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Theorem minveclem3b 25356

Description: Lemma for minvec 25364. 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 4029	. . . . . 6 ⊢ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)} ⊆ 𝑌
3		minvec.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))
4	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑌 ∈ (LSubSp‘𝑈))
5		elpw2g 5273	. . . . . . 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 17	. . . . . 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 258	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)} ∈ 𝒫 𝑌)
8	7	fmpttd 7054	. . . 4 ⊢ (𝜑 → (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}):ℝ⁺⟶𝒫 𝑌)
9	8	frnd 6664	. . 3 ⊢ (𝜑 → ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) ⊆ 𝒫 𝑌)
10	1, 9	eqsstrid 3969	. 2 ⊢ (𝜑 → 𝐹 ⊆ 𝒫 𝑌)
11		1rp 12896	. . . . . 6 ⊢ 1 ∈ ℝ⁺
12		eqid 2733	. . . . . . 7 ⊢ (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
13	12, 7	dmmptd 6631	. . . . . 6 ⊢ (𝜑 → dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = ℝ⁺)
14	11, 13	eleqtrrid 2840	. . . . 5 ⊢ (𝜑 → 1 ∈ dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}))
15	14	ne0d 4291	. . . 4 ⊢ (𝜑 → dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) ≠ ∅)
16		dm0rn0 5868	. . . . . 6 ⊢ (dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = ∅ ↔ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = ∅)
17	1	eqeq1i 2738	. . . . . 6 ⊢ (𝐹 = ∅ ↔ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = ∅)
18	16, 17	bitr4i 278	. . . . 5 ⊢ (dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = ∅ ↔ 𝐹 = ∅)
19	18	necon3bii 2981	. . . 4 ⊢ (dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) ≠ ∅ ↔ 𝐹 ≠ ∅)
20	15, 19	sylib 218	. . 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 25353	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆 ∈ ℝ)
31	30	resqcld 14034	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑆↑2) ∈ ℝ)
32		ltaddrp 12931	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆↑2) ∈ ℝ ∧ 𝑟 ∈ ℝ⁺) → (𝑆↑2) < ((𝑆↑2) + 𝑟))
33	31, 32	sylan 580	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (𝑆↑2) < ((𝑆↑2) + 𝑟))
34	31	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (𝑆↑2) ∈ ℝ)
35		rpre 12901	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ)
36	35	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ)
37	34, 36	readdcld 11148	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((𝑆↑2) + 𝑟) ∈ ℝ)
38	37	recnd 11147	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((𝑆↑2) + 𝑟) ∈ ℂ)
39	38	sqsqrtd 15351	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((√‘((𝑆↑2) + 𝑟))↑2) = ((𝑆↑2) + 𝑟))
40	33, 39	breqtrrd 5121	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (𝑆↑2) < ((√‘((𝑆↑2) + 𝑟))↑2))
41	21, 22, 23, 24, 3, 25, 26, 27, 28	minveclem1 25352	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
42	41	simp1d 1142	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑅 ⊆ ℝ)
43	42	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑅 ⊆ ℝ)
44	41	simp2d 1143	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑅 ≠ ∅)
45	44	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑅 ≠ ∅)
46		0re 11121	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ
47	41	simp3d 1144	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)
48		breq1 5096	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 0 → (𝑦 ≤ 𝑤 ↔ 0 ≤ 𝑤))
49	48	ralbidv 3156	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 0 → (∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤 ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
50	49	rspcev 3573	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℝ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) → ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤)
51	46, 47, 50	sylancr 587	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤)
52	51	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤)
53		infrecl 12111	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤) → inf(𝑅, ℝ, < ) ∈ ℝ)
54	43, 45, 52, 53	syl3anc 1373	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → inf(𝑅, ℝ, < ) ∈ ℝ)
55	29, 54	eqeltrid 2837	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑆 ∈ ℝ)
56		0red 11122	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ∈ ℝ)
