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 25624

Description: Lemma for minvec 25632. 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 4037	. . . . . 6 ⊢ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)} ⊆ 𝑌
3		minvec.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))
4	3	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑌 ∈ (LSubSp‘𝑈))
5		elpw2g 5309	. . . . . . 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 7117	. . . 4 ⊢ (𝜑 → (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}):ℝ⁺⟶𝒫 𝑌)
9	8	frnd 6721	. . 3 ⊢ (𝜑 → ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) ⊆ 𝒫 𝑌)
10	1, 9	eqsstrid 3978	. 2 ⊢ (𝜑 → 𝐹 ⊆ 𝒫 𝑌)
11		1rp 13038	. . . . . 6 ⊢ 1 ∈ ℝ⁺
12		eqid 2766	. . . . . . 7 ⊢ (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
13	12, 7	dmmptd 6687	. . . . . 6 ⊢ (𝜑 → dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = ℝ⁺)
14	11, 13	eleqtrrid 2873	. . . . 5 ⊢ (𝜑 → 1 ∈ dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}))
15	14	ne0d 4298	. . . 4 ⊢ (𝜑 → dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) ≠ ∅)
16		dm0rn0 5919	. . . . . 6 ⊢ (dom (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = ∅ ↔ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = ∅)
17	1	eqeq1i 2771	. . . . . 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 3013	. . . 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 25621	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆 ∈ ℝ)
31	30	resqcld 14181	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑆↑2) ∈ ℝ)
32		ltaddrp 13073	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆↑2) ∈ ℝ ∧ 𝑟 ∈ ℝ⁺) → (𝑆↑2) < ((𝑆↑2) + 𝑟))
33	31, 32	sylan 592	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (𝑆↑2) < ((𝑆↑2) + 𝑟))
34	31	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (𝑆↑2) ∈ ℝ)
35		rpre 13043	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ)
36	35	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ)
37	34, 36	readdcld 11256	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((𝑆↑2) + 𝑟) ∈ ℝ)
38	37	recnd 11255	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((𝑆↑2) + 𝑟) ∈ ℂ)
39	38	sqsqrtd 15519	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ((√‘((𝑆↑2) + 𝑟))↑2) = ((𝑆↑2) + 𝑟))
40	33, 39	breqtrrd 5144	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (𝑆↑2) < ((√‘((𝑆↑2) + 𝑟))↑2))
41	21, 22, 23, 24, 3, 25, 26, 27, 28	minveclem1 25620	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
42	41	simp1d 1160	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑅 ⊆ ℝ)
43	42	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑅 ⊆ ℝ)
44	41	simp2d 1161	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑅 ≠ ∅)
45	44	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑅 ≠ ∅)
46		0re 11228	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ
47	41	simp3d 1162	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)
48		breq1 5117	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 0 → (𝑦 ≤ 𝑤 ↔ 0 ≤ 𝑤))
49	48	ralbidv 3191	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 0 → (∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤 ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
50	49	rspcev 3584	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℝ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) → ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤)
51	46, 47, 50	sylancr 599	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤)
52	51	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤)
53		infrecl 12215	. . . . . . . . . . . . . . . . 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 2870	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑆 ∈ ℝ)
56		0red 11229	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ∈ ℝ)
57	55	sqge0d 14193	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ (𝑆↑2))
58	56, 34, 37, 57, 33	lelttrd 11386	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 < ((𝑆↑2) + 𝑟))
59	56, 37, 58	ltled 11376	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ ((𝑆↑2) + 𝑟))
60	37, 59	resqrtcld 15495	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (√‘((𝑆↑2) + 𝑟)) ∈ ℝ)
61	47	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)
62		infregelb 12217	. . . . . . . . . . . . . . . . . 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 5158	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ 𝑆)
66	37, 59	sqrtge0d 15498	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ (√‘((𝑆↑2) + 𝑟)))
67	55, 60, 65, 66	lt2sqd 14312	. . . . . . . . . . . . . 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 11375	. . . . . . . . . . . . 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 5122	. . . . . . . . . . . . 13 ⊢ ((√‘((𝑆↑2) + 𝑟)) ≤ 𝑆 ↔ (√‘((𝑆↑2) + 𝑟)) ≤ inf(𝑅, ℝ, < ))
72		infregelb 12217	. . . . . . . . . . . . . . 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 3324	. . . . . . . . . . . . . . 15 ⊢ (∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + 𝑟)) ≤ 𝑤 ↔ ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))(√‘((𝑆↑2) + 𝑟)) ≤ 𝑤)
75		fvex 6901	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁‘(𝐴 − 𝑦)) ∈ V
76	75	rgenw 3086	. . . . . . . . . . . . . . . 16 ⊢ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑦)) ∈ V
77		eqid 2766	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))
78		breq2 5118	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (𝑁‘(𝐴 − 𝑦)) → ((√‘((𝑆↑2) + 𝑟)) ≤ 𝑤 ↔ (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦))))
