minveclem3b - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > minveclem3b		Structured version Visualization version GIF version

Theorem minveclem3b 25568

Description: Lemma for minvec 25576. The set of vectors within a fixed distance of the infimum forms a filter base. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 3-Oct-2020.)

**Hypotheses**
Ref	Expression
minvec.x	⊢ 𝑋 = (Base‘𝑈)
minvec.m	⊢ − = (-_g‘𝑈)
minvec.n	⊢ 𝑁 = (norm‘𝑈)
minvec.u	⊢ (𝜑 → 𝑈 ∈ ℂPreHil)
minvec.y	⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))
minvec.w	⊢ (𝜑 → (𝑈 ↾_s 𝑌) ∈ CMetSp)
minvec.a	⊢ (𝜑 → 𝐴 ∈ 𝑋)
minvec.j	⊢ 𝐽 = (TopOpen‘𝑈)
minvec.r	⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))
minvec.s	⊢ 𝑆 = inf(𝑅, ℝ, < )
minvec.d	⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))
minvec.f	⊢ 𝐹 = ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})

**Assertion**
Ref	Expression
minveclem3b	⊢ (𝜑 → 𝐹 ∈ (fBas‘𝑌))

Distinct variable groups: 𝑦, − 𝑦,𝑟,𝐴 𝐽,𝑟,𝑦 𝑦,𝐹 𝑦,𝑁 𝜑,𝑟,𝑦 𝑦,𝑅 𝑦,𝑈 𝑋,𝑟,𝑦 𝑌,𝑟,𝑦 𝐷,𝑟,𝑦 𝑆,𝑟,𝑦 Allowed substitution hints: 𝑅(𝑟) 𝑈(𝑟) 𝐹(𝑟) − (𝑟) 𝑁(𝑟)

Proof of Theorem minveclem3b Dummy variables 𝑤 𝑠 𝑡 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		minvec.f	. . 3 ⊢ 𝐹 = ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
2		ssrab2 4035	. . . . . 6 ⊢ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)} ⊆ 𝑌
3		minvec.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))
4	3	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑌 ∈ (LSubSp‘𝑈))
5		elpw2g 5305	. . . . . . 7 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → ({𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)} ∈ 𝒫 𝑌 ↔ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)} ⊆ 𝑌))
6	4, 5	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ({𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)} ∈ 𝒫 𝑌 ↔ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)} ⊆ 𝑌))
7	2, 6	mpbiri 261	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)} ∈ 𝒫 𝑌)
8	7	fmpttd 7112	. . . 4 ⊢ (𝜑 → (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}):ℝ⁺⟶𝒫 𝑌)
9	8	frnd 6716	. . 3 ⊢ (𝜑 → ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) ⊆ 𝒫 𝑌)
10	1, 9	eqsstrid 3976	. 2 ⊢ (𝜑 → 𝐹 ⊆ 𝒫 𝑌)
11		1rp 13021	. . . . . 6 ⊢ 1 ∈ ℝ⁺
12		eqid 2763	. . . . . . 7 ⊢ (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
13	12, 7	dmmptd 6682	. . . . . 6 ⊢ (𝜑 → dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = ℝ⁺)
14	11, 13	eleqtrrid 2870	. . . . 5 ⊢ (𝜑 → 1 ∈ dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}))
15	14	ne0d 4296	. . . 4 ⊢ (𝜑 → dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) ≠ ∅)
16		dm0rn0 5916	. . . . . 6 ⊢ (dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = ∅ ↔ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = ∅)
17	1	eqeq1i 2768	. . . . . 6 ⊢ (𝐹 = ∅ ↔ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = ∅)
18	16, 17	bitr4i 281	. . . . 5 ⊢ (dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = ∅ ↔ 𝐹 = ∅)
19	18	necon3bii 3010	. . . 4 ⊢ (dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) ≠ ∅ ↔ 𝐹 ≠ ∅)
20	15, 19	sylib 221	. . 3 ⊢ (𝜑 → 𝐹 ≠ ∅)
21		minvec.x	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑋 = (Base‘𝑈)
22		minvec.m	. . . . . . . . . . . . . . . . . 18 ⊢ − = (-_g‘𝑈)
23		minvec.n	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑁 = (norm‘𝑈)
24		minvec.u	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)
25		minvec.w	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑈 ↾_s 𝑌) ∈ CMetSp)
26		minvec.a	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
27		minvec.j	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐽 = (TopOpen‘𝑈)
28		minvec.r	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))
29		minvec.s	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑆 = inf(𝑅, ℝ, < )
30	21, 22, 23, 24, 3, 25, 26, 27, 28, 29	minveclem4c 25565	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆 ∈ ℝ)
