ipcnlem2 - Metamath Proof Explorer

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Theorem ipcnlem2 25192

Description: The inner product operation of a subcomplex pre-Hilbert space is continuous. (Contributed by Mario Carneiro, 13-Oct-2015.)

**Hypotheses**
Ref	Expression
ipcn.v	⊢ 𝑉 = (Base‘𝑊)
ipcn.h	⊢ , = (·_𝑖‘𝑊)
ipcn.d	⊢ 𝐷 = (dist‘𝑊)
ipcn.n	⊢ 𝑁 = (norm‘𝑊)
ipcn.t	⊢ 𝑇 = ((𝑅 / 2) / ((𝑁‘𝐴) + 1))
ipcn.u	⊢ 𝑈 = ((𝑅 / 2) / ((𝑁‘𝐵) + 𝑇))
ipcn.w	⊢ (𝜑 → 𝑊 ∈ ℂPreHil)
ipcn.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ipcn.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)
ipcn.r	⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
ipcn.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
ipcn.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
ipcn.1	⊢ (𝜑 → (𝐴𝐷𝑋) < 𝑈)
ipcn.2	⊢ (𝜑 → (𝐵𝐷𝑌) < 𝑇)

**Assertion**
Ref	Expression
ipcnlem2	⊢ (𝜑 → (abs‘((𝐴 , 𝐵) − (𝑋 , 𝑌))) < 𝑅)

**Proof of Theorem ipcnlem2**
Step	Hyp	Ref	Expression
1		ipcn.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ ℂPreHil)
2		ipcn.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		ipcn.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
4		ipcn.v	. . . 4 ⊢ 𝑉 = (Base‘𝑊)
5		ipcn.h	. . . 4 ⊢ , = (·_𝑖‘𝑊)
6	4, 5	cphipcl 25139	. . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 , 𝐵) ∈ ℂ)
7	1, 2, 3, 6	syl3anc 1368	. 2 ⊢ (𝜑 → (𝐴 , 𝐵) ∈ ℂ)
8		ipcn.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
9		ipcn.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
10	4, 5	cphipcl 25139	. . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 , 𝑌) ∈ ℂ)
11	1, 8, 9, 10	syl3anc 1368	. 2 ⊢ (𝜑 → (𝑋 , 𝑌) ∈ ℂ)
12	4, 5	cphipcl 25139	. . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝐴 , 𝑌) ∈ ℂ)
13	1, 2, 9, 12	syl3anc 1368	. 2 ⊢ (𝜑 → (𝐴 , 𝑌) ∈ ℂ)
14		ipcn.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
15	14	rpred 13056	. 2 ⊢ (𝜑 → 𝑅 ∈ ℝ)
16	7, 13	subcld 11609	. . . 4 ⊢ (𝜑 → ((𝐴 , 𝐵) − (𝐴 , 𝑌)) ∈ ℂ)
17	16	abscld 15423	. . 3 ⊢ (𝜑 → (abs‘((𝐴 , 𝐵) − (𝐴 , 𝑌))) ∈ ℝ)
18		cphnlm 25120	. . . . . . . . 9 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
19	1, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ NrmMod)
20		nlmngp 24614	. . . . . . . 8 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)
21	19, 20	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ NrmGrp)
22		ipcn.n	. . . . . . . 8 ⊢ 𝑁 = (norm‘𝑊)
23	4, 22	nmcl 24545	. . . . . . 7 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) ∈ ℝ)
24	21, 2, 23	syl2anc 582	. . . . . 6 ⊢ (𝜑 → (𝑁‘𝐴) ∈ ℝ)
25	4, 22	nmge0 24546	. . . . . . 7 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝐴 ∈ 𝑉) → 0 ≤ (𝑁‘𝐴))
26	21, 2, 25	syl2anc 582	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝑁‘𝐴))
27	24, 26	ge0p1rpd 13086	. . . . 5 ⊢ (𝜑 → ((𝑁‘𝐴) + 1) ∈ ℝ⁺)
28	27	rpred 13056	. . . 4 ⊢ (𝜑 → ((𝑁‘𝐴) + 1) ∈ ℝ)
29		ngpms 24529	. . . . . 6 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ MetSp)
30	21, 29	syl 17	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ MetSp)
31		ipcn.d	. . . . . 6 ⊢ 𝐷 = (dist‘𝑊)