57	55	sqge0d 14046	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ (𝑆↑2))
58	56, 34, 37, 57, 33	lelttrd 11278	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 < ((𝑆↑2) + 𝑟))
59	56, 37, 58	ltled 11268	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ ((𝑆↑2) + 𝑟))
60	37, 59	resqrtcld 15327	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (√‘((𝑆↑2) + 𝑟)) ∈ ℝ)
61	47	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)
62		infregelb 12113	. . . . . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (0 ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
64	61, 63	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ inf(𝑅, ℝ, < ))
65	64, 29	breqtrrdi 5135	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ 𝑆)
66	37, 59	sqrtge0d 15330	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ (√‘((𝑆↑2) + 𝑟)))
67	55, 60, 65, 66	lt2sqd 14165	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (𝑆 < (√‘((𝑆↑2) + 𝑟)) ↔ (𝑆↑2) < ((√‘((𝑆↑2) + 𝑟))↑2)))
68	40, 67	mpbird 257	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑆 < (√‘((𝑆↑2) + 𝑟)))
69	55, 60	ltnled 11267	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (𝑆 < (√‘((𝑆↑2) + 𝑟)) ↔ ¬ (√‘((𝑆↑2) + 𝑟)) ≤ 𝑆))
70	68, 69	mpbid 232	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ¬ (√‘((𝑆↑2) + 𝑟)) ≤ 𝑆)
71	29	breq2i 5101	. . . . . . . . . . . . 13 ⊢ ((√‘((𝑆↑2) + 𝑟)) ≤ 𝑆 ↔ (√‘((𝑆↑2) + 𝑟)) ≤ inf(𝑅, ℝ, < ))
72		infregelb 12113	. . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((√‘((𝑆↑2) + 𝑟)) ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + 𝑟)) ≤ 𝑤))
74	28	raleqi 3291	. . . . . . . . . . . . . . 15 ⊢ (∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + 𝑟)) ≤ 𝑤 ↔ ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))(√‘((𝑆↑2) + 𝑟)) ≤ 𝑤)
75		fvex 6841	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁‘(𝐴 − 𝑦)) ∈ V
76	75	rgenw 3052	. . . . . . . . . . . . . . . 16 ⊢ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑦)) ∈ V
77		eqid 2733	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))
78		breq2 5097	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (𝑁‘(𝐴 − 𝑦)) → ((√‘((𝑆↑2) + 𝑟)) ≤ 𝑤 ↔ (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦))))
79	77, 78	ralrnmptw 7033	. . . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . 14 ⊢ (∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + 𝑟)) ≤ 𝑤 ↔ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)))
82	73, 81	bitrdi 287	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((√‘((𝑆↑2) + 𝑟)) ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦))))
83	71, 82	bitrid 283	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((√‘((𝑆↑2) + 𝑟)) ≤ 𝑆 ↔ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦))))
84	70, 83	mtbid 324	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ¬ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)))
85		rexnal 3085	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝑌 ¬ (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)) ↔ ¬ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)))
86	84, 85	sylibr 234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ∃𝑦 ∈ 𝑌 ¬ (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)))
87	60	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (√‘((𝑆↑2) + 𝑟)) ∈ ℝ)
88		cphngp 25101	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ NrmGrp)
89	24, 88	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑈 ∈ NrmGrp)
90		ngpms 24516	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ NrmGrp → 𝑈 ∈ MetSp)
91		minvec.d	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))
92	21, 91	msmet 24373	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ MetSp → 𝐷 ∈ (Met‘𝑋))
93	89, 90, 92	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
94	93	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝐷 ∈ (Met‘𝑋))
95	26	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋)
96		eqid 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
97	21, 96	lssss 20871	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋)
98	4, 97	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑌 ⊆ 𝑋)
99	98	sselda 3930	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
100		metcl 24248	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴𝐷𝑦) ∈ ℝ)