31	30	resqcld 14163	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑆↑2) ∈ ℝ)
32		ltaddrp 13056	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆↑2) ∈ ℝ ∧ 𝑟 ∈ ℝ⁺) → (𝑆↑2) < ((𝑆↑2) + 𝑟))
33	31, 32	sylan 591	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (𝑆↑2) < ((𝑆↑2) + 𝑟))
34	31	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (𝑆↑2) ∈ ℝ)
35		rpre 13026	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ)
36	35	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ)
37	34, 36	readdcld 11239	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((𝑆↑2) + 𝑟) ∈ ℝ)
38	37	recnd 11238	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((𝑆↑2) + 𝑟) ∈ ℂ)
39	38	sqsqrtd 15495	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((√‘((𝑆↑2) + 𝑟))↑2) = ((𝑆↑2) + 𝑟))
40	33, 39	breqtrrd 5140	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (𝑆↑2) < ((√‘((𝑆↑2) + 𝑟))↑2))
41	21, 22, 23, 24, 3, 25, 26, 27, 28	minveclem1 25564	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
42	41	simp1d 1160	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑅 ⊆ ℝ)
43	42	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑅 ⊆ ℝ)
44	41	simp2d 1161	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑅 ≠ ∅)
45	44	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑅 ≠ ∅)
46		0re 11211	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ
47	41	simp3d 1162	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)
48		breq1 5113	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 0 → (𝑦 ≤ 𝑤 ↔ 0 ≤ 𝑤))
49	48	ralbidv 3188	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 0 → (∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤 ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
50	49	rspcev 3582	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℝ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) → ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤)
51	46, 47, 50	sylancr 598	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤)
52	51	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤)
53		infrecl 12198	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤) → inf(𝑅, ℝ, < ) ∈ ℝ)
54	43, 45, 52, 53	syl3anc 1398	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → inf(𝑅, ℝ, < ) ∈ ℝ)
55	29, 54	eqeltrid 2867	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑆 ∈ ℝ)
56		0red 11212	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ∈ ℝ)
57	55	sqge0d 14175	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ (𝑆↑2))
58	56, 34, 37, 57, 33	lelttrd 11369	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 < ((𝑆↑2) + 𝑟))
59	56, 37, 58	ltled 11359	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ ((𝑆↑2) + 𝑟))
60	37, 59	resqrtcld 15471	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (√‘((𝑆↑2) + 𝑟)) ∈ ℝ)
61	47	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)
62		infregelb 12200	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤) ∧ 0 ∈ ℝ) → (0 ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
63	43, 45, 52, 56, 62	syl31anc 1400	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (0 ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
64	61, 63	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ inf(𝑅, ℝ, < ))
65	64, 29	breqtrrdi 5154	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ 𝑆)
66	37, 59	sqrtge0d 15474	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ (√‘((𝑆↑2) + 𝑟)))
67	55, 60, 65, 66	lt2sqd 14294	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (𝑆 < (√‘((𝑆↑2) + 𝑟)) ↔ (𝑆↑2) < ((√‘((𝑆↑2) + 𝑟))↑2)))
68	40, 67	mpbird 260	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑆 < (√‘((𝑆↑2) + 𝑟)))
69	55, 60	ltnled 11358	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (𝑆 < (√‘((𝑆↑2) + 𝑟)) ↔ ¬ (√‘((𝑆↑2) + 𝑟)) ≤ 𝑆))
70	68, 69	mpbid 235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ¬ (√‘((𝑆↑2) + 𝑟)) ≤ 𝑆)
71	29	breq2i 5118	. . . . . . . . . . . . 13 ⊢ ((√‘((𝑆↑2) + 𝑟)) ≤ 𝑆 ↔ (√‘((𝑆↑2) + 𝑟)) ≤ inf(𝑅, ℝ, < ))
