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Theorem minveclem4 24949

Description: Lemma for minvec 24953. 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 24947	. . 3 ⊢ (𝜑 → 𝑃 ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌))
15	14	elin2d 4200	. 2 ⊢ (𝜑 → 𝑃 ∈ 𝑌)
16	11	oveqi 7422	. . . . . . 7 ⊢ (𝐴𝐷𝑃) = (𝐴((dist‘𝑈) ↾ (𝑋 × 𝑋))𝑃)
17	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	minveclem4b 24948	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ 𝑋)
18	7, 17	ovresd 7574	. . . . . . 7 ⊢ (𝜑 → (𝐴((dist‘𝑈) ↾ (𝑋 × 𝑋))𝑃) = (𝐴(dist‘𝑈)𝑃))
19	16, 18	eqtrid 2785	. . . . . 6 ⊢ (𝜑 → (𝐴𝐷𝑃) = (𝐴(dist‘𝑈)𝑃))
20		cphngp 24690	. . . . . . . 8 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ NrmGrp)
21	4, 20	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ NrmGrp)
22		eqid 2733	. . . . . . . 8 ⊢ (dist‘𝑈) = (dist‘𝑈)
23	3, 1, 2, 22	ngpds 24113	. . . . . . 7 ⊢ ((𝑈 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝐴(dist‘𝑈)𝑃) = (𝑁‘(𝐴 − 𝑃)))
24	21, 7, 17, 23	syl3anc 1372	. . . . . 6 ⊢ (𝜑 → (𝐴(dist‘𝑈)𝑃) = (𝑁‘(𝐴 − 𝑃)))
25	19, 24	eqtrd 2773	. . . . 5 ⊢ (𝜑 → (𝐴𝐷𝑃) = (𝑁‘(𝐴 − 𝑃)))
26	25	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑃) = (𝑁‘(𝐴 − 𝑃)))
27		ngpms 24109	. . . . . . . 8 ⊢ (𝑈 ∈ NrmGrp → 𝑈 ∈ MetSp)
28	1, 11	msmet 23963	. . . . . . . 8 ⊢ (𝑈 ∈ MetSp → 𝐷 ∈ (Met‘𝑋))
29	21, 27, 28	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
30		metcl 23838	. . . . . . 7 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝐴𝐷𝑃) ∈ ℝ)
31	29, 7, 17, 30	syl3anc 1372	. . . . . 6 ⊢ (𝜑 → (𝐴𝐷𝑃) ∈ ℝ)
32	31	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑃) ∈ ℝ)
33	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	minveclem4c 24942	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ℝ)
34	33	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑆 ∈ ℝ)
35	21	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑈 ∈ NrmGrp)
36		cphlmod 24691	. . . . . . . . 9 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ LMod)
37	4, 36	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
38	37	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑈 ∈ LMod)
39	7	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋)
40		eqid 2733	. . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
41	1, 40	lssss 20547	. . . . . . . . 9 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋)
42	5, 41	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
43	42	sselda 3983	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
44	1, 2	lmodvsubcl 20517	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴 − 𝑦) ∈ 𝑋)
45	38, 39, 43, 44	syl3anc 1372	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐴 − 𝑦) ∈ 𝑋)
46	1, 3	nmcl 24125	. . . . . 6 ⊢ ((𝑈 ∈ NrmGrp ∧ (𝐴 − 𝑦) ∈ 𝑋) → (𝑁‘(𝐴 − 𝑦)) ∈ ℝ)
47	35, 45, 46	syl2anc 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑁‘(𝐴 − 𝑦)) ∈ ℝ)
48	33, 31	ltnled 11361	. . . . . . . 8 ⊢ (𝜑 → (𝑆 < (𝐴𝐷𝑃) ↔ ¬ (𝐴𝐷𝑃) ≤ 𝑆))
49	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	minveclem3b 24945	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 ∈ (fBas‘𝑌))
50		fbsspw 23336	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 ∈ (fBas‘𝑌) → 𝐹 ⊆ 𝒫 𝑌)
51	49, 50	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹 ⊆ 𝒫 𝑌)
52	42	sspwd 4616	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝒫 𝑌 ⊆ 𝒫 𝑋)
53	51, 52	sstrd 3993	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 ⊆ 𝒫 𝑋)
54	1	fvexi 6906	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑋 ∈ V
55	54	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑋 ∈ V)
56		fbasweak 23369	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ∈ (fBas‘𝑌) ∧ 𝐹 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ V) → 𝐹 ∈ (fBas‘𝑋))
57	49, 53, 55, 56	syl3anc 1372	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 ∈ (fBas‘𝑋))
58	57	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → 𝐹 ∈ (fBas‘𝑋))
59		fgcl 23382	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