101	94, 95, 99, 100	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑦) ∈ ℝ)
102	66	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 0 ≤ (√‘((𝑆↑2) + 𝑟)))
103		metge0 24261	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝑦))
104	94, 95, 99, 103	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 0 ≤ (𝐴𝐷𝑦))
105	87, 101, 102, 104	le2sqd 14166	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((√‘((𝑆↑2) + 𝑟)) ≤ (𝐴𝐷𝑦) ↔ ((√‘((𝑆↑2) + 𝑟))↑2) ≤ ((𝐴𝐷𝑦)↑2)))
106	91	oveqi 7365	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴𝐷𝑦) = (𝐴((dist‘𝑈) ↾ (𝑋 × 𝑋))𝑦)
107	95, 99	ovresd 7519	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝐴((dist‘𝑈) ↾ (𝑋 × 𝑋))𝑦) = (𝐴(dist‘𝑈)𝑦))
108	106, 107	eqtrid 2780	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑦) = (𝐴(dist‘𝑈)𝑦))
109	89	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝑈 ∈ NrmGrp)
110		eqid 2733	. . . . . . . . . . . . . . . . . 18 ⊢ (dist‘𝑈) = (dist‘𝑈)
111	23, 21, 22, 110	ngpds 24520	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴(dist‘𝑈)𝑦) = (𝑁‘(𝐴 − 𝑦)))
112	109, 95, 99, 111	syl3anc 1373	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝐴(dist‘𝑈)𝑦) = (𝑁‘(𝐴 − 𝑦)))
113	108, 112	eqtrd 2768	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑦) = (𝑁‘(𝐴 − 𝑦)))
114	113	breq2d 5105	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((√‘((𝑆↑2) + 𝑟)) ≤ (𝐴𝐷𝑦) ↔ (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦))))
115	39	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((√‘((𝑆↑2) + 𝑟))↑2) = ((𝑆↑2) + 𝑟))
116	115	breq1d 5103	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (((√‘((𝑆↑2) + 𝑟))↑2) ≤ ((𝐴𝐷𝑦)↑2) ↔ ((𝑆↑2) + 𝑟) ≤ ((𝐴𝐷𝑦)↑2)))
117	105, 114, 116	3bitr3d 309	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)) ↔ ((𝑆↑2) + 𝑟) ≤ ((𝐴𝐷𝑦)↑2)))
118	117	notbid 318	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (¬ (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)) ↔ ¬ ((𝑆↑2) + 𝑟) ≤ ((𝐴𝐷𝑦)↑2)))
119	37	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((𝑆↑2) + 𝑟) ∈ ℝ)
120	101	resqcld 14034	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((𝐴𝐷𝑦)↑2) ∈ ℝ)
121	119, 120	letrid 11272	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (((𝑆↑2) + 𝑟) ≤ ((𝐴𝐷𝑦)↑2) ∨ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)))
122	121	ord 864	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (¬ ((𝑆↑2) + 𝑟) ≤ ((𝐴𝐷𝑦)↑2) → ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)))
123	118, 122	sylbid 240	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (¬ (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)) → ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)))
124	123	reximdva 3146	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (∃𝑦 ∈ 𝑌 ¬ (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)) → ∃𝑦 ∈ 𝑌 ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)))
125	86, 124	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ∃𝑦 ∈ 𝑌 ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟))
126		rabn0 4338	. . . . . . . . 9 ⊢ ({𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)} ≠ ∅ ↔ ∃𝑦 ∈ 𝑌 ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟))
127	125, 126	sylibr 234	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)} ≠ ∅)
128	127	necomd 2984	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ∅ ≠ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
129	128	neneqd 2934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ¬ ∅ = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
130	129	nrexdv 3128	. . . . 5 ⊢ (𝜑 → ¬ ∃𝑟 ∈ ℝ⁺ ∅ = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
131	1	eleq2i 2825	. . . . . 6 ⊢ (∅ ∈ 𝐹 ↔ ∅ ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}))
132		0ex 5247	. . . . . . 7 ⊢ ∅ ∈ V
133	12	elrnmpt 5902	. . . . . . 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 275	. . . . 5 ⊢ (∅ ∈ 𝐹 ↔ ∃𝑟 ∈ ℝ⁺ ∅ = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
136	130, 135	sylnibr 329	. . . 4 ⊢ (𝜑 → ¬ ∅ ∈ 𝐹)
137		df-nel 3034	. . . 4 ⊢ (∅ ∉ 𝐹 ↔ ¬ ∅ ∈ 𝐹)
138	136, 137	sylibr 234	. . 3 ⊢ (𝜑 → ∅ ∉ 𝐹)
139	55	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝑆 ∈ ℝ)
140	139	resqcld 14034	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝑆↑2) ∈ ℝ)
141	36	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝑟 ∈ ℝ)
