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Theorem minveclem3b 25481

Description: Lemma for minvec 25489. 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 4103	. . . . . 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 5351	. . . . . . 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 7149	. . . 4 ⊢ (𝜑 → (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}):ℝ⁺⟶𝒫 𝑌)
9	8	frnd 6755	. . 3 ⊢ (𝜑 → ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) ⊆ 𝒫 𝑌)
10	1, 9	eqsstrid 4057	. 2 ⊢ (𝜑 → 𝐹 ⊆ 𝒫 𝑌)
11		1rp 13061	. . . . . 6 ⊢ 1 ∈ ℝ⁺
12		eqid 2740	. . . . . . 7 ⊢ (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
13	12, 7	dmmptd 6725	. . . . . 6 ⊢ (𝜑 → dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = ℝ⁺)
14	11, 13	eleqtrrid 2851	. . . . 5 ⊢ (𝜑 → 1 ∈ dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}))
15	14	ne0d 4365	. . . 4 ⊢ (𝜑 → dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) ≠ ∅)
16		dm0rn0 5949	. . . . . 6 ⊢ (dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = ∅ ↔ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = ∅)
17	1	eqeq1i 2745	. . . . . 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 2999	. . . 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 25478	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆 ∈ ℝ)
31	30	resqcld 14175	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑆↑2) ∈ ℝ)
32		ltaddrp 13094	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆↑2) ∈ ℝ ∧ 𝑟 ∈ ℝ⁺) → (𝑆↑2) < ((𝑆↑2) + 𝑟))
33	31, 32	sylan 579	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (𝑆↑2) < ((𝑆↑2) + 𝑟))
34	31	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (𝑆↑2) ∈ ℝ)
35		rpre 13065	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ)
36	35	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ)
37	34, 36	readdcld 11319	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((𝑆↑2) + 𝑟) ∈ ℝ)
38	37	recnd 11318	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((𝑆↑2) + 𝑟) ∈ ℂ)
39	38	sqsqrtd 15488	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((√‘((𝑆↑2) + 𝑟))↑2) = ((𝑆↑2) + 𝑟))
40	33, 39	breqtrrd 5194	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (𝑆↑2) < ((√‘((𝑆↑2) + 𝑟))↑2))
41	21, 22, 23, 24, 3, 25, 26, 27, 28	minveclem1 25477	. . . . . . . . . . . . . . . . . . 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 11292	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ
47	41	simp3d 1144	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)
48		breq1 5169	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 0 → (𝑦 ≤ 𝑤 ↔ 0 ≤ 𝑤))
49	48	ralbidv 3184	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 0 → (∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤 ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
50	49	rspcev 3635	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℝ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) → ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤)
51	46, 47, 50	sylancr 586	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤)
52	51	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤)
53		infrecl 12277	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤) → inf(𝑅, ℝ, < ) ∈ ℝ)
54	43, 45, 52, 53	syl3anc 1371	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → inf(𝑅, ℝ, < ) ∈ ℝ)
55	29, 54	eqeltrid 2848	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑆 ∈ ℝ)
56		0red 11293	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ∈ ℝ)
57	55	sqge0d 14187	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ (𝑆↑2))
58	56, 34, 37, 57, 33	lelttrd 11448	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 < ((𝑆↑2) + 𝑟))
59	56, 37, 58	ltled 11438	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ ((𝑆↑2) + 𝑟))
60	37, 59	resqrtcld 15466	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (√‘((𝑆↑2) + 𝑟)) ∈ ℝ)
61	47	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)
62		infregelb 12279	. . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . 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 5208	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ 𝑆)
66	37, 59	sqrtge0d 15469	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ (√‘((𝑆↑2) + 𝑟)))
67	55, 60, 65, 66	lt2sqd 14305	. . . . . . . . . . . . . 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 11437	. . . . . . . . . . . . 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 5174	. . . . . . . . . . . . 13 ⊢ ((√‘((𝑆↑2) + 𝑟)) ≤ 𝑆 ↔ (√‘((𝑆↑2) + 𝑟)) ≤ inf(𝑅, ℝ, < ))
72		infregelb 12279	. . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((√‘((𝑆↑2) + 𝑟)) ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + 𝑟)) ≤ 𝑤))
74	28	raleqi 3332	. . . . . . . . . . . . . . 15 ⊢ (∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + 𝑟)) ≤ 𝑤 ↔ ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))(√‘((𝑆↑2) + 𝑟)) ≤ 𝑤)
75		fvex 6933	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁‘(𝐴 − 𝑦)) ∈ V
76	75	rgenw 3071	. . . . . . . . . . . . . . . 16 ⊢ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑦)) ∈ V
77		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))
78		breq2 5170	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (𝑁‘(𝐴 − 𝑦)) → ((√‘((𝑆↑2) + 𝑟)) ≤ 𝑤 ↔ (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦))))
