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Theorem minveclem3b 24497

Description: Lemma for minvec 24505. 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 4009	. . . . . 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 5263	. . . . . . 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 257	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)} ∈ 𝒫 𝑌)
8	7	fmpttd 6971	. . . 4 ⊢ (𝜑 → (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}):ℝ⁺⟶𝒫 𝑌)
9	8	frnd 6592	. . 3 ⊢ (𝜑 → ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) ⊆ 𝒫 𝑌)
10	1, 9	eqsstrid 3965	. 2 ⊢ (𝜑 → 𝐹 ⊆ 𝒫 𝑌)
11		1rp 12663	. . . . . 6 ⊢ 1 ∈ ℝ⁺
12		eqid 2738	. . . . . . 7 ⊢ (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
13	12, 7	dmmptd 6562	. . . . . 6 ⊢ (𝜑 → dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = ℝ⁺)
14	11, 13	eleqtrrid 2846	. . . . 5 ⊢ (𝜑 → 1 ∈ dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}))
15	14	ne0d 4266	. . . 4 ⊢ (𝜑 → dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) ≠ ∅)
16		dm0rn0 5823	. . . . . 6 ⊢ (dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = ∅ ↔ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = ∅)
17	1	eqeq1i 2743	. . . . . 6 ⊢ (𝐹 = ∅ ↔ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = ∅)
18	16, 17	bitr4i 277	. . . . 5 ⊢ (dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = ∅ ↔ 𝐹 = ∅)
19	18	necon3bii 2995	. . . 4 ⊢ (dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) ≠ ∅ ↔ 𝐹 ≠ ∅)
20	15, 19	sylib 217	. . 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 24494	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆 ∈ ℝ)
31	30	resqcld 13893	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑆↑2) ∈ ℝ)
32		ltaddrp 12696	. . . . . . . . . . . . . . . 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 12667	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ)
36	35	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ)
37	34, 36	readdcld 10935	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((𝑆↑2) + 𝑟) ∈ ℝ)
38	37	recnd 10934	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((𝑆↑2) + 𝑟) ∈ ℂ)
39	38	sqsqrtd 15079	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((√‘((𝑆↑2) + 𝑟))↑2) = ((𝑆↑2) + 𝑟))
40	33, 39	breqtrrd 5098	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (𝑆↑2) < ((√‘((𝑆↑2) + 𝑟))↑2))
41	21, 22, 23, 24, 3, 25, 26, 27, 28	minveclem1 24493	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
42	41	simp1d 1140	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑅 ⊆ ℝ)
43	42	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑅 ⊆ ℝ)
44	41	simp2d 1141	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑅 ≠ ∅)
45	44	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑅 ≠ ∅)
46		0re 10908	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ
47	41	simp3d 1142	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)
48		breq1 5073	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 0 → (𝑦 ≤ 𝑤 ↔ 0 ≤ 𝑤))
49	48	ralbidv 3120	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 0 → (∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤 ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
50	49	rspcev 3552	. . . . . . . . . . . . . . . . . . 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 11887	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤) → inf(𝑅, ℝ, < ) ∈ ℝ)
54	43, 45, 52, 53	syl3anc 1369	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → inf(𝑅, ℝ, < ) ∈ ℝ)
55	29, 54	eqeltrid 2843	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑆 ∈ ℝ)
56		0red 10909	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ∈ ℝ)
57	55	sqge0d 13894	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ (𝑆↑2))
58	56, 34, 37, 57, 33	lelttrd 11063	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 < ((𝑆↑2) + 𝑟))
59	56, 37, 58	ltled 11053	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ ((𝑆↑2) + 𝑟))
60	37, 59	resqrtcld 15057	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (√‘((𝑆↑2) + 𝑟)) ∈ ℝ)
61	47	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)
62		infregelb 11889	. . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (0 ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
64	61, 63	mpbird 256	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ inf(𝑅, ℝ, < ))
65	64, 29	breqtrrdi 5112	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ 𝑆)
66	37, 59	sqrtge0d 15060	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ (√‘((𝑆↑2) + 𝑟)))
67	55, 60, 65, 66	lt2sqd 13901	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (𝑆 < (√‘((𝑆↑2) + 𝑟)) ↔ (𝑆↑2) < ((√‘((𝑆↑2) + 𝑟))↑2)))
