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Theorem minveclem4 24596

Description: Lemma for minvec 24600. The convergent point of the Cauchy sequence 𝐹 attains the minimum distance, and so is closer to 𝐴 than any other point in 𝑌. (Contributed by Mario Carneiro, 7-May-2014.) (Revised 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)})
minvec.p	⊢ 𝑃 = ∪ (𝐽 fLim (𝑋filGen𝐹))
minvec.t	⊢ 𝑇 = (((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) − (𝑆↑2))

**Assertion**
Ref	Expression
minveclem4	⊢ (𝜑 → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))

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

Proof of Theorem minveclem4 Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		minvec.x	. . . 4 ⊢ 𝑋 = (Base‘𝑈)
2		minvec.m	. . . 4 ⊢ − = (-_g‘𝑈)
3		minvec.n	. . . 4 ⊢ 𝑁 = (norm‘𝑈)
4		minvec.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)
5		minvec.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))
6		minvec.w	. . . 4 ⊢ (𝜑 → (𝑈 ↾_s 𝑌) ∈ CMetSp)
7		minvec.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
8		minvec.j	. . . 4 ⊢ 𝐽 = (TopOpen‘𝑈)
9		minvec.r	. . . 4 ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))
10		minvec.s	. . . 4 ⊢ 𝑆 = inf(𝑅, ℝ, < )
11		minvec.d	. . . 4 ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))
12		minvec.f	. . . 4 ⊢ 𝐹 = ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
13		minvec.p	. . . 4 ⊢ 𝑃 = ∪ (𝐽 fLim (𝑋filGen𝐹))
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	minveclem4a 24594	. . 3 ⊢ (𝜑 → 𝑃 ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌))
15	14	elin2d 4133	. 2 ⊢ (𝜑 → 𝑃 ∈ 𝑌)
16	11	oveqi 7288	. . . . . . 7 ⊢ (𝐴𝐷𝑃) = (𝐴((dist‘𝑈) ↾ (𝑋 × 𝑋))𝑃)
17	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	minveclem4b 24595	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ 𝑋)
18	7, 17	ovresd 7439	. . . . . . 7 ⊢ (𝜑 → (𝐴((dist‘𝑈) ↾ (𝑋 × 𝑋))𝑃) = (𝐴(dist‘𝑈)𝑃))
19	16, 18	eqtrid 2790	. . . . . 6 ⊢ (𝜑 → (𝐴𝐷𝑃) = (𝐴(dist‘𝑈)𝑃))
20		cphngp 24337	. . . . . . . 8 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ NrmGrp)
21	4, 20	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ NrmGrp)
22		eqid 2738	. . . . . . . 8 ⊢ (dist‘𝑈) = (dist‘𝑈)
23	3, 1, 2, 22	ngpds 23760	. . . . . . 7 ⊢ ((𝑈 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝐴(dist‘𝑈)𝑃) = (𝑁‘(𝐴 − 𝑃)))
24	21, 7, 17, 23	syl3anc 1370	. . . . . 6 ⊢ (𝜑 → (𝐴(dist‘𝑈)𝑃) = (𝑁‘(𝐴 − 𝑃)))
25	19, 24	eqtrd 2778	. . . . 5 ⊢ (𝜑 → (𝐴𝐷𝑃) = (𝑁‘(𝐴 − 𝑃)))
26	25	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑃) = (𝑁‘(𝐴 − 𝑃)))
27		ngpms 23756	. . . . . . . 8 ⊢ (𝑈 ∈ NrmGrp → 𝑈 ∈ MetSp)
28	1, 11	msmet 23610	. . . . . . . 8 ⊢ (𝑈 ∈ MetSp → 𝐷 ∈ (Met‘𝑋))
29	21, 27, 28	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
30		metcl 23485	. . . . . . 7 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝐴𝐷𝑃) ∈ ℝ)
31	29, 7, 17, 30	syl3anc 1370	. . . . . 6 ⊢ (𝜑 → (𝐴𝐷𝑃) ∈ ℝ)
32	31	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑃) ∈ ℝ)
33	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	minveclem4c 24589	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ℝ)
34	33	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑆 ∈ ℝ)
35	21	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑈 ∈ NrmGrp)
36		cphlmod 24338	. . . . . . . . 9 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ LMod)
37	4, 36	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
38	37	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑈 ∈ LMod)
39	7	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋)
40		eqid 2738	. . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
41	1, 40	lssss 20198	. . . . . . . . 9 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋)
42	5, 41	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
43	42	sselda 3921	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
44	1, 2	lmodvsubcl 20168	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴 − 𝑦) ∈ 𝑋)
45	38, 39, 43, 44	syl3anc 1370	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐴 − 𝑦) ∈ 𝑋)
46	1, 3	nmcl 23772	. . . . . 6 ⊢ ((𝑈 ∈ NrmGrp ∧ (𝐴 − 𝑦) ∈ 𝑋) → (𝑁‘(𝐴 − 𝑦)) ∈ ℝ)
