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Theorem bldisj 24408

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 1197	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))
2		simpr1 1195	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → 𝑅 ∈ ℝ^*)
3		simpr2 1196	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → 𝑆 ∈ ℝ^*)
4	2, 3	xaddcld 13343	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → (𝑅 +_𝑒 𝑆) ∈ ℝ^*)
5		xmetcl 24341	. . . . . 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 11327	. . . 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 3967	. . . 4 ⊢ (𝑥 ∈ ((𝑃(ball‘𝐷)𝑅) ∩ (𝑄(ball‘𝐷)𝑆)) ↔ (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ∧ 𝑥 ∈ (𝑄(ball‘𝐷)𝑆)))
10		simpl1 1192	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → 𝐷 ∈ (∞Met‘𝑋))
11		simpl2 1193	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → 𝑃 ∈ 𝑋)
12		elbl 24398	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
13	10, 11, 2, 12	syl3anc 1373	. . . . . . 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 1194	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → 𝑄 ∈ 𝑋)
15		elbl 24398	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑄 ∈ 𝑋 ∧ 𝑆 ∈ ℝ^*) → (𝑥 ∈ (𝑄(ball‘𝐷)𝑆) ↔ (𝑥 ∈ 𝑋 ∧ (𝑄𝐷𝑥) < 𝑆)))
16	10, 14, 3, 15	syl3anc 1373	. . . . . . 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 632	. . . . . 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 676	. . . . . 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 24341	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑃𝐷𝑥) ∈ ℝ^*)
24	20, 21, 22, 23	syl3anc 1373	. . . . . . . 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 24341	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑄 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑄𝐷𝑥) ∈ ℝ^*)
27	20, 25, 22, 26	syl3anc 1373	. . . . . . . 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 13302	. . . . . . . 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 24363	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑃𝐷𝑄) ≤ ((𝑃𝐷𝑥) +_𝑒 (𝑄𝐷𝑥)))
33	20, 21, 25, 22, 32	syl13anc 1374	. . . . . . . 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 13343	. . . . . . . . 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 13198	. . . . . . . . 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 1373	. . . . . . . 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 695	. . . . . . 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 4407	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 1540 ∈ wcel 2108 ∩ cin 3950 ∅c0 4333 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 +_𝑒 cxad 13152 ∞Metcxmet 21349 ballcbl 21351

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-xneg 13154 df-xadd 13155 df-psmet 21356 df-xmet 21357 df-bl 21359

This theorem is referenced by: bl2in 24410 blcld 24518 methaus 24533 metnrmlem3 24883 cntotbnd 37803 heiborlem6 37823

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