72		infregelb 12200	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤) ∧ (√‘((𝑆↑2) + 𝑟)) ∈ ℝ) → ((√‘((𝑆↑2) + 𝑟)) ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + 𝑟)) ≤ 𝑤))
73	43, 45, 52, 60, 72	syl31anc 1400	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((√‘((𝑆↑2) + 𝑟)) ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + 𝑟)) ≤ 𝑤))
74	28	raleqi 3321	. . . . . . . . . . . . . . 15 ⊢ (∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + 𝑟)) ≤ 𝑤 ↔ ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))(√‘((𝑆↑2) + 𝑟)) ≤ 𝑤)
75		fvex 6896	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁‘(𝐴 − 𝑦)) ∈ V
76	75	rgenw 3083	. . . . . . . . . . . . . . . 16 ⊢ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑦)) ∈ V
77		eqid 2763	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))
78		breq2 5114	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (𝑁‘(𝐴 − 𝑦)) → ((√‘((𝑆↑2) + 𝑟)) ≤ 𝑤 ↔ (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦))))
79	77, 78	ralrnmptw 7091	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑦)) ∈ V → (∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))(√‘((𝑆↑2) + 𝑟)) ≤ 𝑤 ↔ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦))))
80	76, 79	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))(√‘((𝑆↑2) + 𝑟)) ≤ 𝑤 ↔ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)))
81	74, 80	bitri 278	. . . . . . . . . . . . . 14 ⊢ (∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + 𝑟)) ≤ 𝑤 ↔ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)))
82	73, 81	bitrdi 290	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((√‘((𝑆↑2) + 𝑟)) ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦))))
83	71, 82	bitrid 286	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((√‘((𝑆↑2) + 𝑟)) ≤ 𝑆 ↔ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦))))
84	70, 83	mtbid 327	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ¬ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)))
85		rexnal 3117	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝑌 ¬ (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)) ↔ ¬ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)))
86	84, 85	sylibr 237	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ∃𝑦 ∈ 𝑌 ¬ (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)))
87	60	adantr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (√‘((𝑆↑2) + 𝑟)) ∈ ℝ)
88		cphngp 25313	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ NrmGrp)
89	24, 88	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑈 ∈ NrmGrp)
90		ngpms 24738	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ NrmGrp → 𝑈 ∈ MetSp)
91		minvec.d	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))
92	21, 91	msmet 24595	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ MetSp → 𝐷 ∈ (Met‘𝑋))
93	89, 90, 92	3syl 19	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
94	93	ad2antrr 738	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝐷 ∈ (Met‘𝑋))
95	26	ad2antrr 738	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋)
96		eqid 2763	. . . . . . . . . . . . . . . . . . 19 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
97	21, 96	lssss 21038	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋)
98	4, 97	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑌 ⊆ 𝑋)
99	98	sselda 3938	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
100		metcl 24470	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴𝐷𝑦) ∈ ℝ)
101	94, 95, 99, 100	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑦) ∈ ℝ)
102	66	adantr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 0 ≤ (√‘((𝑆↑2) + 𝑟)))
103		metge0 24483	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝑦))
104	94, 95, 99, 103	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 0 ≤ (𝐴𝐷𝑦))
105	87, 101, 102, 104	le2sqd 14295	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((√‘((𝑆↑2) + 𝑟)) ≤ (𝐴𝐷𝑦) ↔ ((√‘((𝑆↑2) + 𝑟))↑2) ≤ ((𝐴𝐷𝑦)↑2)))
