xblss2 - Metamath Proof Explorer

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Theorem xblss2 22017

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 22019 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	⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
xblss2.2	⊢ (𝜑 → 𝑃 ∈ 𝑋)
xblss2.3	⊢ (𝜑 → 𝑄 ∈ 𝑋)
xblss2.4	⊢ (𝜑 → 𝑅 ∈ ℝ^*)
xblss2.5	⊢ (𝜑 → 𝑆 ∈ 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.

Step	Hyp	Ref	Expression
1		xblss2.1	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
2		xblss2.2	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝑋)
3		xblss2.4	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℝ^*)
4		elbl 22003	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
5	1, 2, 3, 4	syl3anc 1318	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
6	5	simprbda 651	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ 𝑋)
7	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝐷 ∈ (∞Met‘𝑋))
8		xblss2.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ 𝑋)
9	8	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑄 ∈ 𝑋)
10		xmetcl 21946	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑄 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑄𝐷𝑥) ∈ ℝ^*)
11	7, 9, 6, 10	syl3anc 1318	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑄𝐷𝑥) ∈ ℝ^*)
12	11	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → (𝑄𝐷𝑥) ∈ ℝ^*)
13		xblss2.6	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃𝐷𝑄) ∈ ℝ)
14	13	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑄) ∈ ℝ)
15	14	rexrd 9968	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑄) ∈ ℝ^*)
16	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑅 ∈ ℝ^*)
17	15, 16	xaddcld 12003	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) +_𝑒 𝑅) ∈ ℝ^*)
18	17	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → ((𝑃𝐷𝑄) +_𝑒 𝑅) ∈ ℝ^*)
19		xblss2.5	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ ℝ^*)
20	19	ad2antrr 758	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → 𝑆 ∈ ℝ^*)
21	2	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑃 ∈ 𝑋)
22		xmetcl 21946	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑃𝐷𝑥) ∈ ℝ^*)
23	7, 21, 6, 22	syl3anc 1318	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑥) ∈ ℝ^*)
24	15, 23	xaddcld 12003	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) +_𝑒 (𝑃𝐷𝑥)) ∈ ℝ^*)
25		xmettri2 21955	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑄𝐷𝑥) ≤ ((𝑃𝐷𝑄) +_𝑒 (𝑃𝐷𝑥)))
26	7, 21, 9, 6, 25	syl13anc 1320	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑄𝐷𝑥) ≤ ((𝑃𝐷𝑄) +_𝑒 (𝑃𝐷𝑥)))
27	5	simplbda 652	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑥) < 𝑅)
28		xltadd2 11959	. . . . . . . . . 10 ⊢ (((𝑃𝐷𝑥) ∈ ℝ^* ∧ 𝑅 ∈ ℝ^* ∧ (𝑃𝐷𝑄) ∈ ℝ) → ((𝑃𝐷𝑥) < 𝑅 ↔ ((𝑃𝐷𝑄) +_𝑒 (𝑃𝐷𝑥)) < ((𝑃𝐷𝑄) +_𝑒 𝑅)))
29	23, 16, 14, 28	syl3anc 1318	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑥) < 𝑅 ↔ ((𝑃𝐷𝑄) +_𝑒 (𝑃𝐷𝑥)) < ((𝑃𝐷𝑄) +_𝑒 𝑅)))
30	27, 29	mpbid 221	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) +_𝑒 (𝑃𝐷𝑥)) < ((𝑃𝐷𝑄) +_𝑒 𝑅))
31	11, 24, 17, 26, 30	xrlelttrd 11867	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑄𝐷𝑥) < ((𝑃𝐷𝑄) +_𝑒 𝑅))
32	31	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → (𝑄𝐷𝑥) < ((𝑃𝐷𝑄) +_𝑒 𝑅))
33	19	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑆 ∈ ℝ^*)
