xblss2 - Intuitionistic Logic Explorer

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Theorem xblss2 15128

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 15130 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 15114	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
5	1, 2, 3, 4	syl3anc 1273	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
6	5	simprbda 383	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ 𝑋)
7	1	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝐷 ∈ (∞Met‘𝑋))
8		xblss2.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ 𝑋)
9	8	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑄 ∈ 𝑋)
10		xmetcl 15075	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑄 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑄𝐷𝑥) ∈ ℝ^*)
11	7, 9, 6, 10	syl3anc 1273	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑄𝐷𝑥) ∈ ℝ^*)
12	11	adantr 276	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → (𝑄𝐷𝑥) ∈ ℝ^*)
13		xblss2.6	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃𝐷𝑄) ∈ ℝ)
14	13	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑄) ∈ ℝ)
15	14	rexrd 8228	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑄) ∈ ℝ^*)
16	3	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑅 ∈ ℝ^*)
17	15, 16	xaddcld 10118	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) +_𝑒 𝑅) ∈ ℝ^*)
18	17	adantr 276	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → ((𝑃𝐷𝑄) +_𝑒 𝑅) ∈ ℝ^*)
19		xblss2.5	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ ℝ^*)
20	19	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → 𝑆 ∈ ℝ^*)
21	2	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑃 ∈ 𝑋)
22		xmetcl 15075	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑃𝐷𝑥) ∈ ℝ^*)
23	7, 21, 6, 22	syl3anc 1273	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑥) ∈ ℝ^*)
24	15, 23	xaddcld 10118	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) +_𝑒 (𝑃𝐷𝑥)) ∈ ℝ^*)
25		xmettri2 15084	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑄𝐷𝑥) ≤ ((𝑃𝐷𝑄) +_𝑒 (𝑃𝐷𝑥)))
26	7, 21, 9, 6, 25	syl13anc 1275	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑄𝐷𝑥) ≤ ((𝑃𝐷𝑄) +_𝑒 (𝑃𝐷𝑥)))
27	5	simplbda 384	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑥) < 𝑅)
28		xltadd2 10111	. . . . . . . . . 10 ⊢ (((𝑃𝐷𝑥) ∈ ℝ^* ∧ 𝑅 ∈ ℝ^* ∧ (𝑃𝐷𝑄) ∈ ℝ) → ((𝑃𝐷𝑥) < 𝑅 ↔ ((𝑃𝐷𝑄) +_𝑒 (𝑃𝐷𝑥)) < ((𝑃𝐷𝑄) +_𝑒 𝑅)))
29	23, 16, 14, 28	syl3anc 1273	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑥) < 𝑅 ↔ ((𝑃𝐷𝑄) +_𝑒 (𝑃𝐷𝑥)) < ((𝑃𝐷𝑄) +_𝑒 𝑅)))
30	27, 29	mpbid 147	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) +_𝑒 (𝑃𝐷𝑥)) < ((𝑃𝐷𝑄) +_𝑒 𝑅))
31	11, 24, 17, 26, 30	xrlelttrd 10044	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑄𝐷𝑥) < ((𝑃𝐷𝑄) +_𝑒 𝑅))
32	31	adantr 276	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → (𝑄𝐷𝑥) < ((𝑃𝐷𝑄) +_𝑒 𝑅))
33	19	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑆 ∈ ℝ^*)
34	16	xnegcld 10089	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → -_𝑒𝑅 ∈ ℝ^*)
35	33, 34	xaddcld 10118	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑆 +_𝑒 -_𝑒𝑅) ∈ ℝ^*)
36		xblss2.7	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃𝐷𝑄) ≤ (𝑆 +_𝑒 -_𝑒𝑅))
37	36	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑄) ≤ (𝑆 +_𝑒 -_𝑒𝑅))
38		xleadd1a 10107	. . . . . . . . 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 1276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) +_𝑒 𝑅) ≤ ((𝑆 +_𝑒 -_𝑒𝑅) +_𝑒 𝑅))