142	120, 140, 141	lesubadd2d 11723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟 ↔ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)))
143	142	rabbidva 3402	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟} = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
144	143	mpteq2dva 5186	. . . . . . . . . 10 ⊢ (𝜑 → (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) = (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}))
145	144	rneqd 5882	. . . . . . . . 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 2787	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}))
147	146	eleq2d 2819	. . . . . . 7 ⊢ (𝜑 → (𝑢 ∈ 𝐹 ↔ 𝑢 ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟})))
148		breq2 5097	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑠 → ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟 ↔ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠))
149	148	rabbidv 3403	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟} = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠})
150	149	cbvmptv 5197	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) = (𝑠 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠})
151	150	elrnmpt 5902	. . . . . . . 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 3442	. . . . . . 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 287	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ 𝐹 ↔ ∃𝑠 ∈ ℝ⁺ 𝑢 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠}))
154	146	eleq2d 2819	. . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ 𝐹 ↔ 𝑣 ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟})))
155		breq2 5097	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑡 → ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟 ↔ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡))
156	155	rabbidv 3403	. . . . . . . . . 10 ⊢ (𝑟 = 𝑡 → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟} = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡})
157	156	cbvmptv 5197	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) = (𝑡 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡})
158	157	elrnmpt 5902	. . . . . . . 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 3442	. . . . . . 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 287	. . . . . 6 ⊢ (𝜑 → (𝑣 ∈ 𝐹 ↔ ∃𝑡 ∈ ℝ⁺ 𝑣 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡}))
161	153, 160	anbi12d 632	. . . . 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 3205	. . . . . 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 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝐷 ∈ (Met‘𝑋))
164	26	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋)
165	3, 97	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
166	165	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → 𝑌 ⊆ 𝑋)
167	166	sselda 3930	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
168	163, 164, 167, 100	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑦) ∈ ℝ)
169	168	resqcld 14034	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → ((𝐴𝐷𝑦)↑2) ∈ ℝ)
170	31	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → (𝑆↑2) ∈ ℝ)
171	169, 170	resubcld 11552	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ∈ ℝ)
172		simplrl 776	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝑠 ∈ ℝ⁺)
173	172	rpred 12936	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝑠 ∈ ℝ)
174		simplrr 777	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝑡 ∈ ℝ⁺)
175	174	rpred 12936	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝑡 ∈ ℝ)
176		lemin 13093	. . . . . . . . . . . 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 1373	. . . . . . . . . . 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 3402	. . . . . . . . . 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 4520	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺) → if(𝑠 ≤ 𝑡, 𝑠, 𝑡) ∈ ℝ⁺)
180	3	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → 𝑌 ∈ (LSubSp‘𝑈))
181		rabexg 5277	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ if(𝑠 ≤ 𝑡, 𝑠, 𝑡)} ∈ V)
182	180, 181	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ if(𝑠 ≤ 𝑡, 𝑠, 𝑡)} ∈ V)
183		eqid 2733	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) = (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟})