34	16	xnegcld 12002	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → -_𝑒𝑅 ∈ ℝ^*)
35	33, 34	xaddcld 12003	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑆 +_𝑒 -_𝑒𝑅) ∈ ℝ^*)
36		xblss2.7	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃𝐷𝑄) ≤ (𝑆 +_𝑒 -_𝑒𝑅))
37	36	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑄) ≤ (𝑆 +_𝑒 -_𝑒𝑅))
38		xleadd1a 11955	. . . . . . . . 9 ⊢ ((((𝑃𝐷𝑄) ∈ ℝ^* ∧ (𝑆 +_𝑒 -_𝑒𝑅) ∈ ℝ^* ∧ 𝑅 ∈ ℝ^*) ∧ (𝑃𝐷𝑄) ≤ (𝑆 +_𝑒 -_𝑒𝑅)) → ((𝑃𝐷𝑄) +_𝑒 𝑅) ≤ ((𝑆 +_𝑒 -_𝑒𝑅) +_𝑒 𝑅))
39	15, 35, 16, 37, 38	syl31anc 1321	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) +_𝑒 𝑅) ≤ ((𝑆 +_𝑒 -_𝑒𝑅) +_𝑒 𝑅))
40	39	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → ((𝑃𝐷𝑄) +_𝑒 𝑅) ≤ ((𝑆 +_𝑒 -_𝑒𝑅) +_𝑒 𝑅))
41		xnpcan 11954	. . . . . . . 8 ⊢ ((𝑆 ∈ ℝ^* ∧ 𝑅 ∈ ℝ) → ((𝑆 +_𝑒 -_𝑒𝑅) +_𝑒 𝑅) = 𝑆)
42	33, 41	sylan 487	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → ((𝑆 +_𝑒 -_𝑒𝑅) +_𝑒 𝑅) = 𝑆)
43	40, 42	breqtrd 4609	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → ((𝑃𝐷𝑄) +_𝑒 𝑅) ≤ 𝑆)
44	12, 18, 20, 32, 43	xrltletrd 11868	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → (𝑄𝐷𝑥) < 𝑆)
45	27	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑃𝐷𝑥) < 𝑅)
46	36	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑃𝐷𝑄) ≤ (𝑆 +_𝑒 -_𝑒𝑅))
47		0xr 9965	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ^*
48	47	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ∈ ℝ^*)
49		xmetge0 21959	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → 0 ≤ (𝑃𝐷𝑄))
50	7, 21, 9, 49	syl3anc 1318	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ (𝑃𝐷𝑄))
51	48, 15, 35, 50, 37	xrletrd 11869	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ (𝑆 +_𝑒 -_𝑒𝑅))
52		ge0nemnf 11878	. . . . . . . . . . . . . 14 ⊢ (((𝑆 +_𝑒 -_𝑒𝑅) ∈ ℝ^* ∧ 0 ≤ (𝑆 +_𝑒 -_𝑒𝑅)) → (𝑆 +_𝑒 -_𝑒𝑅) ≠ -∞)
53	35, 51, 52	syl2anc 691	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑆 +_𝑒 -_𝑒𝑅) ≠ -∞)
54	53	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 +_𝑒 -_𝑒𝑅) ≠ -∞)
55	19	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑆 ∈ ℝ^*)
56		xaddmnf1 11933	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ ℝ^* ∧ 𝑆 ≠ +∞) → (𝑆 +_𝑒 -∞) = -∞)
57	56	ex 449	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ ℝ^* → (𝑆 ≠ +∞ → (𝑆 +_𝑒 -∞) = -∞))
58	55, 57	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 ≠ +∞ → (𝑆 +_𝑒 -∞) = -∞))
59		simpr 476	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑅 = +∞)
60		xnegeq 11912	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 = +∞ → -_𝑒𝑅 = -_𝑒+∞)
61	59, 60	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → -_𝑒𝑅 = -_𝑒+∞)
62		xnegpnf 11914	. . . . . . . . . . . . . . . . 17 ⊢ -_𝑒+∞ = -∞
63	61, 62	syl6eq 2660	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → -_𝑒𝑅 = -∞)
64	63	oveq2d 6565	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 +_𝑒 -_𝑒𝑅) = (𝑆 +_𝑒 -∞))
65	64	eqeq1d 2612	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑆 +_𝑒 -_𝑒𝑅) = -∞ ↔ (𝑆 +_𝑒 -∞) = -∞))
66	58, 65	sylibrd 248	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 ≠ +∞ → (𝑆 +_𝑒 -_𝑒𝑅) = -∞))
67	66	necon1d 2804	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑆 +_𝑒 -_𝑒𝑅) ≠ -∞ → 𝑆 = +∞))
68	54, 67	mpd 15	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑆 = +∞)
69	68, 63	oveq12d 6567	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 +_𝑒 -_𝑒𝑅) = (+∞ +_𝑒 -∞))