60	58, 59	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
61		ssfg 23376	. . . . . . . . . . . . . . . . 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 11243	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((𝐴𝐷𝑃) + 𝑆) ∈ ℝ)
65	64	rehalfcld 12459	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (((𝐴𝐷𝑃) + 𝑆) / 2) ∈ ℝ)
66	65	resqcld 14090	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) ∈ ℝ)
67	33	resqcld 14090	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑆↑2) ∈ ℝ)
68	66, 67	resubcld 11642	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) − (𝑆↑2)) ∈ ℝ)
69	68	adantr 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → (((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) − (𝑆↑2)) ∈ ℝ)
70	33, 31, 33	ltadd1d 11807	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑆 < (𝐴𝐷𝑃) ↔ (𝑆 + 𝑆) < ((𝐴𝐷𝑃) + 𝑆)))
71	33	recnd 11242	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑆 ∈ ℂ)
72	71	2timesd 12455	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (2 · 𝑆) = (𝑆 + 𝑆))
73	72	breq1d 5159	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((2 · 𝑆) < ((𝐴𝐷𝑃) + 𝑆) ↔ (𝑆 + 𝑆) < ((𝐴𝐷𝑃) + 𝑆)))
74		2re 12286	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 2 ∈ ℝ
75		2pos 12315	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 0 < 2
76	74, 75	pm3.2i 472	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (2 ∈ ℝ ∧ 0 < 2)
77	76	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (2 ∈ ℝ ∧ 0 < 2))
78		ltmuldiv2 12088	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑆 ∈ ℝ ∧ ((𝐴𝐷𝑃) + 𝑆) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((2 · 𝑆) < ((𝐴𝐷𝑃) + 𝑆) ↔ 𝑆 < (((𝐴𝐷𝑃) + 𝑆) / 2)))
79	33, 64, 77, 78	syl3anc 1372	. . . . . . . . . . . . . . . . . . . . . . 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 24941	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
82	81	simp3d 1145	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)
83	81	simp1d 1143	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑅 ⊆ ℝ)
84	81	simp2d 1144	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑅 ≠ ∅)
85		0re 11216	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 ∈ ℝ
86		breq1 5152	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑤 ↔ 0 ≤ 𝑤))
87	86	ralbidv 3178	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = 0 → (∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤 ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
88	87	rspcev 3613	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((0 ∈ ℝ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤)
89	85, 82, 88	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤)
90	85	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 0 ∈ ℝ)
91		infregelb 12198	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (0 ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
93	82, 92	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 0 ≤ inf(𝑅, ℝ, < ))
94	93, 10	breqtrrdi 5191	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 0 ≤ 𝑆)
95		metge0 23851	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝑃))
96	29, 7, 17, 95	syl3anc 1372	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 0 ≤ (𝐴𝐷𝑃))
97	31, 33, 96, 94	addge0d 11790	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 0 ≤ ((𝐴𝐷𝑃) + 𝑆))
98		divge0 12083	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐴𝐷𝑃) + 𝑆) ∈ ℝ ∧ 0 ≤ ((𝐴𝐷𝑃) + 𝑆)) ∧ (2 ∈ ℝ ∧ 0 < 2)) → 0 ≤ (((𝐴𝐷𝑃) + 𝑆) / 2))
99	64, 97, 77, 98	syl21anc 837	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 0 ≤ (((𝐴𝐷𝑃) + 𝑆) / 2))
100	33, 65, 94, 99	lt2sqd 14219	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑆 < (((𝐴𝐷𝑃) + 𝑆) / 2) ↔ (𝑆↑2) < ((((𝐴𝐷𝑃) + 𝑆) / 2)↑2)))
101	67, 66	posdifd 11801	. . . . . . . . . . . . . . . . . . . . . 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 478	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → 0 < (((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) − (𝑆↑2)))