47	35, 45, 46	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑁‘(𝐴 − 𝑦)) ∈ ℝ)
48	33, 31	ltnled 11122	. . . . . . . 8 ⊢ (𝜑 → (𝑆 < (𝐴𝐷𝑃) ↔ ¬ (𝐴𝐷𝑃) ≤ 𝑆))
49	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	minveclem3b 24592	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 ∈ (fBas‘𝑌))
50		fbsspw 22983	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 ∈ (fBas‘𝑌) → 𝐹 ⊆ 𝒫 𝑌)
51	49, 50	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹 ⊆ 𝒫 𝑌)
52	42	sspwd 4548	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝒫 𝑌 ⊆ 𝒫 𝑋)
53	51, 52	sstrd 3931	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 ⊆ 𝒫 𝑋)
54	1	fvexi 6788	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑋 ∈ V
55	54	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑋 ∈ V)
56		fbasweak 23016	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ∈ (fBas‘𝑌) ∧ 𝐹 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ V) → 𝐹 ∈ (fBas‘𝑋))
57	49, 53, 55, 56	syl3anc 1370	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 ∈ (fBas‘𝑋))
58	57	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → 𝐹 ∈ (fBas‘𝑋))
59		fgcl 23029	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
60	58, 59	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
61		ssfg 23023	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
62	58, 61	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → 𝐹 ⊆ (𝑋filGen𝐹))
63		minvec.t	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑇 = (((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) − (𝑆↑2))
64	31, 33	readdcld 11004	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((𝐴𝐷𝑃) + 𝑆) ∈ ℝ)
65	64	rehalfcld 12220	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (((𝐴𝐷𝑃) + 𝑆) / 2) ∈ ℝ)
66	65	resqcld 13965	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) ∈ ℝ)
67	33	resqcld 13965	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑆↑2) ∈ ℝ)
68	66, 67	resubcld 11403	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) − (𝑆↑2)) ∈ ℝ)
69	68	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → (((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) − (𝑆↑2)) ∈ ℝ)
70	33, 31, 33	ltadd1d 11568	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑆 < (𝐴𝐷𝑃) ↔ (𝑆 + 𝑆) < ((𝐴𝐷𝑃) + 𝑆)))
71	33	recnd 11003	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑆 ∈ ℂ)
72	71	2timesd 12216	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (2 · 𝑆) = (𝑆 + 𝑆))
73	72	breq1d 5084	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((2 · 𝑆) < ((𝐴𝐷𝑃) + 𝑆) ↔ (𝑆 + 𝑆) < ((𝐴𝐷𝑃) + 𝑆)))
74		2re 12047	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 2 ∈ ℝ
75		2pos 12076	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 0 < 2
76	74, 75	pm3.2i 471	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (2 ∈ ℝ ∧ 0 < 2)
77	76	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (2 ∈ ℝ ∧ 0 < 2))
78		ltmuldiv2 11849	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑆 ∈ ℝ ∧ ((𝐴𝐷𝑃) + 𝑆) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((2 · 𝑆) < ((𝐴𝐷𝑃) + 𝑆) ↔ 𝑆 < (((𝐴𝐷𝑃) + 𝑆) / 2)))
79	33, 64, 77, 78	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((2 · 𝑆) < ((𝐴𝐷𝑃) + 𝑆) ↔ 𝑆 < (((𝐴𝐷𝑃) + 𝑆) / 2)))
80	70, 73, 79	3bitr2d 307	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑆 < (𝐴𝐷𝑃) ↔ 𝑆 < (((𝐴𝐷𝑃) + 𝑆) / 2)))
81	1, 2, 3, 4, 5, 6, 7, 8, 9	minveclem1 24588	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
82	81	simp3d 1143	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)
83	81	simp1d 1141	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑅 ⊆ ℝ)
84	81	simp2d 1142	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑅 ≠ ∅)
85		0re 10977	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 ∈ ℝ
86		breq1 5077	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑤 ↔ 0 ≤ 𝑤))
87	86	ralbidv 3112	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = 0 → (∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤 ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
88	87	rspcev 3561	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((0 ∈ ℝ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤)
