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Theorem bldisj 23296

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 12891	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → (𝑅 +_𝑒 𝑆) ∈ ℝ^*)
5		xmetcl 23229	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → (𝑃𝐷𝑄) ∈ ℝ^*)
6	5	adantr 484	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → (𝑃𝐷𝑄) ∈ ℝ^*)
7	4, 6	xrlenltd 10899	. . . 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 235	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → ¬ (𝑃𝐷𝑄) < (𝑅 +_𝑒 𝑆))
9		elin 3882	. . . 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 23286	. . . . . . . 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 1195	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → 𝑄 ∈ 𝑋)
15		elbl 23286	. . . . . . . 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 634	. . . . . 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 292	. . . . 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 484	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ (∞Met‘𝑋))
21	11	adantr 484	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → 𝑃 ∈ 𝑋)
22		simpr 488	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
23		xmetcl 23229	. . . . . . . . 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 484	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → 𝑄 ∈ 𝑋)
26		xmetcl 23229	. . . . . . . . 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 484	. . . . . . . 8 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ ℝ^*)
29	3	adantr 484	. . . . . . . 8 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → 𝑆 ∈ ℝ^*)
30		xlt2add 12850	. . . . . . . 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 23251	. . . . . . . . 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 484	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → (𝑃𝐷𝑄) ∈ ℝ^*)
35	24, 27	xaddcld 12891	. . . . . . . . 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 484	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → (𝑅 +_𝑒 𝑆) ∈ ℝ^*)
37		xrlelttr 12746	. . . . . . . . 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 457	. . . . 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 243	. . . 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	syl5bi 245	. . 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 201	. 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 4319	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 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∩ cin 3865 ∅c0 4237 class class class wbr 5053 ‘cfv 6380 (class class class)co 7213 ℝ^*cxr 10866 < clt 10867 ≤ cle 10868 +_𝑒 cxad 12702 ∞Metcxmet 20348 ballcbl 20350

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-1st 7761 df-2nd 7762 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-xneg 12704 df-xadd 12705 df-psmet 20355 df-xmet 20356 df-bl 20358

This theorem is referenced by: bl2in 23298 blcld 23403 methaus 23418 metnrmlem3 23758 cntotbnd 35691 heiborlem6 35711

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