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Theorem bldisj 24373

Description: Two balls are disjoint if the center-to-center distance is more than the sum of the radii. (Contributed by Mario Carneiro, 30-Dec-2013.)

**Assertion**
Ref	Expression
bldisj	⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → ((𝑃(ball‘𝐷)𝑅) ∩ (𝑄(ball‘𝐷)𝑆)) = ∅)

Proof of Theorem bldisj Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr3 1198	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))
2		simpr1 1196	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → 𝑅 ∈ ℝ^*)
3		simpr2 1197	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → 𝑆 ∈ ℝ^*)
4	2, 3	xaddcld 13244	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → (𝑅 +_𝑒 𝑆) ∈ ℝ^*)
5		xmetcl 24306	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → (𝑃𝐷𝑄) ∈ ℝ^*)
6	5	adantr 480	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → (𝑃𝐷𝑄) ∈ ℝ^*)
7	4, 6	xrlenltd 11202	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → ((𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄) ↔ ¬ (𝑃𝐷𝑄) < (𝑅 +_𝑒 𝑆)))
8	1, 7	mpbid 232	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → ¬ (𝑃𝐷𝑄) < (𝑅 +_𝑒 𝑆))
9		elin 3906	. . . 4 ⊢ (𝑥 ∈ ((𝑃(ball‘𝐷)𝑅) ∩ (𝑄(ball‘𝐷)𝑆)) ↔ (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ∧ 𝑥 ∈ (𝑄(ball‘𝐷)𝑆)))
10		simpl1 1193	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → 𝐷 ∈ (∞Met‘𝑋))
11		simpl2 1194	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → 𝑃 ∈ 𝑋)
12		elbl 24363	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
13	10, 11, 2, 12	syl3anc 1374	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
14		simpl3 1195	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → 𝑄 ∈ 𝑋)
15		elbl 24363	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑄 ∈ 𝑋 ∧ 𝑆 ∈ ℝ^*) → (𝑥 ∈ (𝑄(ball‘𝐷)𝑆) ↔ (𝑥 ∈ 𝑋 ∧ (𝑄𝐷𝑥) < 𝑆)))
16	10, 14, 3, 15	syl3anc 1374	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → (𝑥 ∈ (𝑄(ball‘𝐷)𝑆) ↔ (𝑥 ∈ 𝑋 ∧ (𝑄𝐷𝑥) < 𝑆)))
17	13, 16	anbi12d 633	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → ((𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ∧ 𝑥 ∈ (𝑄(ball‘𝐷)𝑆)) ↔ ((𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅) ∧ (𝑥 ∈ 𝑋 ∧ (𝑄𝐷𝑥) < 𝑆))))
18		anandi 677	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ ((𝑃𝐷𝑥) < 𝑅 ∧ (𝑄𝐷𝑥) < 𝑆)) ↔ ((𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅) ∧ (𝑥 ∈ 𝑋 ∧ (𝑄𝐷𝑥) < 𝑆)))
19	17, 18	bitr4di 289	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → ((𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ∧ 𝑥 ∈ (𝑄(ball‘𝐷)𝑆)) ↔ (𝑥 ∈ 𝑋 ∧ ((𝑃𝐷𝑥) < 𝑅 ∧ (𝑄𝐷𝑥) < 𝑆))))
20	10	adantr 480	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ (∞Met‘𝑋))
21	11	adantr 480	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → 𝑃 ∈ 𝑋)
22		simpr 484	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
23		xmetcl 24306	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑃𝐷𝑥) ∈ ℝ^*)
24	20, 21, 22, 23	syl3anc 1374	. . . . . . . 8 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → (𝑃𝐷𝑥) ∈ ℝ^*)
25	14	adantr 480	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → 𝑄 ∈ 𝑋)
26		xmetcl 24306	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑄 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑄𝐷𝑥) ∈ ℝ^*)