68	40, 67	mpbird 256	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑆 < (√‘((𝑆↑2) + 𝑟)))
69	55, 60	ltnled 11052	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (𝑆 < (√‘((𝑆↑2) + 𝑟)) ↔ ¬ (√‘((𝑆↑2) + 𝑟)) ≤ 𝑆))
70	68, 69	mpbid 231	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ¬ (√‘((𝑆↑2) + 𝑟)) ≤ 𝑆)
71	29	breq2i 5078	. . . . . . . . . . . . 13 ⊢ ((√‘((𝑆↑2) + 𝑟)) ≤ 𝑆 ↔ (√‘((𝑆↑2) + 𝑟)) ≤ inf(𝑅, ℝ, < ))
72		infregelb 11889	. . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((√‘((𝑆↑2) + 𝑟)) ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + 𝑟)) ≤ 𝑤))
74	28	raleqi 3337	. . . . . . . . . . . . . . 15 ⊢ (∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + 𝑟)) ≤ 𝑤 ↔ ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))(√‘((𝑆↑2) + 𝑟)) ≤ 𝑤)
75		fvex 6769	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁‘(𝐴 − 𝑦)) ∈ V
76	75	rgenw 3075	. . . . . . . . . . . . . . . 16 ⊢ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑦)) ∈ V
77		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))
78		breq2 5074	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (𝑁‘(𝐴 − 𝑦)) → ((√‘((𝑆↑2) + 𝑟)) ≤ 𝑤 ↔ (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦))))
79	77, 78	ralrnmptw 6952	. . . . . . . . . . . . . . . 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 274	. . . . . . . . . . . . . 14 ⊢ (∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + 𝑟)) ≤ 𝑤 ↔ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)))
82	73, 81	bitrdi 286	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((√‘((𝑆↑2) + 𝑟)) ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦))))
83	71, 82	syl5bb 282	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((√‘((𝑆↑2) + 𝑟)) ≤ 𝑆 ↔ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦))))
84	70, 83	mtbid 323	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ¬ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)))
85		rexnal 3165	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝑌 ¬ (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)) ↔ ¬ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)))
86	84, 85	sylibr 233	. . . . . . . . . 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 24242	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ NrmGrp)
89	24, 88	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑈 ∈ NrmGrp)
90		ngpms 23662	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ NrmGrp → 𝑈 ∈ MetSp)
91		minvec.d	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))
92	21, 91	msmet 23518	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ MetSp → 𝐷 ∈ (Met‘𝑋))
93	89, 90, 92	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
94	93	ad2antrr 722	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝐷 ∈ (Met‘𝑋))
95	26	ad2antrr 722	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋)
96		eqid 2738	. . . . . . . . . . . . . . . . . . 19 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
97	21, 96	lssss 20113	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋)
98	4, 97	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑌 ⊆ 𝑋)
99	98	sselda 3917	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
100		metcl 23393	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴𝐷𝑦) ∈ ℝ)
101	94, 95, 99, 100	syl3anc 1369	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑦) ∈ ℝ)
102	66	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 0 ≤ (√‘((𝑆↑2) + 𝑟)))
103		metge0 23406	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝑦))
104	94, 95, 99, 103	syl3anc 1369	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 0 ≤ (𝐴𝐷𝑦))
105	87, 101, 102, 104	le2sqd 13902	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((√‘((𝑆↑2) + 𝑟)) ≤ (𝐴𝐷𝑦) ↔ ((√‘((𝑆↑2) + 𝑟))↑2) ≤ ((𝐴𝐷𝑦)↑2)))
106	91	oveqi 7268	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴𝐷𝑦) = (𝐴((dist‘𝑈) ↾ (𝑋 × 𝑋))𝑦)
107	95, 99	ovresd 7417	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝐴((dist‘𝑈) ↾ (𝑋 × 𝑋))𝑦) = (𝐴(dist‘𝑈)𝑦))
108	106, 107	syl5eq 2791	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑦) = (𝐴(dist‘𝑈)𝑦))
109	89	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝑈 ∈ NrmGrp)
110		eqid 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (dist‘𝑈) = (dist‘𝑈)
111	23, 21, 22, 110	ngpds 23666	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴(dist‘𝑈)𝑦) = (𝑁‘(𝐴 − 𝑦)))
112	109, 95, 99, 111	syl3anc 1369	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝐴(dist‘𝑈)𝑦) = (𝑁‘(𝐴 − 𝑦)))
113	108, 112	eqtrd 2778	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑦) = (𝑁‘(𝐴 − 𝑦)))