40	39	adantr 276	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → ((𝑃𝐷𝑄) +_𝑒 𝑅) ≤ ((𝑆 +_𝑒 -_𝑒𝑅) +_𝑒 𝑅))
41		xnpcan 10106	. . . . . . . 8 ⊢ ((𝑆 ∈ ℝ^* ∧ 𝑅 ∈ ℝ) → ((𝑆 +_𝑒 -_𝑒𝑅) +_𝑒 𝑅) = 𝑆)
42	33, 41	sylan 283	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → ((𝑆 +_𝑒 -_𝑒𝑅) +_𝑒 𝑅) = 𝑆)
43	40, 42	breqtrd 4114	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → ((𝑃𝐷𝑄) +_𝑒 𝑅) ≤ 𝑆)
44	12, 18, 20, 32, 43	xrltletrd 10045	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 ∈ ℝ) → (𝑄𝐷𝑥) < 𝑆)
45	27	adantr 276	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑃𝐷𝑥) < 𝑅)
46	36	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑃𝐷𝑄) ≤ (𝑆 +_𝑒 -_𝑒𝑅))
47	19	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑆 ∈ ℝ^*)
48		xrpnfdc 10076	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ ℝ^* → DECID 𝑆 = +∞)
49	47, 48	syl 14	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → DECID 𝑆 = +∞)
50		0xr 8225	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ^*
51	50	a1i 9	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ∈ ℝ^*)
52		xmetge0 15088	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → 0 ≤ (𝑃𝐷𝑄))
53	7, 21, 9, 52	syl3anc 1273	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ (𝑃𝐷𝑄))
54	51, 15, 35, 53, 37	xrletrd 10046	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ (𝑆 +_𝑒 -_𝑒𝑅))
55		ge0nemnf 10058	. . . . . . . . . . . . . 14 ⊢ (((𝑆 +_𝑒 -_𝑒𝑅) ∈ ℝ^* ∧ 0 ≤ (𝑆 +_𝑒 -_𝑒𝑅)) → (𝑆 +_𝑒 -_𝑒𝑅) ≠ -∞)
56	35, 54, 55	syl2anc 411	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑆 +_𝑒 -_𝑒𝑅) ≠ -∞)
57	56	adantr 276	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 +_𝑒 -_𝑒𝑅) ≠ -∞)
58		xaddmnf1 10082	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ ℝ^* ∧ 𝑆 ≠ +∞) → (𝑆 +_𝑒 -∞) = -∞)
59	58	ex 115	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ ℝ^* → (𝑆 ≠ +∞ → (𝑆 +_𝑒 -∞) = -∞))
60	47, 59	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 ≠ +∞ → (𝑆 +_𝑒 -∞) = -∞))
61		simpr 110	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑅 = +∞)
62		xnegeq 10061	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 = +∞ → -_𝑒𝑅 = -_𝑒+∞)
63	61, 62	syl 14	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → -_𝑒𝑅 = -_𝑒+∞)
64		xnegpnf 10062	. . . . . . . . . . . . . . . . . 18 ⊢ -_𝑒+∞ = -∞
65	63, 64	eqtrdi 2280	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → -_𝑒𝑅 = -∞)
66	65	oveq2d 6033	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 +_𝑒 -_𝑒𝑅) = (𝑆 +_𝑒 -∞))
67	66	eqeq1d 2240	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑆 +_𝑒 -_𝑒𝑅) = -∞ ↔ (𝑆 +_𝑒 -∞) = -∞))
68	60, 67	sylibrd 169	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 ≠ +∞ → (𝑆 +_𝑒 -_𝑒𝑅) = -∞))
69	68	a1d 22	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (DECID 𝑆 = +∞ → (𝑆 ≠ +∞ → (𝑆 +_𝑒 -_𝑒𝑅) = -∞)))
70	69	necon1ddc 2480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (DECID 𝑆 = +∞ → ((𝑆 +_𝑒 -_𝑒𝑅) ≠ -∞ → 𝑆 = +∞)))
71	49, 57, 70	mp2d 47	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑆 = +∞)
72	71, 65	oveq12d 6035	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 +_𝑒 -_𝑒𝑅) = (+∞ +_𝑒 -∞))
73		pnfaddmnf 10084	. . . . . . . . . 10 ⊢ (+∞ +_𝑒 -∞) = 0
74	72, 73	eqtrdi 2280	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑆 +_𝑒 -_𝑒𝑅) = 0)
75	46, 74	breqtrd 4114	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑃𝐷𝑄) ≤ 0)
76	53	biantrud 304	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) ≤ 0 ↔ ((𝑃𝐷𝑄) ≤ 0 ∧ 0 ≤ (𝑃𝐷𝑄))))