27	20, 25, 22, 26	syl3anc 1374	. . . . . . . 8 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → (𝑄𝐷𝑥) ∈ ℝ^*)
28	2	adantr 480	. . . . . . . 8 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ ℝ^*)
29	3	adantr 480	. . . . . . . 8 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → 𝑆 ∈ ℝ^*)
30		xlt2add 13203	. . . . . . . 8 ⊢ ((((𝑃𝐷𝑥) ∈ ℝ^* ∧ (𝑄𝐷𝑥) ∈ ℝ^) ∧ (𝑅 ∈ ℝ^ ∧ 𝑆 ∈ ℝ^*)) → (((𝑃𝐷𝑥) < 𝑅 ∧ (𝑄𝐷𝑥) < 𝑆) → ((𝑃𝐷𝑥) +_𝑒 (𝑄𝐷𝑥)) < (𝑅 +_𝑒 𝑆)))
31	24, 27, 28, 29, 30	syl22anc 839	. . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → (((𝑃𝐷𝑥) < 𝑅 ∧ (𝑄𝐷𝑥) < 𝑆) → ((𝑃𝐷𝑥) +_𝑒 (𝑄𝐷𝑥)) < (𝑅 +_𝑒 𝑆)))
32		xmettri3 24328	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑃𝐷𝑄) ≤ ((𝑃𝐷𝑥) +_𝑒 (𝑄𝐷𝑥)))
33	20, 21, 25, 22, 32	syl13anc 1375	. . . . . . . 8 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → (𝑃𝐷𝑄) ≤ ((𝑃𝐷𝑥) +_𝑒 (𝑄𝐷𝑥)))
34	6	adantr 480	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → (𝑃𝐷𝑄) ∈ ℝ^*)
35	24, 27	xaddcld 13244	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → ((𝑃𝐷𝑥) +_𝑒 (𝑄𝐷𝑥)) ∈ ℝ^*)
36	4	adantr 480	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → (𝑅 +_𝑒 𝑆) ∈ ℝ^*)
37		xrlelttr 13098	. . . . . . . . 9 ⊢ (((𝑃𝐷𝑄) ∈ ℝ^* ∧ ((𝑃𝐷𝑥) +_𝑒 (𝑄𝐷𝑥)) ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ∈ ℝ^*) → (((𝑃𝐷𝑄) ≤ ((𝑃𝐷𝑥) +_𝑒 (𝑄𝐷𝑥)) ∧ ((𝑃𝐷𝑥) +_𝑒 (𝑄𝐷𝑥)) < (𝑅 +_𝑒 𝑆)) → (𝑃𝐷𝑄) < (𝑅 +_𝑒 𝑆)))
38	34, 35, 36, 37	syl3anc 1374	. . . . . . . 8 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → (((𝑃𝐷𝑄) ≤ ((𝑃𝐷𝑥) +_𝑒 (𝑄𝐷𝑥)) ∧ ((𝑃𝐷𝑥) +_𝑒 (𝑄𝐷𝑥)) < (𝑅 +_𝑒 𝑆)) → (𝑃𝐷𝑄) < (𝑅 +_𝑒 𝑆)))
39	33, 38	mpand 696	. . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → (((𝑃𝐷𝑥) +_𝑒 (𝑄𝐷𝑥)) < (𝑅 +_𝑒 𝑆) → (𝑃𝐷𝑄) < (𝑅 +_𝑒 𝑆)))
40	31, 39	syld 47	. . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → (((𝑃𝐷𝑥) < 𝑅 ∧ (𝑄𝐷𝑥) < 𝑆) → (𝑃𝐷𝑄) < (𝑅 +_𝑒 𝑆)))
41	40	expimpd 453	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → ((𝑥 ∈ 𝑋 ∧ ((𝑃𝐷𝑥) < 𝑅 ∧ (𝑄𝐷𝑥) < 𝑆)) → (𝑃𝐷𝑄) < (𝑅 +_𝑒 𝑆)))
42	19, 41	sylbid 240	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → ((𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ∧ 𝑥 ∈ (𝑄(ball‘𝐷)𝑆)) → (𝑃𝐷𝑄) < (𝑅 +_𝑒 𝑆)))
43	9, 42	biimtrid 242	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → (𝑥 ∈ ((𝑃(ball‘𝐷)𝑅) ∩ (𝑄(ball‘𝐷)𝑆)) → (𝑃𝐷𝑄) < (𝑅 +_𝑒 𝑆)))
44	8, 43	mtod 198	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → ¬ 𝑥 ∈ ((𝑃(ball‘𝐷)𝑅) ∩ (𝑄(ball‘𝐷)𝑆)))
45	44	eq0rdv 4348	1 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → ((𝑃(ball‘𝐷)𝑅) ∩ (𝑄(ball‘𝐷)𝑆)) = ∅)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∩ cin 3889 ∅c0 4274 class class class wbr 5086 ‘cfv 6492 (class class class)co 7360 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 +_𝑒 cxad 13052 ∞Metcxmet 21329 ballcbl 21331

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5519 df-po 5532 df-so 5533 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-1st 7935 df-2nd 7936 df-er 8636 df-map 8768 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-xneg 13054 df-xadd 13055 df-psmet 21336 df-xmet 21337 df-bl 21339

This theorem is referenced by: bl2in 24375 blcld 24480 methaus 24495 metnrmlem3 24837 cntotbnd 38131 heiborlem6 38151

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