104	69, 103	elrpd 13013	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → (((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) − (𝑆↑2)) ∈ ℝ⁺)
105	63, 104	eqeltrid 2838	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → 𝑇 ∈ ℝ⁺)
106	5	adantr 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → 𝑌 ∈ (LSubSp‘𝑈))
107		rabexg 5332	. . . . . . . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) = (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
110		oveq2 7417	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑟 = 𝑇 → ((𝑆↑2) + 𝑟) = ((𝑆↑2) + 𝑇))
111	110	breq2d 5161	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑟 = 𝑇 → (((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟) ↔ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑇)))
112	111	rabbidv 3441	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 = 𝑇 → {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)} = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑇)})
113	109, 112	elrnmpt1s 5957	. . . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . . 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 2845	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑇)} ∈ 𝐹)
116	62, 115	sseldd 3984	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑇)} ∈ (𝑋filGen𝐹))
117		ssrab2 4078	. . . . . . . . . . . . . . . 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 7420	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑆↑2) + 𝑇) = ((𝑆↑2) + (((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) − (𝑆↑2)))
120	67	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → (𝑆↑2) ∈ ℝ)
121	120	recnd 11242	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → (𝑆↑2) ∈ ℂ)
122	65	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → (((𝐴𝐷𝑃) + 𝑆) / 2) ∈ ℝ)
123	122	resqcld 14090	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → ((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) ∈ ℝ)
124	123	recnd 11242	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → ((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) ∈ ℂ)
125	121, 124	pncan3d 11574	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → ((𝑆↑2) + (((((𝐴𝐷𝑃) + 𝑆) / 2)↑2) − (𝑆↑2))) = ((((𝐴𝐷𝑃) + 𝑆) / 2)↑2))
126	119, 125	eqtrid 2785	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → ((𝑆↑2) + 𝑇) = ((((𝐴𝐷𝑃) + 𝑆) / 2)↑2))
127	126	breq2d 5161	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → (((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑇) ↔ ((𝐴𝐷𝑦)↑2) ≤ ((((𝐴𝐷𝑃) + 𝑆) / 2)↑2)))
128	29	ad2antrr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → 𝐷 ∈ (Met‘𝑋))
129	7	ad2antrr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋)
130	43	adantlr 714	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
131		metcl 23838	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴𝐷𝑦) ∈ ℝ)
132	128, 129, 130, 131	syl3anc 1372	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑦) ∈ ℝ)
133		metge0 23851	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝑦))
134	128, 129, 130, 133	syl3anc 1372	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → 0 ≤ (𝐴𝐷𝑦))
135	99	ad2antrr 725	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → 0 ≤ (((𝐴𝐷𝑃) + 𝑆) / 2))
136	132, 122, 134, 135	le2sqd 14220	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → ((𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2) ↔ ((𝐴𝐷𝑦)↑2) ≤ ((((𝐴𝐷𝑃) + 𝑆) / 2)↑2)))
137	127, 136	bitr4d 282	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) ∧ 𝑦 ∈ 𝑌) → (((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑇) ↔ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)))
138	137	rabbidva 3440	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑇)} = {𝑦 ∈ 𝑌 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)})
139	42	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → 𝑌 ⊆ 𝑋)
140		rabss2 4076	. . . . . . . . . . . . . . . . 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 4021	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑇)} ⊆ {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)})
