑅 ∈ ℝ) → ((𝑆 +𝑒 -𝑒𝑅) +𝑒 𝑅) = 𝑆)
4233, 41sylan 281 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → ((𝑆 +𝑒 -𝑒𝑅) +𝑒 𝑅) = 𝑆)
4340, 42breqtrd 3962 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → ((𝑃𝐷𝑄) +𝑒 𝑅) ≤ 𝑆)
4412, 18, 20, 32, 43xrltletrd 9624 . . . . 5 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → (𝑄𝐷𝑥) < 𝑆)
4527adantr 274 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑃𝐷𝑥) < 𝑅)
4636ad2antrr 480 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑃𝐷𝑄) ≤ (𝑆 +𝑒 -𝑒𝑅))
4719ad2antrr 480 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑆 ∈ ℝ*)
48 xrpnfdc 9655 . . . . . . . . . . . . 13 (𝑆 ∈ ℝ*DECID 𝑆 = +∞)
4947, 48syl 14 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → DECID 𝑆 = +∞)
50 0xr 7836 . . . . . . . . . . . . . . . 16 0 ∈ ℝ*
5150a1i 9 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ∈ ℝ*)
52 xmetge0 12573 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑄𝑋) → 0 ≤ (𝑃𝐷𝑄))
537, 21, 9, 52syl3anc 1217 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ (𝑃𝐷𝑄))
5451, 15, 35, 53, 37xrletrd 9625 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ (𝑆 +𝑒 -𝑒𝑅))
55 ge0nemnf 9637 . . . . . . . . . . . . . 14 (((𝑆 +𝑒 -𝑒𝑅) ∈ ℝ* ∧ 0 ≤ (𝑆 +𝑒 -𝑒𝑅)) → (𝑆 +𝑒 -𝑒𝑅) ≠ -∞)
5635, 54, 55syl2anc 409 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑆 +𝑒 -𝑒𝑅) ≠ -∞)
5756adantr 274 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 +𝑒 -𝑒𝑅) ≠ -∞)
58 xaddmnf1 9661 . . . . . . . . . . . . . . . . 17 ((𝑆 ∈ ℝ*𝑆 ≠ +∞) → (𝑆 +𝑒 -∞) = -∞)
5958ex 114 . . . . . . . . . . . . . . . 16 (𝑆 ∈ ℝ* → (𝑆 ≠ +∞ → (𝑆 +𝑒 -∞) = -∞))
6047, 59syl 14 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 ≠ +∞ → (𝑆 +𝑒 -∞) = -∞))
61 simpr 109 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑅 = +∞)
62 xnegeq 9640 . . . . . . . . . . . . . . . . . . 19 (𝑅 = +∞ → -𝑒𝑅 = -𝑒+∞)
6361, 62syl 14 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → -𝑒𝑅 = -𝑒+∞)
64 xnegpnf 9641 . . . . . . . . . . . . . . . . . 18 -𝑒+∞ = -∞
6563, 64eqtrdi 2189 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → -𝑒𝑅 = -∞)
6665oveq2d 5798 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 +𝑒 -𝑒𝑅) = (𝑆 +𝑒 -∞))
6766eqeq1d 2149 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑆 +𝑒 -𝑒𝑅) = -∞ ↔ (𝑆 +𝑒 -∞) = -∞))
6860, 67sylibrd 168 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 ≠ +∞ → (𝑆 +𝑒 -𝑒𝑅) = -∞))
6968a1d 22 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (DECID 𝑆 = +∞ → (𝑆 ≠ +∞ → (𝑆 +𝑒 -𝑒𝑅) = -∞)))
7069necon1ddc 2387 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (DECID 𝑆 = +∞ → ((𝑆 +𝑒 -𝑒𝑅) ≠ -∞ → 𝑆 = +∞)))
7149, 57, 70mp2d 47 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑆 = +∞)
7271, 65oveq12d 5800 . . . . . . . . . 10 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 +𝑒 -𝑒𝑅) = (+∞ +𝑒 -∞))
73 pnfaddmnf 9663 . . . . . . . . . 10 (+∞ +𝑒 -∞) = 0
7472, 73eqtrdi 2189 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 +𝑒 -𝑒𝑅) = 0)
7546, 74breqtrd 3962 . . . . . . . 8 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑃𝐷𝑄) ≤ 0)
7653biantrud 302 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) ≤ 0 ↔ ((𝑃𝐷𝑄) ≤ 0 ∧ 0 ≤ (𝑃𝐷𝑄))))