79	77, 78	ralrnmptw 7128	. . . . . . . . . . . . . . . 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 3106	. . . . . . . . . . 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 25226	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ NrmGrp)
89	24, 88	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑈 ∈ NrmGrp)
90		ngpms 24634	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ NrmGrp → 𝑈 ∈ MetSp)
91		minvec.d	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))
92	21, 91	msmet 24488	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ MetSp → 𝐷 ∈ (Met‘𝑋))
93	89, 90, 92	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
94	93	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝐷 ∈ (Met‘𝑋))
95	26	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋)
96		eqid 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
97	21, 96	lssss 20957	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋)
98	4, 97	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑌 ⊆ 𝑋)
99	98	sselda 4008	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
100		metcl 24363	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴𝐷𝑦) ∈ ℝ)
101	94, 95, 99, 100	syl3anc 1371	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑦) ∈ ℝ)
102	66	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 0 ≤ (√‘((𝑆↑2) + 𝑟)))
103		metge0 24376	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝑦))
104	94, 95, 99, 103	syl3anc 1371	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 0 ≤ (𝐴𝐷𝑦))
105	87, 101, 102, 104	le2sqd 14306	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((√‘((𝑆↑2) + 𝑟)) ≤ (𝐴𝐷𝑦) ↔ ((√‘((𝑆↑2) + 𝑟))↑2) ≤ ((𝐴𝐷𝑦)↑2)))
106	91	oveqi 7461	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴𝐷𝑦) = (𝐴((dist‘𝑈) ↾ (𝑋 × 𝑋))𝑦)
107	95, 99	ovresd 7617	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝐴((dist‘𝑈) ↾ (𝑋 × 𝑋))𝑦) = (𝐴(dist‘𝑈)𝑦))
108	106, 107	eqtrid 2792	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑦) = (𝐴(dist‘𝑈)𝑦))
109	89	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝑈 ∈ NrmGrp)
110		eqid 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (dist‘𝑈) = (dist‘𝑈)
111	23, 21, 22, 110	ngpds 24638	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴(dist‘𝑈)𝑦) = (𝑁‘(𝐴 − 𝑦)))
112	109, 95, 99, 111	syl3anc 1371	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝐴(dist‘𝑈)𝑦) = (𝑁‘(𝐴 − 𝑦)))
113	108, 112	eqtrd 2780	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑦) = (𝑁‘(𝐴 − 𝑦)))
114	113	breq2d 5178	. . . . . . . . . . . . . 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 5176	. . . . . . . . . . . . . 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 14175	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((𝐴𝐷𝑦)↑2) ∈ ℝ)
121	119, 120	letrid 11442	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (((𝑆↑2) + 𝑟) ≤ ((𝐴𝐷𝑦)↑2) ∨ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)))
122	121	ord 863	. . . . . . . . . . . 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 3174	. . . . . . . . . 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 4412	. . . . . . . . 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 3002	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ∅ ≠ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
129	128	neneqd 2951	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ¬ ∅ = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
130	129	nrexdv 3155	. . . . 5 ⊢ (𝜑 → ¬ ∃𝑟 ∈ ℝ⁺ ∅ = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
131	1	eleq2i 2836	. . . . . 6 ⊢ (∅ ∈ 𝐹 ↔ ∅ ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}))
132		0ex 5325	. . . . . . 7 ⊢ ∅ ∈ V
133	12	elrnmpt 5981	. . . . . . 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 3053	. . . 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 14175	. . . . . . . . . . . . 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 11889	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟 ↔ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)))
143	142	rabbidva 3450	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟} = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
144	143	mpteq2dva 5266	. . . . . . . . . 10 ⊢ (𝜑 → (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) = (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}))
145	144	rneqd 5963	. . . . . . . . 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 2799	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}))
147	146	eleq2d 2830	. . . . . . 7 ⊢ (𝜑 → (𝑢 ∈ 𝐹 ↔ 𝑢 ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟})))
148		breq2 5170	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑠 → ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟 ↔ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠))
149	148	rabbidv 3451	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟} = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠})
150	149	cbvmptv 5279	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) = (𝑠 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠})