106	91	oveqi 7425	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴𝐷𝑦) = (𝐴((dist‘𝑈) ↾ (𝑋 × 𝑋))𝑦)
107	95, 99	ovresd 7579	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝐴((dist‘𝑈) ↾ (𝑋 × 𝑋))𝑦) = (𝐴(dist‘𝑈)𝑦))
108	106, 107	eqtrid 2810	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑦) = (𝐴(dist‘𝑈)𝑦))
109	89	ad2antrr 738	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝑈 ∈ NrmGrp)
110		eqid 2763	. . . . . . . . . . . . . . . . . 18 ⊢ (dist‘𝑈) = (dist‘𝑈)
111	23, 21, 22, 110	ngpds 24742	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴(dist‘𝑈)𝑦) = (𝑁‘(𝐴 − 𝑦)))
112	109, 95, 99, 111	syl3anc 1398	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝐴(dist‘𝑈)𝑦) = (𝑁‘(𝐴 − 𝑦)))
113	108, 112	eqtrd 2798	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑦) = (𝑁‘(𝐴 − 𝑦)))
114	113	breq2d 5122	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((√‘((𝑆↑2) + 𝑟)) ≤ (𝐴𝐷𝑦) ↔ (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦))))
115	39	adantr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((√‘((𝑆↑2) + 𝑟))↑2) = ((𝑆↑2) + 𝑟))
116	115	breq1d 5120	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (((√‘((𝑆↑2) + 𝑟))↑2) ≤ ((𝐴𝐷𝑦)↑2) ↔ ((𝑆↑2) + 𝑟) ≤ ((𝐴𝐷𝑦)↑2)))
117	105, 114, 116	3bitr3d 312	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)) ↔ ((𝑆↑2) + 𝑟) ≤ ((𝐴𝐷𝑦)↑2)))
118	117	notbid 321	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (¬ (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)) ↔ ¬ ((𝑆↑2) + 𝑟) ≤ ((𝐴𝐷𝑦)↑2)))
119	37	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((𝑆↑2) + 𝑟) ∈ ℝ)
120	101	resqcld 14163	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((𝐴𝐷𝑦)↑2) ∈ ℝ)
121	119, 120	letrid 11363	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (((𝑆↑2) + 𝑟) ≤ ((𝐴𝐷𝑦)↑2) ∨ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)))
122	121	ord 877	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (¬ ((𝑆↑2) + 𝑟) ≤ ((𝐴𝐷𝑦)↑2) → ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)))
123	118, 122	sylbid 243	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (¬ (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)) → ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)))
124	123	reximdva 3178	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (∃𝑦 ∈ 𝑌 ¬ (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)) → ∃𝑦 ∈ 𝑌 ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)))
125	86, 124	mpd 16	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ∃𝑦 ∈ 𝑌 ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟))
126		rabn0 4347	. . . . . . . . 9 ⊢ ({𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)} ≠ ∅ ↔ ∃𝑦 ∈ 𝑌 ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟))
127	125, 126	sylibr 237	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)} ≠ ∅)
128	127	necomd 3013	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ∅ ≠ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
129	128	neneqd 2963	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ¬ ∅ = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
130	129	nrexdv 3160	. . . . 5 ⊢ (𝜑 → ¬ ∃𝑟 ∈ ℝ⁺ ∅ = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
131	1	eleq2i 2855	. . . . . 6 ⊢ (∅ ∈ 𝐹 ↔ ∅ ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}))
132		0ex 5271	. . . . . . 7 ⊢ ∅ ∈ V
133	12	elrnmpt 5950	. . . . . . 7 ⊢ (∅ ∈ V → (∅ ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) ↔ ∃𝑟 ∈ ℝ⁺ ∅ = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}))
134	132, 133	ax-mp 5	. . . . . 6 ⊢ (∅ ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) ↔ ∃𝑟 ∈ ℝ⁺ ∅ = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
135	131, 134	bitri 278	. . . . 5 ⊢ (∅ ∈ 𝐹 ↔ ∃𝑟 ∈ ℝ⁺ ∅ = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
136	130, 135	sylnibr 332	. . . 4 ⊢ (𝜑 → ¬ ∅ ∈ 𝐹)
137		df-nel 3065	. . . 4 ⊢ (∅ ∉ 𝐹 ↔ ¬ ∅ ∈ 𝐹)
138	136, 137	sylibr 237	. . 3 ⊢ (𝜑 → ∅ ∉ 𝐹)
139	55	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝑆 ∈ ℝ)