79	77, 78	ralrnmptw 7096	. . . . . . . . . . . . . . . 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 3120	. . . . . . . . . . 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 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (√‘((𝑆↑2) + 𝑟)) ∈ ℝ)
88		cphngp 25369	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ NrmGrp)
89	24, 88	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑈 ∈ NrmGrp)
90		ngpms 24794	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ NrmGrp → 𝑈 ∈ MetSp)
91		minvec.d	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))
92	21, 91	msmet 24651	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ MetSp → 𝐷 ∈ (Met‘𝑋))
93	89, 90, 92	3syl 19	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
94	93	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝐷 ∈ (Met‘𝑋))
95	26	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋)
96		eqid 2766	. . . . . . . . . . . . . . . . . . 19 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
97	21, 96	lssss 21094	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋)
98	4, 97	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝑌 ⊆ 𝑋)
99	98	sselda 3940	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
100		metcl 24526	. . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 0 ≤ (√‘((𝑆↑2) + 𝑟)))
103		metge0 24539	. . . . . . . . . . . . . . . 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 14313	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((√‘((𝑆↑2) + 𝑟)) ≤ (𝐴𝐷𝑦) ↔ ((√‘((𝑆↑2) + 𝑟))↑2) ≤ ((𝐴𝐷𝑦)↑2)))
106	91	oveqi 7436	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴𝐷𝑦) = (𝐴((dist‘𝑈) ↾ (𝑋 × 𝑋))𝑦)
107	95, 99	ovresd 7590	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝐴((dist‘𝑈) ↾ (𝑋 × 𝑋))𝑦) = (𝐴(dist‘𝑈)𝑦))
108	106, 107	eqtrid 2813	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑦) = (𝐴(dist‘𝑈)𝑦))
109	89	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝑈 ∈ NrmGrp)
110		eqid 2766	. . . . . . . . . . . . . . . . . 18 ⊢ (dist‘𝑈) = (dist‘𝑈)
111	23, 21, 22, 110	ngpds 24798	. . . . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑦) = (𝑁‘(𝐴 − 𝑦)))
114	113	breq2d 5126	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((√‘((𝑆↑2) + 𝑟)) ≤ (𝐴𝐷𝑦) ↔ (√‘((𝑆↑2) + 𝑟)) ≤ (𝑁‘(𝐴 − 𝑦))))
115	39	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((√‘((𝑆↑2) + 𝑟))↑2) = ((𝑆↑2) + 𝑟))
116	115	breq1d 5124	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((𝑆↑2) + 𝑟) ∈ ℝ)
120	101	resqcld 14181	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((𝐴𝐷𝑦)↑2) ∈ ℝ)
121	119, 120	letrid 11380	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (((𝑆↑2) + 𝑟) ≤ ((𝐴𝐷𝑦)↑2) ∨ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)))
122	121	ord 878	. . . . . . . . . . . 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 3181	. . . . . . . . . 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 4349	. . . . . . . . 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 3016	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ∅ ≠ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
129	128	neneqd 2966	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ¬ ∅ = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
130	129	nrexdv 3163	. . . . 5 ⊢ (𝜑 → ¬ ∃𝑟 ∈ ℝ⁺ ∅ = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
131	1	eleq2i 2858	. . . . . 6 ⊢ (∅ ∈ 𝐹 ↔ ∅ ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}))
132		0ex 5275	. . . . . . 7 ⊢ ∅ ∈ V
133	12	elrnmpt 5953	. . . . . . 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 3068	. . . 4 ⊢ (∅ ∉ 𝐹 ↔ ¬ ∅ ∈ 𝐹)
138	136, 137	sylibr 237	. . 3 ⊢ (𝜑 → ∅ ∉ 𝐹)
139	55	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝑆 ∈ ℝ)
140	139	resqcld 14181	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → (𝑆↑2) ∈ ℝ)
141	36	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → 𝑟 ∈ ℝ)
142	120, 140, 141	lesubadd2d 11831	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑌) → ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟 ↔ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)))
143	142	rabbidva 3425	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟} = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
144	143	mpteq2dva 5209	. . . . . . . . . 10 ⊢ (𝜑 → (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) = (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}))
145	144	rneqd 5933	. . . . . . . . 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 2820	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}))
147	146	eleq2d 2852	. . . . . . 7 ⊢ (𝜑 → (𝑢 ∈ 𝐹 ↔ 𝑢 ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟})))
148		breq2 5118	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑠 → ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟 ↔ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠))
149	148	rabbidv 3426	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟} = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠})
150	149	cbvmptv 5220	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) = (𝑠 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠})
151	150	elrnmpt 5953	. . . . . . . 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 3463	. . . . . . 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 2852	. . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ 𝐹 ↔ 𝑣 ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟})))
155		breq2 5118	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑡 → ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟 ↔ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡))