89	85, 82, 88	sylancr 587	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤)
90	85	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 0 ∈ ℝ)
91		infregelb 11959	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤) ∧ 0 ∈ ℝ) → (0 ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
92	83, 84, 89, 90, 91	syl31anc 1372	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (0 ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
93	82, 92	mpbird 256	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 0 ≤ inf(𝑅, ℝ, < ))
94	93, 10	breqtrrdi 5116	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 0 ≤ 𝑆)
95		metge0 23498	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝑃))
96	29, 7, 17, 95	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 0 ≤ (𝐴𝐷𝑃))
97	31, 33, 96, 94	addge0d 11551	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 0 ≤ ((𝐴𝐷𝑃) + 𝑆))
98		divge0 11844	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐴𝐷𝑃) + 𝑆) ∈ ℝ ∧ 0 ≤ ((𝐴𝐷𝑃) + 𝑆)) ∧ (2 ∈ ℝ ∧ 0 < 2)) → 0 ≤ (((𝐴𝐷𝑃) + 𝑆) / 2))
99	64, 97, 77, 98	syl21anc 835	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 0 ≤ (((𝐴𝐷𝑃) + 𝑆) / 2))
100	33, 65, 94, 99	lt2sqd 13973	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑆 < (((𝐴𝐷𝑃) + 𝑆) / 2) ↔ (𝑆↑2) < ((((𝐴𝐷𝑃) + 𝑆) / 2)↑2)))
101	67, 66	posdifd 11562	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((𝑆↑2) < ((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) ↔ 0 < (((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) − (𝑆↑2))))
102	80, 100, 101	3bitrd 305	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑆 < (𝐴𝐷𝑃) ↔ 0 < (((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) − (𝑆↑2))))
103	102	biimpa 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → 0 < (((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) − (𝑆↑2)))
104	69, 103	elrpd 12769	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → (((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) − (𝑆↑2)) ∈ ℝ⁺)
105	63, 104	eqeltrid 2843	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → 𝑇 ∈ ℝ⁺)
106	5	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → 𝑌 ∈ (LSubSp‘𝑈))
107		rabexg 5255	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑇)} ∈ V)
108	106, 107	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑇)} ∈ V)
109		eqid 2738	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
110		oveq2 7283	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑟 = 𝑇 → ((𝑆↑2) + 𝑟) = ((𝑆↑2) + 𝑇))
111	110	breq2d 5086	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑟 = 𝑇 → (((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟) ↔ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑇)))
112	111	rabbidv 3414	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 = 𝑇 → {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)} = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑇)})
113	109, 112	elrnmpt1s 5866	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑇 ∈ ℝ⁺ ∧ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑇)} ∈ V) → {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑇)} ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}))
114	105, 108, 113	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑇)} ∈ ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}))
115	114, 12	eleqtrrdi 2850	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑇)} ∈ 𝐹)
116	62, 115	sseldd 3922	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑇)} ∈ (𝑋filGen𝐹))
117		ssrab2 4013	. . . . . . . . . . . . . . . 16 ⊢ {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)} ⊆ 𝑋
118	117	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)} ⊆ 𝑋)
119	63	oveq2i 7286	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑆↑2) + 𝑇) = ((𝑆↑2) + (((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) − (𝑆↑2)))
120	67	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → (𝑆↑2) ∈ ℝ)
121	120	recnd 11003	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → (𝑆↑2) ∈ ℂ)
122	65	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → (((𝐴𝐷𝑃) + 𝑆) / 2) ∈ ℝ)