184		breq2 5097	. . . . . . . . . . . . . 14 ⊢ (𝑟 = if(𝑠 ≤ 𝑡, 𝑠, 𝑡) → ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟 ↔ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ if(𝑠 ≤ 𝑡, 𝑠, 𝑡)))
185	184	rabbidv 3403	. . . . . . . . . . . . 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 5903	. . . . . . . . . . . 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 686	. . . . . . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → 𝐹 = ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}))
189	187, 188	eleqtrrd 2836	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ if(𝑠 ≤ 𝑡, 𝑠, 𝑡)} ∈ 𝐹)
190	178, 189	eqeltrrd 2834	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → {𝑦 ∈ 𝑌 ∣ ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠 ∧ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡)} ∈ 𝐹)
191		ineq12 4164	. . . . . . . . . . 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 4265	. . . . . . . . . . 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 2784	. . . . . . . . . 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 2818	. . . . . . . . 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 247	. . . . . . . 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 3441	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
197	196	inex1 5257	. . . . . . . . . 10 ⊢ (𝑢 ∩ 𝑣) ∈ V
198	197	pwid 4571	. . . . . . . . 9 ⊢ (𝑢 ∩ 𝑣) ∈ 𝒫 (𝑢 ∩ 𝑣)
199		inelcm 4414	. . . . . . . . 9 ⊢ (((𝑢 ∩ 𝑣) ∈ 𝐹 ∧ (𝑢 ∩ 𝑣) ∈ 𝒫 (𝑢 ∩ 𝑣)) → (𝐹 ∩ 𝒫 (𝑢 ∩ 𝑣)) ≠ ∅)
200	198, 199	mpan2 691	. . . . . . . 8 ⊢ ((𝑢 ∩ 𝑣) ∈ 𝐹 → (𝐹 ∩ 𝒫 (𝑢 ∩ 𝑣)) ≠ ∅)
201	195, 200	syl6 35	. . . . . . 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 3190	. . . . . 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 243	. . . . 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 240	. . . 4 ⊢ (𝜑 → ((𝑢 ∈ 𝐹 ∧ 𝑣 ∈ 𝐹) → (𝐹 ∩ 𝒫 (𝑢 ∩ 𝑣)) ≠ ∅))
205	204	ralrimivv 3174	. . 3 ⊢ (𝜑 → ∀𝑢 ∈ 𝐹 ∀𝑣 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑢 ∩ 𝑣)) ≠ ∅)
206	20, 138, 205	3jca 1128	. 2 ⊢ (𝜑 → (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑢 ∈ 𝐹 ∀𝑣 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑢 ∩ 𝑣)) ≠ ∅))
207		isfbas 23745	. . 3 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → (𝐹 ∈ (fBas‘𝑌) ↔ (𝐹 ⊆ 𝒫 𝑌 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑢 ∈ 𝐹 ∀𝑣 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑢 ∩ 𝑣)) ≠ ∅))))
208	3, 207	syl 17	. 2 ⊢ (𝜑 → (𝐹 ∈ (fBas‘𝑌) ↔ (𝐹 ⊆ 𝒫 𝑌 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑢 ∈ 𝐹 ∀𝑣 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑢 ∩ 𝑣)) ≠ ∅))))
209	10, 206, 208	mpbir2and 713	1 ⊢ (𝜑 → 𝐹 ∈ (fBas‘𝑌))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2929 ∉ wnel 3033 ∀wral 3048 ∃wrex 3057 {crab 3396 Vcvv 3437 ∩ cin 3897 ⊆ wss 3898 ∅c0 4282 ifcif 4474 𝒫 cpw 4549 class class class wbr 5093 ↦ cmpt 5174 × cxp 5617 dom cdm 5619 ran crn 5620 ↾ cres 5621 ‘cfv 6486 (class class class)co 7352 infcinf 9332 ℝcr 11012 0cc0 11013 1c1 11014 + caddc 11016 < clt 11153 ≤ cle 11154 − cmin 11351 2c2 12187 ℝ⁺crp 12892 ↑cexp 13970 √csqrt 15142 Basecbs 17122 ↾_s cress 17143 distcds 17172 TopOpenctopn 17327 -_gcsg 18850 LSubSpclss 20866 Metcmet 21279 fBascfbas 21281 MetSpcms 24234 normcnm 24492 NrmGrpcngp 24493 ℂPreHilccph 25094 CMetSpccms 25260

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-1st 7927 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-er 8628 df-map 8758 df-en 8876 df-dom 8877 df-sdom 8878 df-sup 9333 df-inf 9334 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-2 12195 df-3 12196 df-n0 12389 df-z 12476 df-uz 12739 df-q 12849 df-rp 12893 df-xneg 13013 df-xadd 13014 df-xmul 13015 df-seq 13911 df-exp 13971 df-cj 15008 df-re 15009 df-im 15010 df-sqrt 15144 df-abs 15145 df-0g 17347 df-topgen 17349 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-grp 18851 df-minusg 18852 df-sbg 18853 df-lmod 20797 df-lss 20867 df-psmet 21285 df-xmet 21286 df-met 21287 df-bl 21288 df-mopn 21289 df-fbas 21290 df-top 22810 df-topon 22827 df-topsp 22849 df-bases 22862 df-xms 24236 df-ms 24237 df-nm 24498 df-ngp 24499 df-nlm 24502 df-cph 25096

This theorem is referenced by: minveclem3 25357 minveclem4a 25358 minveclem4 25360

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