156	155	rabbidv 3426	. . . . . . . . . 10 ⊢ (𝑟 = 𝑡 → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟} = {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡})
157	156	cbvmptv 5220	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) = (𝑡 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡})
158	157	elrnmpt 5953	. . . . . . . 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 3463	. . . . . . 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 644	. . . . 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 3240	. . . . . 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 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝐷 ∈ (Met‘𝑋))
164	26	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋)
165	3, 97	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
166	165	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → 𝑌 ⊆ 𝑋)
167	166	sselda 3940	. . . . . . . . . . . . . . 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 14181	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → ((𝐴𝐷𝑦)↑2) ∈ ℝ)
170	31	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → (𝑆↑2) ∈ ℝ)
171	169, 170	resubcld 11660	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ∈ ℝ)
172		simplrl 789	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝑠 ∈ ℝ⁺)
173	172	rpred 13078	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝑠 ∈ ℝ)
174		simplrr 790	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝑡 ∈ ℝ⁺)
175	174	rpred 13078	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝑌) → 𝑡 ∈ ℝ)
176		lemin 13236	. . . . . . . . . . . 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 3425	. . . . . . . . . 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 4538	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺) → if(𝑠 ≤ 𝑡, 𝑠, 𝑡) ∈ ℝ⁺)
180	3	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → 𝑌 ∈ (LSubSp‘𝑈))
181		rabexg 5313	. . . . . . . . . . . . 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 2766	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}) = (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟})
184		breq2 5118	. . . . . . . . . . . . . 14 ⊢ (𝑟 = if(𝑠 ≤ 𝑡, 𝑠, 𝑡) → ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟 ↔ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ if(𝑠 ≤ 𝑡, 𝑠, 𝑡)))
185	184	rabbidv 3426	. . . . . . . . . . . . 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 5954	. . . . . . . . . . . 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 699	. . . . . . . . . . 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 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → 𝐹 = ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑟}))
189	187, 188	eleqtrrd 2869	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → {𝑦 ∈ 𝑌 ∣ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ if(𝑠 ≤ 𝑡, 𝑠, 𝑡)} ∈ 𝐹)
190	178, 189	eqeltrrd 2867	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑠 ∈ ℝ⁺ ∧ 𝑡 ∈ ℝ⁺)) → {𝑦 ∈ 𝑌 ∣ ((((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑠 ∧ (((𝐴𝐷𝑦)↑2) − (𝑆↑2)) ≤ 𝑡)} ∈ 𝐹)
191		ineq12 4171	. . . . . . . . . . 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 4272	. . . . . . . . . . 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 2817	. . . . . . . . . 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 2851	. . . . . . . . 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 3462	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
197	196	inex1 5291	. . . . . . . . . 10 ⊢ (𝑢 ∩ 𝑣) ∈ V
198	197	pwid 4590	. . . . . . . . 9 ⊢ (𝑢 ∩ 𝑣) ∈ 𝒫 (𝑢 ∩ 𝑣)
199		inelcm 4428	. . . . . . . . 9 ⊢ (((𝑢 ∩ 𝑣) ∈ 𝐹 ∧ (𝑢 ∩ 𝑣) ∈ 𝒫 (𝑢 ∩ 𝑣)) → (𝐹 ∩ 𝒫 (𝑢 ∩ 𝑣)) ≠ ∅)
200	198, 199	mpan2 704	. . . . . . . 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 3225	. . . . . 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 3209	. . 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 24023	. . 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 726	1 ⊢ (𝜑 → 𝐹 ∈ (fBas‘𝑌))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ≠ wne 2961 ∉ wnel 3067 ∀wral 3082 ∃wrex 3092 {crab 3419 Vcvv 3458 ∩ cin 3907 ⊆ wss 3908 ∅c0 4289 ifcif 4492 𝒫 cpw 4567 class class class wbr 5114 ↦ cmpt 5197 × cxp 5664 dom cdm 5666 ran crn 5667 ↾ cres 5668 ‘cfv 6543 (class class class)co 7423 infcinf 9411 ℝcr 11117 0cc0 11118 1c1 11119 + caddc 11121 < clt 11261 ≤ cle 11262 − cmin 11459 2c2 12313 ℝ⁺crp 13034 ↑cexp 14117 √csqrt 15310 Basecbs 17294 ↾_s cress 17315 distcds 17344 TopOpenctopn 17499 -_gcsg 19033 LSubSpclss 21089 Metcmet 21545 fBascfbas 21547 MetSpcms 24512 normcnm 24770 NrmGrpcngp 24771 ℂPreHilccph 25362 CMetSpccms 25528

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-er 8703 df-map 8835 df-en 8953 df-dom 8954 df-sdom 8955 df-sup 9412 df-inf 9413 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-n0 12523 df-z 12610 df-uz 12881 df-q 12991 df-rp 13035 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-seq 14058 df-exp 14118 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-0g 17519 df-topgen 17521 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-grp 19034 df-minusg 19035 df-sbg 19036 df-lmod 21020 df-lss 21090 df-psmet 21551 df-xmet 21552 df-met 21553 df-bl 21554 df-mopn 21555 df-fbas 21556 df-top 23088 df-topon 23105 df-topsp 23127 df-bases 23140 df-xms 24514 df-ms 24515 df-nm 24776 df-ngp 24777 df-nlm 24780 df-cph 25364

This theorem is used by: minveclem3 25625 minveclem4a 25626 minveclem4 25628

W3C validator