123	122	resqcld 13965	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → ((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) ∈ ℝ)
124	123	recnd 11003	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → ((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) ∈ ℂ)
125	121, 124	pncan3d 11335	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → ((𝑆↑2) + (((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) − (𝑆↑2))) = ((((𝐴𝐷𝑃) + 𝑆) / 2)↑2))
126	119, 125	eqtrid 2790	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → ((𝑆↑2) + 𝑇) = ((((𝐴𝐷𝑃) + 𝑆) / 2)↑2))
127	126	breq2d 5086	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → (((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑇) ↔ ((𝐴𝐷𝑦)↑2) ≤ ((((𝐴𝐷𝑃) + 𝑆) / 2)↑2)))
128	29	ad2antrr 723	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → 𝐷 ∈ (Met‘𝑋))
129	7	ad2antrr 723	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋)
130	43	adantlr 712	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
131		metcl 23485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴𝐷𝑦) ∈ ℝ)
132	128, 129, 130, 131	syl3anc 1370	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑦) ∈ ℝ)
133		metge0 23498	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝑦))
134	128, 129, 130, 133	syl3anc 1370	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → 0 ≤ (𝐴𝐷𝑦))
135	99	ad2antrr 723	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → 0 ≤ (((𝐴𝐷𝑃) + 𝑆) / 2))
136	132, 122, 134, 135	le2sqd 13974	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → ((𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2) ↔ ((𝐴𝐷𝑦)↑2) ≤ ((((𝐴𝐷𝑃) + 𝑆) / 2)↑2)))
137	127, 136	bitr4d 281	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → (((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑇) ↔ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)))
138	137	rabbidva 3413	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑇)} = {𝑦 ∈ 𝑌 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)})
139	42	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → 𝑌 ⊆ 𝑋)
140		rabss2 4011	. . . . . . . . . . . . . . . . 17 ⊢ (𝑌 ⊆ 𝑋 → {𝑦 ∈ 𝑌 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)} ⊆ {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)})
141	139, 140	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → {𝑦 ∈ 𝑌 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)} ⊆ {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)})
142	138, 141	eqsstrd 3959	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑇)} ⊆ {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)})
143		filss 23004	. . . . . . . . . . . . . . 15 ⊢ (((𝑋filGen𝐹) ∈ (Fil‘𝑋) ∧ ({𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑇)} ∈ (𝑋filGen𝐹) ∧ {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)} ⊆ 𝑋 ∧ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑇)} ⊆ {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)})) → {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)} ∈ (𝑋filGen𝐹))
144	60, 116, 118, 142, 143	syl13anc 1371	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)} ∈ (𝑋filGen𝐹))
145		flimclsi 23129	. . . . . . . . . . . . . 14 ⊢ ({𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)} ∈ (𝑋filGen𝐹) → (𝐽 fLim (𝑋filGen𝐹)) ⊆ ((cls‘𝐽)‘{𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)}))
146	144, 145	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → (𝐽 fLim (𝑋filGen𝐹)) ⊆ ((cls‘𝐽)‘{𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)}))
147	14	elin1d 4132	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ (𝐽 fLim (𝑋filGen𝐹)))
148	147	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → 𝑃 ∈ (𝐽 fLim (𝑋filGen𝐹)))
149	146, 148	sseldd 3922	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → 𝑃 ∈ ((cls‘𝐽)‘{𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)}))
150		ngpxms 23757	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ NrmGrp → 𝑈 ∈ ∞MetSp)
151	1, 11	xmsxmet 23609	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ ∞MetSp → 𝐷 ∈ (∞Met‘𝑋))