143		filss 23357	. . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)} ∈ (𝑋filGen𝐹))
145		flimclsi 23482	. . . . . . . . . . . . . 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 4199	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ (𝐽 fLim (𝑋filGen𝐹)))
148	147	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → 𝑃 ∈ (𝐽 fLim (𝑋filGen𝐹)))
149	146, 148	sseldd 3984	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → 𝑃 ∈ ((cls‘𝐽)‘{𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)}))
150		ngpxms 24110	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ NrmGrp → 𝑈 ∈ ∞MetSp)
151	1, 11	xmsxmet 23962	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ ∞MetSp → 𝐷 ∈ (∞Met‘𝑋))
152	21, 150, 151	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
153	152	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → 𝐷 ∈ (∞Met‘𝑋))
154	7	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → 𝐴 ∈ 𝑋)
155	65	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → (((𝐴𝐷𝑃) + 𝑆) / 2) ∈ ℝ)
156	155	rexrd 11264	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → (((𝐴𝐷𝑃) + 𝑆) / 2) ∈ ℝ^*)
157		eqid 2733	. . . . . . . . . . . . . . . 16 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
158		eqid 2733	. . . . . . . . . . . . . . . 16 ⊢ {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)} = {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)}
159	157, 158	blcld 24014	. . . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)} ∈ (Clsd‘(MetOpen‘𝐷)))
161	8, 1, 11	xmstopn 23957	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))
162	21, 150, 161	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐽 = (MetOpen‘𝐷))
163	162	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → 𝐽 = (MetOpen‘𝐷))
164	163	fveq2d 6896	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → (Clsd‘𝐽) = (Clsd‘(MetOpen‘𝐷)))
165	160, 164	eleqtrrd 2837	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)} ∈ (Clsd‘𝐽))
166		cldcls 22546	. . . . . . . . . . . . 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 2836	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → 𝑃 ∈ {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)})
169		oveq2 7417	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑃 → (𝐴𝐷𝑦) = (𝐴𝐷𝑃))
170	169	breq1d 5159	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑃 → ((𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2) ↔ (𝐴𝐷𝑃) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)))
171	170	elrab 3684	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ {𝑦 ∈ 𝑋 ∣ (𝐴𝐷𝑦) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)} ↔ (𝑃 ∈ 𝑋 ∧ (𝐴𝐷𝑃) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)))
172	171	simprbi 498	. . . . . . . . . . 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 11809	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴𝐷𝑃) ≤ 𝑆 ↔ ((𝐴𝐷𝑃) + (𝐴𝐷𝑃)) ≤ ((𝐴𝐷𝑃) + 𝑆)))
175	31	recnd 11242	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴𝐷𝑃) ∈ ℂ)
176	175	2timesd 12455	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 · (𝐴𝐷𝑃)) = ((𝐴𝐷𝑃) + (𝐴𝐷𝑃)))
177	176	breq1d 5159	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2 · (𝐴𝐷𝑃)) ≤ ((𝐴𝐷𝑃) + 𝑆) ↔ ((𝐴𝐷𝑃) + (𝐴𝐷𝑃)) ≤ ((𝐴𝐷𝑃) + 𝑆)))
178		lemuldiv2 12095	. . . . . . . . . . . . . 14 ⊢ (((𝐴𝐷𝑃) ∈ ℝ ∧ ((𝐴𝐷𝑃) + 𝑆) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((2 · (𝐴𝐷𝑃)) ≤ ((𝐴𝐷𝑃) + 𝑆) ↔ (𝐴𝐷𝑃) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)))
179	76, 178	mp3an3 1451	. . . . . . . . . . . . 13 ⊢ (((𝐴𝐷𝑃) ∈ ℝ ∧ ((𝐴𝐷𝑃) + 𝑆) ∈ ℝ) → ((2 · (𝐴𝐷𝑃)) ≤ ((𝐴𝐷𝑃) + 𝑆) ↔ (𝐴𝐷𝑃) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)))
180	31, 64, 179	syl2anc 585	. . . . . . . . . . . 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 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴𝐷𝑃) ≤ (((𝐴𝐷𝑃) + 𝑆) / 2)) → (𝐴𝐷𝑃) ≤ 𝑆)
183	173, 182	syldan 592	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑆 < (𝐴𝐷𝑃)) → (𝐴𝐷𝑃) ≤ 𝑆)
184	183	ex 414	. . . . . . . 8 ⊢ (𝜑 → (𝑆 < (𝐴𝐷𝑃) → (𝐴𝐷𝑃) ≤ 𝑆))