151	150	elrnmpt 5981	. . . . . . . 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 3493	. . . . . . 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 2830	. . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ 𝐹 ↔ 𝑣 ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟})))
155		breq2 5170	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑡 → ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟 ↔ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡))
156	155	rabbidv 3451	. . . . . . . . . 10 ⊢ (𝑟 = 𝑡 → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟} = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡})
157	156	cbvmptv 5279	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) = (𝑡 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡})
158	157	elrnmpt 5981	. . . . . . . 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 3493	. . . . . . 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 631	. . . . 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 3235	. . . . . 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 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝐷 ∈ (Met‘𝑋))
164	26	ad2antrr 725	. . . . . . . . . . . . . . 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 4008	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
168	163, 164, 167, 100	syl3anc 1371	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑦) ∈ ℝ)
169	168	resqcld 14175	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → ((𝐴𝐷𝑦)↑2) ∈ ℝ)
170	31	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → (𝑆↑2) ∈ ℝ)
171	169, 170	resubcld 11718	. . . . . . . . . . . 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 13099	. . . . . . . . . . . 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 13099	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝑡 ∈ ℝ)
176		lemin 13254	. . . . . . . . . . . 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 1371	. . . . . . . . . . 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 3450	. . . . . . . . . 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 4593	. . . . . . . . . . . 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 5355	. . . . . . . . . . . . 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 2740	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) = (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟})
184		breq2 5170	. . . . . . . . . . . . . 14 ⊢ (𝑟 = if(𝑠 ≤ 𝑡, 𝑠, 𝑡) → ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟 ↔ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ if(𝑠 ≤ 𝑡, 𝑠, 𝑡)))
185	184	rabbidv 3451	. . . . . . . . . . . . 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 5982	. . . . . . . . . . . 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 685	. . . . . . . . . . 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 2847	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ if(𝑠 ≤ 𝑡, 𝑠, 𝑡)} ∈ 𝐹)
190	178, 189	eqeltrrd 2845	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → {𝑦 ∈ 𝑌 ∣ ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠 ∧ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡)} ∈ 𝐹)
191		ineq12 4236	. . . . . . . . . . 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 4335	. . . . . . . . . . 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 2796	. . . . . . . . . 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 2829	. . . . . . . . 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 3492	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
197	196	inex1 5335	. . . . . . . . . 10 ⊢ (𝑢 ∩ 𝑣) ∈ V
198	197	pwid 4644	. . . . . . . . 9 ⊢ (𝑢 ∩ 𝑣) ∈ 𝒫 (𝑢 ∩ 𝑣)
199		inelcm 4488	. . . . . . . . 9 ⊢ (((𝑢 ∩ 𝑣) ∈ 𝐹 ∧ (𝑢 ∩ 𝑣) ∈ 𝒫 (𝑢 ∩ 𝑣)) → (𝐹 ∩ 𝒫 (𝑢 ∩ 𝑣)) ≠ ∅)
200	198, 199	mpan2 690	. . . . . . . 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 3219	. . . . . 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 3206	. . 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 23858	. . 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 712	1 ⊢ (𝜑 → 𝐹 ∈ (fBas‘𝑌))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∉ wnel 3052 ∀wral 3067 ∃wrex 3076 {crab 3443 Vcvv 3488 ∩ cin 3975 ⊆ wss 3976 ∅c0 4352 ifcif 4548 𝒫 cpw 4622 class class class wbr 5166 ↦ cmpt 5249 × cxp 5698 dom cdm 5700 ran crn 5701 ↾ cres 5702 ‘cfv 6573 (class class class)co 7448 infcinf 9510 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 < clt 11324 ≤ cle 11325 − cmin 11520 2c2 12348 ℝ⁺crp 13057 ↑cexp 14112 √csqrt 15282 Basecbs 17258 ↾_s cress 17287 distcds 17320 TopOpenctopn 17481 -_gcsg 18975 LSubSpclss 20952 Metcmet 21373 fBascfbas 21375 MetSpcms 24349 normcnm 24610 NrmGrpcngp 24611 ℂPreHilccph 25219 CMetSpccms 25385

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-sup 9511 df-inf 9512 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-seq 14053 df-exp 14113 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-0g 17501 df-topgen 17503 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-grp 18976 df-minusg 18977 df-sbg 18978 df-lmod 20882 df-lss 20953 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-xms 24351 df-ms 24352 df-nm 24616 df-ngp 24617 df-nlm 24620 df-cph 25221

This theorem is referenced by: minveclem3 25482 minveclem4a 25483 minveclem4 25485

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