77 xrletri3 9618 . . . . . . . . . . 11 (((𝑃𝐷𝑄) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((𝑃𝐷𝑄) = 0 ↔ ((𝑃𝐷𝑄) ≤ 0 ∧ 0 ≤ (𝑃𝐷𝑄))))
7815, 50, 77sylancl 410 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) = 0 ↔ ((𝑃𝐷𝑄) ≤ 0 ∧ 0 ≤ (𝑃𝐷𝑄))))
79 xmeteq0 12567 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑄𝑋) → ((𝑃𝐷𝑄) = 0 ↔ 𝑃 = 𝑄))
807, 21, 9, 79syl3anc 1217 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) = 0 ↔ 𝑃 = 𝑄))
8176, 78, 803bitr2d 215 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) ≤ 0 ↔ 𝑃 = 𝑄))
8281adantr 274 . . . . . . . 8 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑃𝐷𝑄) ≤ 0 ↔ 𝑃 = 𝑄))
8375, 82mpbid 146 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑃 = 𝑄)
8483oveq1d 5797 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑃𝐷𝑥) = (𝑄𝐷𝑥))
8561, 71eqtr4d 2176 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑅 = 𝑆)
8645, 84, 853brtr3d 3967 . . . . 5 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑄𝐷𝑥) < 𝑆)
87 xmetge0 12573 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑥𝑋) → 0 ≤ (𝑃𝐷𝑥))
887, 21, 6, 87syl3anc 1217 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ (𝑃𝐷𝑥))
8951, 23, 16, 88, 27xrlelttrd 9623 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 < 𝑅)
9051, 16, 89xrltled 9615 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ 𝑅)
91 ge0nemnf 9637 . . . . . . . 8 ((𝑅 ∈ ℝ* ∧ 0 ≤ 𝑅) → 𝑅 ≠ -∞)
9216, 90, 91syl2anc 409 . . . . . . 7 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑅 ≠ -∞)
9316, 92jca 304 . . . . . 6 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑅 ∈ ℝ*𝑅 ≠ -∞))
94 xrnemnf 9594 . . . . . 6 ((𝑅 ∈ ℝ*𝑅 ≠ -∞) ↔ (𝑅 ∈ ℝ ∨ 𝑅 = +∞))
9593, 94sylib 121 . . . . 5 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑅 ∈ ℝ ∨ 𝑅 = +∞))
9644, 86, 95mpjaodan 788 . . . 4 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑄𝐷𝑥) < 𝑆)
97 elbl 12599 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑄𝑋𝑆 ∈ ℝ*) → (𝑥 ∈ (𝑄(ball‘𝐷)𝑆) ↔ (𝑥𝑋 ∧ (𝑄𝐷𝑥) < 𝑆)))
987, 9, 33, 97syl3anc 1217 . . . 4 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥 ∈ (𝑄(ball‘𝐷)𝑆) ↔ (𝑥𝑋 ∧ (𝑄𝐷𝑥) < 𝑆)))
996, 96, 98mpbir2and 929 . . 3 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ (𝑄(ball‘𝐷)𝑆))
10099ex 114 . 2 (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) → 𝑥 ∈ (𝑄(ball‘𝐷)𝑆)))
101100ssrdv 3108 1 (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698  DECID wdc 820   = wceq 1332   ∈ wcel 1481   ≠ wne 2309   ⊆ wss 3076   class class class wbr 3937  ‘cfv 5131  (class class class)co 5782  ℝcr 7643  0cc0 7644  +∞cpnf 7821  -∞cmnf 7822  ℝ*cxr 7823   < clt 7824   ≤ cle 7825  -𝑒cxne 9586   +𝑒 cxad 9587  ∞Metcxmet 12188  ballcbl 12190 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761 This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-po 4226  df-iso 4227  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-map 6552  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-2 8803  df-xneg 9589  df-xadd 9590  df-psmet 12195  df-xmet 12196  df-bl 12198 This theorem is referenced by:  blss2  12615  ssbl  12634
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