140	139	resqcld 14163	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝑆↑2) ∈ ℝ)
141	36	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝑟 ∈ ℝ)
142	120, 140, 141	lesubadd2d 11814	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟 ↔ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)))
143	142	rabbidva 3422	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟} = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
144	143	mpteq2dva 5205	. . . . . . . . . 10 ⊢ (𝜑 → (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) = (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}))
145	144	rneqd 5930	. . . . . . . . 9 ⊢ (𝜑 → ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) = ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}))
146	1, 145	eqtr4id 2817	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}))
147	146	eleq2d 2849	. . . . . . 7 ⊢ (𝜑 → (𝑢 ∈ 𝐹 ↔ 𝑢 ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟})))
148		breq2 5114	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑠 → ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟 ↔ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠))
149	148	rabbidv 3423	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟} = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠})
150	149	cbvmptv 5216	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) = (𝑠 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠})
151	150	elrnmpt 5950	. . . . . . . 8 ⊢ (𝑢 ∈ V → (𝑢 ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) ↔ ∃𝑠 ∈ ℝ⁺ 𝑢 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠}))
152	151	elv 3460	. . . . . . 7 ⊢ (𝑢 ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) ↔ ∃𝑠 ∈ ℝ⁺ 𝑢 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠})
153	147, 152	bitrdi 290	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ 𝐹 ↔ ∃𝑠 ∈ ℝ⁺ 𝑢 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠}))
154	146	eleq2d 2849	. . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ 𝐹 ↔ 𝑣 ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟})))
155		breq2 5114	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑡 → ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟 ↔ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡))
156	155	rabbidv 3423	. . . . . . . . . 10 ⊢ (𝑟 = 𝑡 → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟} = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡})
157	156	cbvmptv 5216	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) = (𝑡 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡})
158	157	elrnmpt 5950	. . . . . . . 8 ⊢ (𝑣 ∈ V → (𝑣 ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) ↔ ∃𝑡 ∈ ℝ⁺ 𝑣 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡}))
159	158	elv 3460	. . . . . . 7 ⊢ (𝑣 ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) ↔ ∃𝑡 ∈ ℝ⁺ 𝑣 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡})
160	154, 159	bitrdi 290	. . . . . 6 ⊢ (𝜑 → (𝑣 ∈ 𝐹 ↔ ∃𝑡 ∈ ℝ⁺ 𝑣 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡}))
161	153, 160	anbi12d 643	. . . . 5 ⊢ (𝜑 → ((𝑢 ∈ 𝐹 ∧ 𝑣 ∈ 𝐹) ↔ (∃𝑠 ∈ ℝ⁺ 𝑢 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠} ∧ ∃𝑡 ∈ ℝ⁺ 𝑣 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡})))
162		reeanv 3237	. . . . . 6 ⊢ (∃𝑠 ∈ ℝ⁺ ∃𝑡 ∈ ℝ⁺ (𝑢 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠} ∧ 𝑣 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡}) ↔ (∃𝑠 ∈ ℝ⁺ 𝑢 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠} ∧ ∃𝑡 ∈ ℝ⁺ 𝑣 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡}))
163	93	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝐷 ∈ (Met‘𝑋))
164	26	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋)
165	3, 97	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
166	165	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → 𝑌 ⊆ 𝑋)
167	166	sselda 3938	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