77		xrletri3 10038	. . . . . . . . . . 11 ⊢ (((𝑃𝐷𝑄) ∈ ℝ^* ∧ 0 ∈ ℝ^*) → ((𝑃𝐷𝑄) = 0 ↔ ((𝑃𝐷𝑄) ≤ 0 ∧ 0 ≤ (𝑃𝐷𝑄))))
78	15, 50, 77	sylancl 413	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) = 0 ↔ ((𝑃𝐷𝑄) ≤ 0 ∧ 0 ≤ (𝑃𝐷𝑄))))
79		xmeteq0 15082	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → ((𝑃𝐷𝑄) = 0 ↔ 𝑃 = 𝑄))
80	7, 21, 9, 79	syl3anc 1273	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) = 0 ↔ 𝑃 = 𝑄))
81	76, 78, 80	3bitr2d 216	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → ((𝑃𝐷𝑄) ≤ 0 ↔ 𝑃 = 𝑄))
82	81	adantr 276	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → ((𝑃𝐷𝑄) ≤ 0 ↔ 𝑃 = 𝑄))
83	75, 82	mpbid 147	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑃 = 𝑄)
84	83	oveq1d 6032	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑃𝐷𝑥) = (𝑄𝐷𝑥))
85	61, 71	eqtr4d 2267	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → 𝑅 = 𝑆)
86	45, 84, 85	3brtr3d 4119	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) ∧ 𝑅 = +∞) → (𝑄𝐷𝑥) < 𝑆)
87		xmetge0 15088	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → 0 ≤ (𝑃𝐷𝑥))
88	7, 21, 6, 87	syl3anc 1273	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ (𝑃𝐷𝑥))
89	51, 23, 16, 88, 27	xrlelttrd 10044	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 < 𝑅)
90	51, 16, 89	xrltled 10033	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ 𝑅)
91		ge0nemnf 10058	. . . . . . . 8 ⊢ ((𝑅 ∈ ℝ^* ∧ 0 ≤ 𝑅) → 𝑅 ≠ -∞)
92	16, 90, 91	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑅 ≠ -∞)
93	16, 92	jca 306	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑅 ∈ ℝ^* ∧ 𝑅 ≠ -∞))
94		xrnemnf 10011	. . . . . 6 ⊢ ((𝑅 ∈ ℝ^* ∧ 𝑅 ≠ -∞) ↔ (𝑅 ∈ ℝ ∨ 𝑅 = +∞))
95	93, 94	sylib 122	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑅 ∈ ℝ ∨ 𝑅 = +∞))
96	44, 86, 95	mpjaodan 805	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑄𝐷𝑥) < 𝑆)
97		elbl 15114	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑄 ∈ 𝑋 ∧ 𝑆 ∈ ℝ^*) → (𝑥 ∈ (𝑄(ball‘𝐷)𝑆) ↔ (𝑥 ∈ 𝑋 ∧ (𝑄𝐷𝑥) < 𝑆)))
98	7, 9, 33, 97	syl3anc 1273	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥 ∈ (𝑄(ball‘𝐷)𝑆) ↔ (𝑥 ∈ 𝑋 ∧ (𝑄𝐷𝑥) < 𝑆)))
99	6, 96, 98	mpbir2and 952	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ (𝑄(ball‘𝐷)𝑆))
100	99	ex 115	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) → 𝑥 ∈ (𝑄(ball‘𝐷)𝑆)))
101	100	ssrdv 3233	1 ⊢ (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 715 DECID wdc 841 = wceq 1397 ∈ wcel 2202 ≠ wne 2402 ⊆ wss 3200 class class class wbr 4088 ‘cfv 5326 (class class class)co 6017 ℝcr 8030 0cc0 8031 +∞cpnf 8210 -∞cmnf 8211 ℝ^*cxr 8212 < clt 8213 ≤ cle 8214 -_𝑒cxne 10003 +_𝑒 cxad 10004 ∞Metcxmet 14549 ballcbl 14551

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-mulrcl 8130 ax-addcom 8131 ax-mulcom 8132 ax-addass 8133 ax-mulass 8134 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-1rid 8138 ax-0id 8139 ax-rnegex 8140 ax-precex 8141 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145 ax-pre-apti 8146 ax-pre-ltadd 8147 ax-pre-mulgt0 8148

This theorem depends on definitions: df-bi 117 df-stab 838 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-id 4390 df-po 4393 df-iso 4394 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-fv 5334 df-riota 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-1st 6302 df-2nd 6303 df-map 6818 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219 df-sub 8351 df-neg 8352 df-2 9201 df-xneg 10006 df-xadd 10007 df-psmet 14556 df-xmet 14557 df-bl 14559

This theorem is referenced by: blss2 15130 ssbl 15149

W3C validator