185	48, 184	sylbird 260	. . . . . . 7 ⊢ (𝜑 → (¬ (𝐴𝐷𝑃) ≤ 𝑆 → (𝐴𝐷𝑃) ≤ 𝑆))
186	185	pm2.18d 127	. . . . . 6 ⊢ (𝜑 → (𝐴𝐷𝑃) ≤ 𝑆)
187	186	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑃) ≤ 𝑆)
188	83	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑅 ⊆ ℝ)
189	89	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤)
190		simpr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
191		fvex 6905	. . . . . . . . 9 ⊢ (𝑁‘(𝐴 − 𝑦)) ∈ V
192		eqid 2733	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))
193	192	elrnmpt1 5958	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑌 ∧ (𝑁‘(𝐴 − 𝑦)) ∈ V) → (𝑁‘(𝐴 − 𝑦)) ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))))
194	190, 191, 193	sylancl 587	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑁‘(𝐴 − 𝑦)) ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))))
195	194, 9	eleqtrrdi 2845	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑁‘(𝐴 − 𝑦)) ∈ 𝑅)
196		infrelb 12199	. . . . . . 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 1372	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → inf(𝑅, ℝ, < ) ≤ (𝑁‘(𝐴 − 𝑦)))
198	10, 197	eqbrtrid 5184	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑆 ≤ (𝑁‘(𝐴 − 𝑦)))
199	32, 34, 47, 187, 198	letrd 11371	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑃) ≤ (𝑁‘(𝐴 − 𝑦)))
200	26, 199	eqbrtrrd 5173	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑁‘(𝐴 − 𝑃)) ≤ (𝑁‘(𝐴 − 𝑦)))
201	200	ralrimiva 3147	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑃)) ≤ (𝑁‘(𝐴 − 𝑦)))
202		oveq2 7417	. . . . . 6 ⊢ (𝑥 = 𝑃 → (𝐴 − 𝑥) = (𝐴 − 𝑃))
203	202	fveq2d 6896	. . . . 5 ⊢ (𝑥 = 𝑃 → (𝑁‘(𝐴 − 𝑥)) = (𝑁‘(𝐴 − 𝑃)))
204	203	breq1d 5159	. . . 4 ⊢ (𝑥 = 𝑃 → ((𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)) ↔ (𝑁‘(𝐴 − 𝑃)) ≤ (𝑁‘(𝐴 − 𝑦))))
205	204	ralbidv 3178	. . 3 ⊢ (𝑥 = 𝑃 → (∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)) ↔ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑃)) ≤ (𝑁‘(𝐴 − 𝑦))))
206	205	rspcev 3613	. 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 585	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 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 {crab 3433 Vcvv 3475 ⊆ wss 3949 ∅c0 4323 𝒫 cpw 4603 ∪ cuni 4909 class class class wbr 5149 ↦ cmpt 5232 × cxp 5675 ran crn 5678 ↾ cres 5679 ‘cfv 6544 (class class class)co 7409 infcinf 9436 ℝcr 11109 0cc0 11110 + caddc 11113 · cmul 11115 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 − cmin 11444 / cdiv 11871 2c2 12267 ℝ⁺crp 12974 ↑cexp 14027 Basecbs 17144 ↾_s cress 17173 distcds 17206 TopOpenctopn 17367 -_gcsg 18821 LModclmod 20471 LSubSpclss 20542 ∞Metcxmet 20929 Metcmet 20930 fBascfbas 20932 filGencfg 20933 MetOpencmopn 20934 Clsdccld 22520 clsccl 22522 Filcfil 23349 fLim cflim 23438 ∞MetSpcxms 23823 MetSpcms 23824 normcnm 24085 NrmGrpcngp 24086 ℂPreHilccph 24683 CMetSpccms 24849

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-tpos 8211 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fi 9406 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ico 13330 df-icc 13331 df-fz 13485 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-rest 17368 df-0g 17387 df-topgen 17389 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-mhm 18671 df-grp 18822 df-minusg 18823 df-sbg 18824 df-mulg 18951 df-subg 19003 df-ghm 19090 df-cmn 19650 df-abl 19651 df-mgp 19988 df-ur 20005 df-ring 20058 df-cring 20059 df-oppr 20150 df-dvdsr 20171 df-unit 20172 df-invr 20202 df-dvr 20215 df-rnghom 20251 df-subrg 20317 df-drng 20359 df-staf 20453 df-srng 20454 df-lmod 20473 df-lss 20543 df-lmhm 20633 df-lvec 20714 df-sra 20785 df-rgmod 20786 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-phl 21179 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-haus 22819 df-fil 23350 df-flim 23443 df-xms 23826 df-ms 23827 df-nm 24091 df-ngp 24092 df-nlm 24095 df-clm 24579 df-cph 24685 df-cfil 24772 df-cmet 24774 df-cms 24852

This theorem is referenced by: minveclem5 24950

W3C validator