114	113	breq2d 5082	. . . . . . . . . . . . . 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 5080	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (((√‘((𝑆↑2) + 𝑟))↑2) ≤ ((𝐴𝐷𝑦)↑2) ↔ ((𝑆↑2) + 𝑟) ≤ ((𝐴𝐷𝑦)↑2)))
117	105, 114, 116	3bitr3d 308	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)) ↔ ((𝑆↑2) + 𝑟) ≤ ((𝐴𝐷𝑦)↑2)))
118	117	notbid 317	. . . . . . . . . . . 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 13893	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((𝐴𝐷𝑦)↑2) ∈ ℝ)
121	119, 120	letrid 11057	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (((𝑆↑2) + 𝑟) ≤ ((𝐴𝐷𝑦)↑2) ∨ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)))
122	121	ord 860	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (¬ ((𝑆↑2) + 𝑟) ≤ ((𝐴𝐷𝑦)↑2) → ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)))
123	118, 122	sylbid 239	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (¬ (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦)) → ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)))
124	123	reximdva 3202	. . . . . . . . . 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 4316	. . . . . . . . 9 ⊢ ({𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)} ≠ ∅ ↔ ∃𝑦 ∈ 𝑌 ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟))
127	125, 126	sylibr 233	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)} ≠ ∅)
128	127	necomd 2998	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ∅ ≠ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
129	128	neneqd 2947	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ¬ ∅ = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
130	129	nrexdv 3197	. . . . 5 ⊢ (𝜑 → ¬ ∃𝑟 ∈ ℝ⁺ ∅ = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
131	1	eleq2i 2830	. . . . . 6 ⊢ (∅ ∈ 𝐹 ↔ ∅ ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}))
132		0ex 5226	. . . . . . 7 ⊢ ∅ ∈ V
133	12	elrnmpt 5854	. . . . . . 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 274	. . . . 5 ⊢ (∅ ∈ 𝐹 ↔ ∃𝑟 ∈ ℝ⁺ ∅ = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
136	130, 135	sylnibr 328	. . . 4 ⊢ (𝜑 → ¬ ∅ ∈ 𝐹)
137		df-nel 3049	. . . 4 ⊢ (∅ ∉ 𝐹 ↔ ¬ ∅ ∈ 𝐹)
138	136, 137	sylibr 233	. . 3 ⊢ (𝜑 → ∅ ∉ 𝐹)
139	55	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝑆 ∈ ℝ)
140	139	resqcld 13893	. . . . . . . . . . . . 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 11504	. . . . . . . . . . . 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 5170	. . . . . . . . . 10 ⊢ (𝜑 → (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) = (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}))
145	144	rneqd 5836	. . . . . . . . 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 2798	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}))
147	146	eleq2d 2824	. . . . . . 7 ⊢ (𝜑 → (𝑢 ∈ 𝐹 ↔ 𝑢 ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟})))
148		breq2 5074	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑠 → ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟 ↔ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠))
149	148	rabbidv 3404	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟} = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠})
150	149	cbvmptv 5183	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) = (𝑠 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠})
151	150	elrnmpt 5854	. . . . . . . 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 3428	. . . . . . 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 286	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ 𝐹 ↔ ∃𝑠 ∈ ℝ⁺ 𝑢 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠}))
154	146	eleq2d 2824	. . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ 𝐹 ↔ 𝑣 ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟})))
155		breq2 5074	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑡 → ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟 ↔ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡))
156	155	rabbidv 3404	. . . . . . . . . 10 ⊢ (𝑟 = 𝑡 → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟} = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡})
157	156	cbvmptv 5183	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) = (𝑡 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡})
158	157	elrnmpt 5854	. . . . . . . 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 3428	. . . . . . 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 286	. . . . . 6 ⊢ (𝜑 → (𝑣 ∈ 𝐹 ↔ ∃𝑡 ∈ ℝ⁺ 𝑣 = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡}))
161	153, 160	anbi12d 630	. . . . 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 3292	. . . . . 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 722	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝐷 ∈ (Met‘𝑋))
