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Theorem xblss2 22939
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 22941 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 22925 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
51, 2, 3, 4syl3anc 1363 . . . . 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 ∈ (∞Met‘𝑋))
8 xblss2.3 . . . . . . . . 9 (𝜑𝑄𝑋)
98adantr 481 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑄𝑋)
10 xmetcl 22868 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑄𝑋𝑥𝑋) → (𝑄𝐷𝑥) ∈ ℝ*)
117, 9, 6, 10syl3anc 1363 . . . . . . 7 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑄𝐷𝑥) ∈ ℝ*)
1211adantr 481 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → (𝑄𝐷𝑥) ∈ ℝ*)
13 xblss2.6 . . . . . . . . . 10 (𝜑 → (𝑃𝐷𝑄) ∈ ℝ)
1413adantr 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑄) ∈ ℝ)
1514rexrd 10679 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑄) ∈ ℝ*)
163adantr 481 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑅 ∈ ℝ*)
1715, 16xaddcld 12682 . . . . . . 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 xblss2.5 . . . . . . 7 (𝜑𝑆 ∈ ℝ*)
2019ad2antrr 722 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → 𝑆 ∈ ℝ*)
212adantr 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑃𝑋)
22 xmetcl 22868 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑥𝑋) → (𝑃𝐷𝑥) ∈ ℝ*)
237, 21, 6, 22syl3anc 1363 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑥) ∈ ℝ*)
2415, 23xaddcld 12682 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)) ∈ ℝ*)
25 xmettri2 22877 . . . . . . . . 9 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑃𝑋𝑄𝑋𝑥𝑋)) → (𝑄𝐷𝑥) ≤ ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)))
267, 21, 9, 6, 25syl13anc 1364 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑄𝐷𝑥) ≤ ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)))
275simplbda 500 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑥) < 𝑅)
28 xltadd2 12638 . . . . . . . . . 10 (((𝑃𝐷𝑥) ∈ ℝ*𝑅 ∈ ℝ* ∧ (𝑃𝐷𝑄) ∈ ℝ) → ((𝑃𝐷𝑥) < 𝑅 ↔ ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)) < ((𝑃𝐷𝑄) +𝑒 𝑅)))
2923, 16, 14, 28syl3anc 1363 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑥) < 𝑅 ↔ ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)) < ((𝑃𝐷𝑄) +𝑒 𝑅)))
3027, 29mpbid 233 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) +𝑒 (𝑃𝐷𝑥)) < ((𝑃𝐷𝑄) +𝑒 𝑅))
3111, 24, 17, 26, 30xrlelttrd 12541 . . . . . . 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 12681 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → -𝑒𝑅 ∈ ℝ*)
3533, 34xaddcld 12682 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑆 +𝑒 -𝑒𝑅) ∈ ℝ*)
36 xblss2.7 . . . . . . . . . 10 (𝜑 → (𝑃𝐷𝑄) ≤ (𝑆 +𝑒 -𝑒𝑅))
3736adantr 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑄) ≤ (𝑆 +𝑒 -𝑒𝑅))
38 xleadd1a 12634 . . . . . . . . 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 1365 . . . . . . . 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 12633 . . . . . . . 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 5083 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → ((𝑃𝐷𝑄) +𝑒 𝑅) ≤ 𝑆)
4412, 18, 20, 32, 43xrltletrd 12542 . . . . 5 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → (𝑄𝐷𝑥) < 𝑆)
4527adantr 481 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑃𝐷𝑥) < 𝑅)
4636ad2antrr 722 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑃𝐷𝑄) ≤ (𝑆 +𝑒 -𝑒𝑅))
47 0xr 10676 . . . . . . . . . . . . . . . 16 0 ∈ ℝ*
4847a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ∈ ℝ*)
49 xmetge0 22881 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑄𝑋) → 0 ≤ (𝑃𝐷𝑄))
507, 21, 9, 49syl3anc 1363 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ (𝑃𝐷𝑄))
5148, 15, 35, 50, 37xrletrd 12543 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ (𝑆 +𝑒 -𝑒𝑅))
52 ge0nemnf 12554 . . . . . . . . . . . . . 14 (((𝑆 +𝑒 -𝑒𝑅) ∈ ℝ* ∧ 0 ≤ (𝑆 +𝑒 -𝑒𝑅)) → (𝑆 +𝑒 -𝑒𝑅) ≠ -∞)
5335, 51, 52syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑆 +𝑒 -𝑒𝑅) ≠ -∞)
5453adantr 481 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 +𝑒 -𝑒𝑅) ≠ -∞)
5519ad2antrr 722 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑆 ∈ ℝ*)
56 xaddmnf1 12609 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ ℝ*𝑆 ≠ +∞) → (𝑆 +𝑒 -∞) = -∞)
5756ex 413 . . . . . . . . . . . . . . 15 (𝑆 ∈ ℝ* → (𝑆 ≠ +∞ → (𝑆 +𝑒 -∞) = -∞))
