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Theorem bldisj 23010

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 1192	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))
2		simpr1 1190	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → 𝑅 ∈ ℝ^*)
3		simpr2 1191	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → 𝑆 ∈ ℝ^*)
4	2, 3	xaddcld 12697	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → (𝑅 +_𝑒 𝑆) ∈ ℝ^*)
5		xmetcl 22943	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → (𝑃𝐷𝑄) ∈ ℝ^*)
6	5	adantr 483	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → (𝑃𝐷𝑄) ∈ ℝ^*)
7	4, 6	xrlenltd 10709	. . . 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 234	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → ¬ (𝑃𝐷𝑄) < (𝑅 +_𝑒 𝑆))
9		elin 4171	. . . 4 ⊢ (𝑥 ∈ ((𝑃(ball‘𝐷)𝑅) ∩ (𝑄(ball‘𝐷)𝑆)) ↔ (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ∧ 𝑥 ∈ (𝑄(ball‘𝐷)𝑆)))
10		simpl1 1187	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → 𝐷 ∈ (∞Met‘𝑋))
11		simpl2 1188	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → 𝑃 ∈ 𝑋)
12		elbl 23000	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
13	10, 11, 2, 12	syl3anc 1367	. . . . . . 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 1189	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → 𝑄 ∈ 𝑋)
15		elbl 23000	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑄 ∈ 𝑋 ∧ 𝑆 ∈ ℝ^*) → (𝑥 ∈ (𝑄(ball‘𝐷)𝑆) ↔ (𝑥 ∈ 𝑋 ∧ (𝑄𝐷𝑥) < 𝑆)))
16	10, 14, 3, 15	syl3anc 1367	. . . . . . 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 674	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ ((𝑃𝐷𝑥) < 𝑅 ∧ (𝑄𝐷𝑥) < 𝑆)) ↔ ((𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅) ∧ (𝑥 ∈ 𝑋 ∧ (𝑄𝐷𝑥) < 𝑆)))
19	17, 18	syl6bbr 291	. . . . 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 483	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ (∞Met‘𝑋))
21	11	adantr 483	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → 𝑃 ∈ 𝑋)
22		simpr 487	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
23		xmetcl 22943	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑃𝐷𝑥) ∈ ℝ^*)
24	20, 21, 22, 23	syl3anc 1367	. . . . . . . 8 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → (𝑃𝐷𝑥) ∈ ℝ^*)
25	14	adantr 483	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → 𝑄 ∈ 𝑋)
26		xmetcl 22943	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑄 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑄𝐷𝑥) ∈ ℝ^*)
27	20, 25, 22, 26	syl3anc 1367	. . . . . . . 8 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → (𝑄𝐷𝑥) ∈ ℝ^*)
28	2	adantr 483	. . . . . . . 8 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ ℝ^*)
29	3	adantr 483	. . . . . . . 8 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → 𝑆 ∈ ℝ^*)
30		xlt2add 12656	. . . . . . . 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 836	. . . . . . 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 22965	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑃𝐷𝑄) ≤ ((𝑃𝐷𝑥) +_𝑒 (𝑄𝐷𝑥)))
33	20, 21, 25, 22, 32	syl13anc 1368	. . . . . . . 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 483	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → (𝑃𝐷𝑄) ∈ ℝ^*)
35	24, 27	xaddcld 12697	. . . . . . . . 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 483	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → (𝑅 +_𝑒 𝑆) ∈ ℝ^*)
37		xrlelttr 12552	. . . . . . . . 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 1367	. . . . . . . 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 693	. . . . . . 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 456	. . . . 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 242	. . . 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 244	. . 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 200	. 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 4359	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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∩ cin 3937 ∅c0 4293 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℝ^*cxr 10676 < clt 10677 ≤ cle 10678 +_𝑒 cxad 12508 ∞Metcxmet 20532 ballcbl 20534

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-xneg 12510 df-xadd 12511 df-psmet 20539 df-xmet 20540 df-bl 20542

This theorem is referenced by: bl2in 23012 blcld 23117 methaus 23132 metnrmlem3 23471 cntotbnd 35076 heiborlem6 35096

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