70		pnfaddmnf 11935	. . . . . . . . . 10 ⊢ (+∞ +_𝑒 -∞) = 0
71	69, 70	syl6eq 2660	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 +_𝑒 -_𝑒𝑅) = 0)
72	46, 71	breqtrd 4609	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑃𝐷𝑄) ≤ 0)
73	50	biantrud 527	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) ≤ 0 ↔ ((𝑃𝐷𝑄) ≤ 0 ∧ 0 ≤ (𝑃𝐷𝑄))))
74		xrletri3 11861	. . . . . . . . . . 11 ⊢ (((𝑃𝐷𝑄) ∈ ℝ^* ∧ 0 ∈ ℝ^*) → ((𝑃𝐷𝑄) = 0 ↔ ((𝑃𝐷𝑄) ≤ 0 ∧ 0 ≤ (𝑃𝐷𝑄))))
75	15, 47, 74	sylancl 693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) = 0 ↔ ((𝑃𝐷𝑄) ≤ 0 ∧ 0 ≤ (𝑃𝐷𝑄))))
76		xmeteq0 21953	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → ((𝑃𝐷𝑄) = 0 ↔ 𝑃 = 𝑄))
77	7, 21, 9, 76	syl3anc 1318	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) = 0 ↔ 𝑃 = 𝑄))
78	73, 75, 77	3bitr2d 295	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) ≤ 0 ↔ 𝑃 = 𝑄))
79	78	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑃𝐷𝑄) ≤ 0 ↔ 𝑃 = 𝑄))
80	72, 79	mpbid 221	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑃 = 𝑄)
81	80	oveq1d 6564	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑃𝐷𝑥) = (𝑄𝐷𝑥))
82	59, 68	eqtr4d 2647	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑅 = 𝑆)
83	45, 81, 82	3brtr3d 4614	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑄𝐷𝑥) < 𝑆)
84		xmetge0 21959	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → 0 ≤ (𝑃𝐷𝑥))
85	7, 21, 6, 84	syl3anc 1318	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ (𝑃𝐷𝑥))
86	48, 23, 16, 85, 27	xrlelttrd 11867	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 < 𝑅)
87		xrltle 11858	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^*) → (0 < 𝑅 → 0 ≤ 𝑅))
88	47, 16, 87	sylancr 694	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (0 < 𝑅 → 0 ≤ 𝑅))
89	86, 88	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ 𝑅)
90		ge0nemnf 11878	. . . . . . . 8 ⊢ ((𝑅 ∈ ℝ^* ∧ 0 ≤ 𝑅) → 𝑅 ≠ -∞)
91	16, 89, 90	syl2anc 691	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑅 ≠ -∞)
92	16, 91	jca 553	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑅 ∈ ℝ^* ∧ 𝑅 ≠ -∞))
93		xrnemnf 11827	. . . . . 6 ⊢ ((𝑅 ∈ ℝ^* ∧ 𝑅 ≠ -∞) ↔ (𝑅 ∈ ℝ ∨ 𝑅 = +∞))
94	92, 93	sylib 207	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑅 ∈ ℝ ∨ 𝑅 = +∞))
95	44, 83, 94	mpjaodan 823	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑄𝐷𝑥) < 𝑆)
96		elbl 22003	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑄 ∈ 𝑋 ∧ 𝑆 ∈ ℝ^*) → (𝑥 ∈ (𝑄(ball‘𝐷)𝑆) ↔ (𝑥 ∈ 𝑋 ∧ (𝑄𝐷𝑥) < 𝑆)))
97	7, 9, 33, 96	syl3anc 1318	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥 ∈ (𝑄(ball‘𝐷)𝑆) ↔ (𝑥 ∈ 𝑋 ∧ (𝑄𝐷𝑥) < 𝑆)))
98	6, 95, 97	mpbir2and 959	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ (𝑄(ball‘𝐷)𝑆))
99	98	ex 449	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) → 𝑥 ∈ (𝑄(ball‘𝐷)𝑆)))
100	99	ssrdv 3574	1 ⊢ (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ⊆ wss 3540 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 +∞cpnf 9950 -∞cmnf 9951 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 -_𝑒cxne 11819 +_𝑒 cxad 11820 ∞Metcxmt 19552 ballcbl 19554

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-2 10956 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-psmet 19559 df-xmet 19560 df-bl 19562

This theorem is referenced by: blss2 22019 ssbl 22038

W3C validator