168	163, 164, 167, 100	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑦) ∈ ℝ)
169	168	resqcld 14163	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → ((𝐴𝐷𝑦)↑2) ∈ ℝ)
170	31	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → (𝑆↑2) ∈ ℝ)
171	169, 170	resubcld 11643	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ∈ ℝ)
172		simplrl 788	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝑠 ∈ ℝ⁺)
173	172	rpred 13061	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝑠 ∈ ℝ)
174		simplrr 789	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝑡 ∈ ℝ⁺)
175	174	rpred 13061	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝑡 ∈ ℝ)
176		lemin 13219	. . . . . . . . . . . 12 ⊢ (((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ∈ ℝ ∧ 𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ if(𝑠 ≤ 𝑡, 𝑠, 𝑡) ↔ ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠 ∧ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡)))
177	171, 173, 175, 176	syl3anc 1398	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ if(𝑠 ≤ 𝑡, 𝑠, 𝑡) ↔ ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠 ∧ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡)))
178	177	rabbidva 3422	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ if(𝑠 ≤ 𝑡, 𝑠, 𝑡)} = {𝑦 ∈ 𝑌 ∣ ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠 ∧ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡)})
179		ifcl 4534	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺) → if(𝑠 ≤ 𝑡, 𝑠, 𝑡) ∈ ℝ⁺)
180	3	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → 𝑌 ∈ (LSubSp‘𝑈))
181		rabexg 5309	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ if(𝑠 ≤ 𝑡, 𝑠, 𝑡)} ∈ V)
182	180, 181	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ if(𝑠 ≤ 𝑡, 𝑠, 𝑡)} ∈ V)
183		eqid 2763	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) = (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟})
184		breq2 5114	. . . . . . . . . . . . . 14 ⊢ (𝑟 = if(𝑠 ≤ 𝑡, 𝑠, 𝑡) → ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟 ↔ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ if(𝑠 ≤ 𝑡, 𝑠, 𝑡)))
185	184	rabbidv 3423	. . . . . . . . . . . . 13 ⊢ (𝑟 = if(𝑠 ≤ 𝑡, 𝑠, 𝑡) → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟} = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ if(𝑠 ≤ 𝑡, 𝑠, 𝑡)})
186	183, 185	elrnmpt1s 5951	. . . . . . . . . . . 12 ⊢ ((if(𝑠 ≤ 𝑡, 𝑠, 𝑡) ∈ ℝ⁺ ∧ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ if(𝑠 ≤ 𝑡, 𝑠, 𝑡)} ∈ V) → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ if(𝑠 ≤ 𝑡, 𝑠, 𝑡)} ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}))
187	179, 182, 186	syl2an2 698	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ if(𝑠 ≤ 𝑡, 𝑠, 𝑡)} ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}))
188	146	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → 𝐹 = ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}))
189	187, 188	eleqtrrd 2866	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ if(𝑠 ≤ 𝑡, 𝑠, 𝑡)} ∈ 𝐹)
190	178, 189	eqeltrrd 2864	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → {𝑦 ∈ 𝑌 ∣ ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠 ∧ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡)} ∈ 𝐹)
191		ineq12 4169	. . . . . . . . . . 11 ⊢ ((𝑢 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠} ∧ 𝑣 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡}) → (𝑢 ∩ 𝑣) = ({𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠} ∩ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡}))
192		inrab 4270	. . . . . . . . . . 11 ⊢ ({𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠} ∩ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡}) = {𝑦 ∈ 𝑌 ∣ ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠 ∧ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡)}
193	191, 192	eqtrdi 2814	. . . . . . . . . 10 ⊢ ((𝑢 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠} ∧ 𝑣 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡}) → (𝑢 ∩ 𝑣) = {𝑦 ∈ 𝑌 ∣ ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠 ∧ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡)})