152	21, 150, 151	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
153	152	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → 𝐷 ∈ (∞Met‘𝑋))
154	7	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → 𝐴 ∈ 𝑋)
155	65	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → (((𝐴𝐷𝑃) + 𝑆) / 2) ∈ ℝ)
156	155	rexrd 11025	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → (((𝐴𝐷𝑃) + 𝑆) / 2) ∈ ℝ^*)
157		eqid 2738	. . . . . . . . . . . . . . . 16 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
158		eqid 2738	. . . . . . . . . . . . . . . 16 ⊢ {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)} = {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)}
159	157, 158	blcld 23661	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ (((𝐴𝐷𝑃) + 𝑆) / 2) ∈ ℝ^*) → {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)} ∈ (Clsd‘(MetOpen‘𝐷)))
160	153, 154, 156, 159	syl3anc 1370	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)} ∈ (Clsd‘(MetOpen‘𝐷)))
161	8, 1, 11	xmstopn 23604	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))
162	21, 150, 161	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐽 = (MetOpen‘𝐷))
163	162	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → 𝐽 = (MetOpen‘𝐷))
164	163	fveq2d 6778	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → (Clsd‘𝐽) = (Clsd‘(MetOpen‘𝐷)))
165	160, 164	eleqtrrd 2842	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)} ∈ (Clsd‘𝐽))
166		cldcls 22193	. . . . . . . . . . . . 13 ⊢ ({𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)} ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘{𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)}) = {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)})
167	165, 166	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → ((cls‘𝐽)‘{𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)}) = {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)})
168	149, 167	eleqtrd 2841	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → 𝑃 ∈ {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)})
169		oveq2 7283	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑃 → (𝐴𝐷𝑦) = (𝐴𝐷𝑃))
170	169	breq1d 5084	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑃 → ((𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2) ↔ (𝐴𝐷𝑃) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)))
171	170	elrab 3624	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)} ↔ (𝑃 ∈ 𝑋 ∧ (𝐴𝐷𝑃) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)))
172	171	simprbi 497	. . . . . . . . . . 11 ⊢ (𝑃 ∈ {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)} → (𝐴𝐷𝑃) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2))
173	168, 172	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → (𝐴𝐷𝑃) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2))
174	31, 33, 31	leadd2d 11570	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴𝐷𝑃) ≤ 𝑆 ↔ ((𝐴𝐷𝑃) + (𝐴𝐷𝑃)) ≤ ((𝐴𝐷𝑃) + 𝑆)))
175	31	recnd 11003	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴𝐷𝑃) ∈ ℂ)
176	175	2timesd 12216	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 · (𝐴𝐷𝑃)) = ((𝐴𝐷𝑃) + (𝐴𝐷𝑃)))
177	176	breq1d 5084	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2 · (𝐴𝐷𝑃)) ≤ ((𝐴𝐷𝑃) + 𝑆) ↔ ((𝐴𝐷𝑃) + (𝐴𝐷𝑃)) ≤ ((𝐴𝐷𝑃) + 𝑆)))
178		lemuldiv2 11856	. . . . . . . . . . . . . 14 ⊢ (((𝐴𝐷𝑃) ∈ ℝ ∧ ((𝐴𝐷𝑃) + 𝑆) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((2 · (𝐴𝐷𝑃)) ≤ ((𝐴𝐷𝑃) + 𝑆) ↔ (𝐴𝐷𝑃) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)))
179	76, 178	mp3an3 1449	. . . . . . . . . . . . 13 ⊢ (((𝐴𝐷𝑃) ∈ ℝ ∧ ((𝐴𝐷𝑃) + 𝑆) ∈ ℝ) → ((2 · (𝐴𝐷𝑃)) ≤ ((𝐴𝐷𝑃) + 𝑆) ↔ (𝐴𝐷𝑃) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)))
180	31, 64, 179	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2 · (𝐴𝐷𝑃)) ≤ ((𝐴𝐷𝑃) + 𝑆) ↔ (𝐴𝐷𝑃) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)))
181	174, 177, 180	3bitr2d 307	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴𝐷𝑃) ≤ 𝑆 ↔ (𝐴𝐷𝑃) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)))
182	181	biimpar 478	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴𝐷𝑃) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)) → (𝐴𝐷𝑃) ≤ 𝑆)