164	26	ad2antrr 722	. . . . . . . . . . . . . . 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 3917	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
168	163, 164, 167, 100	syl3anc 1369	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑦) ∈ ℝ)
169	168	resqcld 13893	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → ((𝐴𝐷𝑦)↑2) ∈ ℝ)
170	31	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → (𝑆↑2) ∈ ℝ)
171	169, 170	resubcld 11333	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ∈ ℝ)
172		simplrl 773	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝑠 ∈ ℝ⁺)
173	172	rpred 12701	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝑠 ∈ ℝ)
174		simplrr 774	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝑡 ∈ ℝ⁺)
175	174	rpred 12701	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝑡 ∈ ℝ)
176		lemin 12855	. . . . . . . . . . . 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 1369	. . . . . . . . . . 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 4501	. . . . . . . . . . . 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 5250	. . . . . . . . . . . . 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 2738	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) = (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟})
184		breq2 5074	. . . . . . . . . . . . . 14 ⊢ (𝑟 = if(𝑠 ≤ 𝑡, 𝑠, 𝑡) → ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟 ↔ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ if(𝑠 ≤ 𝑡, 𝑠, 𝑡)))
185	184	rabbidv 3404	. . . . . . . . . . . . 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 5855	. . . . . . . . . . . 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 682	. . . . . . . . . . 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 2842	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ if(𝑠 ≤ 𝑡, 𝑠, 𝑡)} ∈ 𝐹)
190	178, 189	eqeltrrd 2840	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → {𝑦 ∈ 𝑌 ∣ ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠 ∧ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡)} ∈ 𝐹)
191		ineq12 4138	. . . . . . . . . . 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 4237	. . . . . . . . . . 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 2795	. . . . . . . . . 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 2823	. . . . . . . . 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 246	. . . . . . . 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 3426	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
197	196	inex1 5236	. . . . . . . . . 10 ⊢ (𝑢 ∩ 𝑣) ∈ V
198	197	pwid 4554	. . . . . . . . 9 ⊢ (𝑢 ∩ 𝑣) ∈ 𝒫 (𝑢 ∩ 𝑣)
199		inelcm 4395	. . . . . . . . 9 ⊢ (((𝑢 ∩ 𝑣) ∈ 𝐹 ∧ (𝑢 ∩ 𝑣) ∈ 𝒫 (𝑢 ∩ 𝑣)) → (𝐹 ∩ 𝒫 (𝑢 ∩ 𝑣)) ≠ ∅)
200	198, 199	mpan2 687	. . . . . . . 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 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	syl5bir 242	. . . . 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 239	. . . 4 ⊢ (𝜑 → ((𝑢 ∈ 𝐹 ∧ 𝑣 ∈ 𝐹) → (𝐹 ∩ 𝒫 (𝑢 ∩ 𝑣)) ≠ ∅))
205	204	ralrimivv 3113	. . 3 ⊢ (𝜑 → ∀𝑢 ∈ 𝐹 ∀𝑣 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑢 ∩ 𝑣)) ≠ ∅)
206	20, 138, 205	3jca 1126	. 2 ⊢ (𝜑 → (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑢 ∈ 𝐹 ∀𝑣 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑢 ∩ 𝑣)) ≠ ∅))
207		isfbas 22888	. . 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 709	1 ⊢ (𝜑 → 𝐹 ∈ (fBas‘𝑌))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∉ wnel 3048 ∀wral 3063 ∃wrex 3064 {crab 3067 Vcvv 3422 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 ifcif 4456 𝒫 cpw 4530 class class class wbr 5070 ↦ cmpt 5153 × cxp 5578 dom cdm 5580 ran crn 5581 ↾ cres 5582 ‘cfv 6418 (class class class)co 7255 infcinf 9130 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 < clt 10940 ≤ cle 10941 − cmin 11135 2c2 11958 ℝ⁺crp 12659 ↑cexp 13710 √csqrt 14872 Basecbs 16840 ↾_s cress 16867 distcds 16897 TopOpenctopn 17049 -_gcsg 18494 LSubSpclss 20108 Metcmet 20496 fBascfbas 20498 MetSpcms 23379 normcnm 23638 NrmGrpcngp 23639 ℂPreHilccph 24235 CMetSpccms 24401

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-0g 17069 df-topgen 17071 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-grp 18495 df-minusg 18496 df-sbg 18497 df-lmod 20040 df-lss 20109 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-xms 23381 df-ms 23382 df-nm 23644 df-ngp 23645 df-nlm 23648 df-cph 24237

This theorem is referenced by: minveclem3 24498 minveclem4a 24499 minveclem4 24501

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