5855, 57syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 ≠ +∞ → (𝑆 +𝑒 -∞) = -∞))
59 simpr 485 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑅 = +∞)
60 xnegeq 12588 . . . . . . . . . . . . . . . . . 18 (𝑅 = +∞ → -𝑒𝑅 = -𝑒+∞)
6159, 60syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → -𝑒𝑅 = -𝑒+∞)
62 xnegpnf 12590 . . . . . . . . . . . . . . . . 17 -𝑒+∞ = -∞
6361, 62syl6eq 2869 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → -𝑒𝑅 = -∞)
6463oveq2d 7161 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 +𝑒 -𝑒𝑅) = (𝑆 +𝑒 -∞))
6564eqeq1d 2820 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑆 +𝑒 -𝑒𝑅) = -∞ ↔ (𝑆 +𝑒 -∞) = -∞))
6658, 65sylibrd 260 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 ≠ +∞ → (𝑆 +𝑒 -𝑒𝑅) = -∞))
6766necon1d 3035 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑆 +𝑒 -𝑒𝑅) ≠ -∞ → 𝑆 = +∞))
6854, 67mpd 15 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑆 = +∞)
6968, 63oveq12d 7163 . . . . . . . . . 10 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 +𝑒 -𝑒𝑅) = (+∞ +𝑒 -∞))
70 pnfaddmnf 12611 . . . . . . . . . 10 (+∞ +𝑒 -∞) = 0
7169, 70syl6eq 2869 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 +𝑒 -𝑒𝑅) = 0)
7246, 71breqtrd 5083 . . . . . . . 8 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑃𝐷𝑄) ≤ 0)
7350biantrud 532 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) ≤ 0 ↔ ((𝑃𝐷𝑄) ≤ 0 ∧ 0 ≤ (𝑃𝐷𝑄))))
74 xrletri3 12535 . . . . . . . . . . 11 (((𝑃𝐷𝑄) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((𝑃𝐷𝑄) = 0 ↔ ((𝑃𝐷𝑄) ≤ 0 ∧ 0 ≤ (𝑃𝐷𝑄))))
7515, 47, 74sylancl 586 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) = 0 ↔ ((𝑃𝐷𝑄) ≤ 0 ∧ 0 ≤ (𝑃𝐷𝑄))))
76 xmeteq0 22875 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑄𝑋) → ((𝑃𝐷𝑄) = 0 ↔ 𝑃 = 𝑄))
777, 21, 9, 76syl3anc 1363 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) = 0 ↔ 𝑃 = 𝑄))
7873, 75, 773bitr2d 308 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) ≤ 0 ↔ 𝑃 = 𝑄))
7978adantr 481 . . . . . . . 8 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑃𝐷𝑄) ≤ 0 ↔ 𝑃 = 𝑄))
8072, 79mpbid 233 . . . . . . 7 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑃 = 𝑄)
8180oveq1d 7160 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑃𝐷𝑥) = (𝑄𝐷𝑥))
8259, 68eqtr4d 2856 . . . . . 6 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑅 = 𝑆)
8345, 81, 823brtr3d 5088 . . . . 5 (((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑄𝐷𝑥) < 𝑆)
84 xmetge0 22881 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑥𝑋) → 0 ≤ (𝑃𝐷𝑥))
857, 21, 6, 84syl3anc 1363 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ (𝑃𝐷𝑥))
8648, 23, 16, 85, 27xrlelttrd 12541 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 < 𝑅)
8748, 16, 86xrltled 12531 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ 𝑅)
88 ge0nemnf 12554 . . . . . . . 8 ((𝑅 ∈ ℝ* ∧ 0 ≤ 𝑅) → 𝑅 ≠ -∞)
8916, 87, 88syl2anc 584 . . . . . . 7 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑅 ≠ -∞)
9016, 89jca 512 . . . . . 6 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑅 ∈ ℝ*𝑅 ≠ -∞))
91 xrnemnf 12500 . . . . . 6 ((𝑅 ∈ ℝ*𝑅 ≠ -∞) ↔ (𝑅 ∈ ℝ ∨ 𝑅 = +∞))
9290, 91sylib 219 . . . . 5 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑅 ∈ ℝ ∨ 𝑅 = +∞))
9344, 83, 92mpjaodan 952 . . . 4 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑄𝐷𝑥) < 𝑆)
94 elbl 22925 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑄𝑋𝑆 ∈ ℝ*) → (𝑥 ∈ (𝑄(ball‘𝐷)𝑆) ↔ (𝑥𝑋 ∧ (𝑄𝐷𝑥) < 𝑆)))
957, 9, 33, 94syl3anc 1363 . . . 4 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥 ∈ (𝑄(ball‘𝐷)𝑆) ↔ (𝑥𝑋 ∧ (𝑄𝐷𝑥) < 𝑆)))
966, 93, 95mpbir2and 709 . . 3 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ (𝑄(ball‘𝐷)𝑆))
9796ex 413 . 2 (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) → 𝑥 ∈ (𝑄(ball‘𝐷)𝑆)))
9897ssrdv 3970 1 (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wcel 2105  wne 3013  wss 3933   class class class wbr 5057  cfv 6348  (class class class)co 7145  cr 10524  0cc0 10525  +∞cpnf 10660  -∞cmnf 10661  *cxr 10662   < clt 10663  cle 10664  -𝑒cxne 12492   +𝑒 cxad 12493  ∞Metcxmet 20458  ballcbl 20460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-2 11688  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-psmet 20465  df-xmet 20466  df-bl 20468
This theorem is referenced by:  blss2  22941  ssbl  22960
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