183	173, 182	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → (𝐴𝐷𝑃) ≤ 𝑆)
184	183	ex 413	. . . . . . . 8 ⊢ (𝜑 → (𝑆 < (𝐴𝐷𝑃) → (𝐴𝐷𝑃) ≤ 𝑆))
185	48, 184	sylbird 259	. . . . . . 7 ⊢ (𝜑 → (¬ (𝐴𝐷𝑃) ≤ 𝑆 → (𝐴𝐷𝑃) ≤ 𝑆))
186	185	pm2.18d 127	. . . . . 6 ⊢ (𝜑 → (𝐴𝐷𝑃) ≤ 𝑆)
187	186	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑃) ≤ 𝑆)
188	83	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑅 ⊆ ℝ)
189	89	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤)
190		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
191		fvex 6787	. . . . . . . . 9 ⊢ (𝑁‘(𝐴 − 𝑦)) ∈ V
192		eqid 2738	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))
193	192	elrnmpt1 5867	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑌 ∧ (𝑁‘(𝐴 − 𝑦)) ∈ V) → (𝑁‘(𝐴 − 𝑦)) ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))))
194	190, 191, 193	sylancl 586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑁‘(𝐴 − 𝑦)) ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))))
195	194, 9	eleqtrrdi 2850	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑁‘(𝐴 − 𝑦)) ∈ 𝑅)
196		infrelb 11960	. . . . . . 7 ⊢ ((𝑅 ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤 ∧ (𝑁‘(𝐴 − 𝑦)) ∈ 𝑅) → inf(𝑅, ℝ, < ) ≤ (𝑁‘(𝐴 − 𝑦)))
197	188, 189, 195, 196	syl3anc 1370	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → inf(𝑅, ℝ, < ) ≤ (𝑁‘(𝐴 − 𝑦)))
198	10, 197	eqbrtrid 5109	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑆 ≤ (𝑁‘(𝐴 − 𝑦)))
199	32, 34, 47, 187, 198	letrd 11132	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑃) ≤ (𝑁‘(𝐴 − 𝑦)))
200	26, 199	eqbrtrrd 5098	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑁‘(𝐴 − 𝑃)) ≤ (𝑁‘(𝐴 − 𝑦)))
201	200	ralrimiva 3103	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑃)) ≤ (𝑁‘(𝐴 − 𝑦)))
202		oveq2 7283	. . . . . 6 ⊢ (𝑥 = 𝑃 → (𝐴 − 𝑥) = (𝐴 − 𝑃))
203	202	fveq2d 6778	. . . . 5 ⊢ (𝑥 = 𝑃 → (𝑁‘(𝐴 − 𝑥)) = (𝑁‘(𝐴 − 𝑃)))
204	203	breq1d 5084	. . . 4 ⊢ (𝑥 = 𝑃 → ((𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)) ↔ (𝑁‘(𝐴 − 𝑃)) ≤ (𝑁‘(𝐴 − 𝑦))))
205	204	ralbidv 3112	. . 3 ⊢ (𝑥 = 𝑃 → (∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)) ↔ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑃)) ≤ (𝑁‘(𝐴 − 𝑦))))
206	205	rspcev 3561	. 2 ⊢ ((𝑃 ∈ 𝑌 ∧ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑃)) ≤ (𝑁‘(𝐴 − 𝑦))) → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))
207	15, 201, 206	syl2anc 584	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 {crab 3068 Vcvv 3432 ⊆ wss 3887 ∅c0 4256 𝒫 cpw 4533 ∪ cuni 4839 class class class wbr 5074 ↦ cmpt 5157 × cxp 5587 ran crn 5590 ↾ cres 5591 ‘cfv 6433 (class class class)co 7275 infcinf 9200 ℝcr 10870 0cc0 10871 + caddc 10874 · cmul 10876 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 − cmin 11205 / cdiv 11632 2c2 12028 ℝ⁺crp 12730 ↑cexp 13782 Basecbs 16912 ↾_s cress 16941 distcds 16971 TopOpenctopn 17132 -_gcsg 18579 LModclmod 20123 LSubSpclss 20193 ∞Metcxmet 20582 Metcmet 20583 fBascfbas 20585 filGencfg 20586 MetOpencmopn 20587 Clsdccld 22167 clsccl 22169 Filcfil 22996 fLim cflim 23085 ∞MetSpcxms 23470 MetSpcms 23471 normcnm 23732 NrmGrpcngp 23733 ℂPreHilccph 24330 CMetSpccms 24496

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fi 9170 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ico 13085 df-icc 13086 df-fz 13240 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-rest 17133 df-0g 17152 df-topgen 17154 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-ghm 18832 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-rnghom 19959 df-drng 19993 df-subrg 20022 df-staf 20105 df-srng 20106 df-lmod 20125 df-lss 20194 df-lmhm 20284 df-lvec 20365 df-sra 20434 df-rgmod 20435 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-phl 20831 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-haus 22466 df-fil 22997 df-flim 23090 df-xms 23473 df-ms 23474 df-nm 23738 df-ngp 23739 df-nlm 23742 df-clm 24226 df-cph 24332 df-cfil 24419 df-cmet 24421 df-cms 24499

This theorem is referenced by: minveclem5 24597

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