194	193	eleq1d 2848	. . . . . . . . 9 ⊢ ((𝑢 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠} ∧ 𝑣 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡}) → ((𝑢 ∩ 𝑣) ∈ 𝐹 ↔ {𝑦 ∈ 𝑌 ∣ ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠 ∧ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡)} ∈ 𝐹))
195	190, 194	syl5ibrcom 250	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → ((𝑢 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠} ∧ 𝑣 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡}) → (𝑢 ∩ 𝑣) ∈ 𝐹))
196		vex 3459	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
197	196	inex1 5287	. . . . . . . . . 10 ⊢ (𝑢 ∩ 𝑣) ∈ V
198	197	pwid 4586	. . . . . . . . 9 ⊢ (𝑢 ∩ 𝑣) ∈ 𝒫 (𝑢 ∩ 𝑣)
199		inelcm 4426	. . . . . . . . 9 ⊢ (((𝑢 ∩ 𝑣) ∈ 𝐹 ∧ (𝑢 ∩ 𝑣) ∈ 𝒫 (𝑢 ∩ 𝑣)) → (𝐹 ∩ 𝒫 (𝑢 ∩ 𝑣)) ≠ ∅)
200	198, 199	mpan2 703	. . . . . . . 8 ⊢ ((𝑢 ∩ 𝑣) ∈ 𝐹 → (𝐹 ∩ 𝒫 (𝑢 ∩ 𝑣)) ≠ ∅)
201	195, 200	syl6 36	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → ((𝑢 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠} ∧ 𝑣 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡}) → (𝐹 ∩ 𝒫 (𝑢 ∩ 𝑣)) ≠ ∅))
202	201	rexlimdvva 3222	. . . . . 6 ⊢ (𝜑 → (∃𝑠 ∈ ℝ⁺ ∃𝑡 ∈ ℝ⁺ (𝑢 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠} ∧ 𝑣 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡}) → (𝐹 ∩ 𝒫 (𝑢 ∩ 𝑣)) ≠ ∅))
203	162, 202	biimtrrid 246	. . . . 5 ⊢ (𝜑 → ((∃𝑠 ∈ ℝ⁺ 𝑢 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠} ∧ ∃𝑡 ∈ ℝ⁺ 𝑣 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡}) → (𝐹 ∩ 𝒫 (𝑢 ∩ 𝑣)) ≠ ∅))
204	161, 203	sylbid 243	. . . 4 ⊢ (𝜑 → ((𝑢 ∈ 𝐹 ∧ 𝑣 ∈ 𝐹) → (𝐹 ∩ 𝒫 (𝑢 ∩ 𝑣)) ≠ ∅))
205	204	ralrimivv 3206	. . 3 ⊢ (𝜑 → ∀𝑢 ∈ 𝐹 ∀𝑣 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑢 ∩ 𝑣)) ≠ ∅)
206	20, 138, 205	3jca 1146	. 2 ⊢ (𝜑 → (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑢 ∈ 𝐹 ∀𝑣 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑢 ∩ 𝑣)) ≠ ∅))
207		isfbas 23967	. . 3 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → (𝐹 ∈ (fBas‘𝑌) ↔ (𝐹 ⊆ 𝒫 𝑌 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑢 ∈ 𝐹 ∀𝑣 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑢 ∩ 𝑣)) ≠ ∅))))
208	3, 207	syl 18	. 2 ⊢ (𝜑 → (𝐹 ∈ (fBas‘𝑌) ↔ (𝐹 ⊆ 𝒫 𝑌 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑢 ∈ 𝐹 ∀𝑣 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑢 ∩ 𝑣)) ≠ ∅))))
209	10, 206, 208	mpbir2and 725	1 ⊢ (𝜑 → 𝐹 ∈ (fBas‘𝑌))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∉ wnel 3064 ∀wral 3079 ∃wrex 3089 {crab 3416 Vcvv 3455 ∩ cin 3905 ⊆ wss 3906 ∅c0 4287 ifcif 4488 𝒫 cpw 4563 class class class wbr 5110 ↦ cmpt 5193 × cxp 5661 dom cdm 5663 ran crn 5664 ↾ cres 5665 ‘cfv 6538 (class class class)co 7412 infcinf 9402 ℝcr 11100 0cc0 11101 1c1 11102 + caddc 11104 < clt 11244 ≤ cle 11245 − cmin 11442 2c2 12296 ℝ⁺crp 13017 ↑cexp 14099 √csqrt 15286 Basecbs 17270 ↾_s cress 17291 distcds 17320 TopOpenctopn 17475 -_gcsg 19003 LSubSpclss 21033 Metcmet 21489 fBascfbas 21491 MetSpcms 24456 normcnm 24714 NrmGrpcngp 24715 ℂPreHilccph 25306 CMetSpccms 25472

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-er 8695 df-map 8827 df-en 8945 df-dom 8946 df-sdom 8947 df-sup 9403 df-inf 9404 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-n0 12506 df-z 12593 df-uz 12864 df-q 12974 df-rp 13018 df-xneg 13138 df-xadd 13139 df-xmul 13140 df-seq 14040 df-exp 14100 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-0g 17495 df-topgen 17497 df-mgm 18699 df-sgrp 18778 df-mnd 18794 df-grp 19004 df-minusg 19005 df-sbg 19006 df-lmod 20964 df-lss 21034 df-psmet 21495 df-xmet 21496 df-met 21497 df-bl 21498 df-mopn 21499 df-fbas 21500 df-top 23032 df-topon 23049 df-topsp 23071 df-bases 23084 df-xms 24458 df-ms 24459 df-nm 24720 df-ngp 24721 df-nlm 24724 df-cph 25308

This theorem is referenced by: minveclem3 25569 minveclem4a 25570 minveclem4 25572

W3C validator