32	4, 31	mscl 24387	. . . . 5 ⊢ ((𝑊 ∈ MetSp ∧ 𝐵 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝐵𝐷𝑌) ∈ ℝ)
33	30, 3, 9, 32	syl3anc 1368	. . . 4 ⊢ (𝜑 → (𝐵𝐷𝑌) ∈ ℝ)
34	28, 33	remulcld 11282	. . 3 ⊢ (𝜑 → (((𝑁‘𝐴) + 1) · (𝐵𝐷𝑌)) ∈ ℝ)
35	15	rehalfcld 12497	. . 3 ⊢ (𝜑 → (𝑅 / 2) ∈ ℝ)
36	24, 33	remulcld 11282	. . . 4 ⊢ (𝜑 → ((𝑁‘𝐴) · (𝐵𝐷𝑌)) ∈ ℝ)
37		eqid 2728	. . . . . . . 8 ⊢ (-_g‘𝑊) = (-_g‘𝑊)
38	5, 4, 37	cphsubdi 25157	. . . . . . 7 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐴 , (𝐵(-_g‘𝑊)𝑌)) = ((𝐴 , 𝐵) − (𝐴 , 𝑌)))
39	1, 2, 3, 9, 38	syl13anc 1369	. . . . . 6 ⊢ (𝜑 → (𝐴 , (𝐵(-_g‘𝑊)𝑌)) = ((𝐴 , 𝐵) − (𝐴 , 𝑌)))
40	39	fveq2d 6906	. . . . 5 ⊢ (𝜑 → (abs‘(𝐴 , (𝐵(-_g‘𝑊)𝑌))) = (abs‘((𝐴 , 𝐵) − (𝐴 , 𝑌))))
41		ngpgrp 24528	. . . . . . . . 9 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ Grp)
42	21, 41	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ Grp)
43	4, 37	grpsubcl 18983	. . . . . . . 8 ⊢ ((𝑊 ∈ Grp ∧ 𝐵 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝐵(-_g‘𝑊)𝑌) ∈ 𝑉)
44	42, 3, 9, 43	syl3anc 1368	. . . . . . 7 ⊢ (𝜑 → (𝐵(-_g‘𝑊)𝑌) ∈ 𝑉)
45	4, 5, 22	ipcau 25186	. . . . . . 7 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ (𝐵(-_g‘𝑊)𝑌) ∈ 𝑉) → (abs‘(𝐴 , (𝐵(-_g‘𝑊)𝑌))) ≤ ((𝑁‘𝐴) · (𝑁‘(𝐵(-_g‘𝑊)𝑌))))
46	1, 2, 44, 45	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐴 , (𝐵(-_g‘𝑊)𝑌))) ≤ ((𝑁‘𝐴) · (𝑁‘(𝐵(-_g‘𝑊)𝑌))))
47	22, 4, 37, 31	ngpds 24533	. . . . . . . 8 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝐵 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝐵𝐷𝑌) = (𝑁‘(𝐵(-_g‘𝑊)𝑌)))
48	21, 3, 9, 47	syl3anc 1368	. . . . . . 7 ⊢ (𝜑 → (𝐵𝐷𝑌) = (𝑁‘(𝐵(-_g‘𝑊)𝑌)))
49	48	oveq2d 7442	. . . . . 6 ⊢ (𝜑 → ((𝑁‘𝐴) · (𝐵𝐷𝑌)) = ((𝑁‘𝐴) · (𝑁‘(𝐵(-_g‘𝑊)𝑌))))
50	46, 49	breqtrrd 5180	. . . . 5 ⊢ (𝜑 → (abs‘(𝐴 , (𝐵(-_g‘𝑊)𝑌))) ≤ ((𝑁‘𝐴) · (𝐵𝐷𝑌)))
51	40, 50	eqbrtrrd 5176	. . . 4 ⊢ (𝜑 → (abs‘((𝐴 , 𝐵) − (𝐴 , 𝑌))) ≤ ((𝑁‘𝐴) · (𝐵𝐷𝑌)))
52		msxms 24380	. . . . . . 7 ⊢ (𝑊 ∈ MetSp → 𝑊 ∈ ∞MetSp)
53	30, 52	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ∞MetSp)
54	4, 31	xmsge0 24389	. . . . . 6 ⊢ ((𝑊 ∈ ∞MetSp ∧ 𝐵 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 0 ≤ (𝐵𝐷𝑌))
55	53, 3, 9, 54	syl3anc 1368	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝐵𝐷𝑌))
56	24	lep1d 12183	. . . . 5 ⊢ (𝜑 → (𝑁‘𝐴) ≤ ((𝑁‘𝐴) + 1))
57	24, 28, 33, 55, 56	lemul1ad 12191	. . . 4 ⊢ (𝜑 → ((𝑁‘𝐴) · (𝐵𝐷𝑌)) ≤ (((𝑁‘𝐴) + 1) · (𝐵𝐷𝑌)))
58	17, 36, 34, 51, 57	letrd 11409	. . 3 ⊢ (𝜑 → (abs‘((𝐴 , 𝐵) − (𝐴 , 𝑌))) ≤ (((𝑁‘𝐴) + 1) · (𝐵𝐷𝑌)))
59		ipcn.2	. . . . 5 ⊢ (𝜑 → (𝐵𝐷𝑌) < 𝑇)
60		ipcn.t	. . . . 5 ⊢ 𝑇 = ((𝑅 / 2) / ((𝑁‘𝐴) + 1))
61	59, 60	breqtrdi 5193	. . . 4 ⊢ (𝜑 → (𝐵𝐷𝑌) < ((𝑅 / 2) / ((𝑁‘𝐴) + 1)))
62	33, 35, 27	ltmuldiv2d 13104	. . . 4 ⊢ (𝜑 → ((((𝑁‘𝐴) + 1) · (𝐵𝐷𝑌)) < (𝑅 / 2) ↔ (𝐵𝐷𝑌) < ((𝑅 / 2) / ((𝑁‘𝐴) + 1))))
63	61, 62	mpbird 256	. . 3 ⊢ (𝜑 → (((𝑁‘𝐴) + 1) · (𝐵𝐷𝑌)) < (𝑅 / 2))
64	17, 34, 35, 58, 63	lelttrd 11410	. 2 ⊢ (𝜑 → (abs‘((𝐴 , 𝐵) − (𝐴 , 𝑌))) < (𝑅 / 2))
65	13, 11	subcld 11609	. . . 4 ⊢ (𝜑 → ((𝐴 , 𝑌) − (𝑋 , 𝑌)) ∈ ℂ)
66	65	abscld 15423	. . 3 ⊢ (𝜑 → (abs‘((𝐴 , 𝑌) − (𝑋 , 𝑌))) ∈ ℝ)
67	4, 31	mscl 24387	. . . . 5 ⊢ ((𝑊 ∈ MetSp ∧ 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝐴𝐷𝑋) ∈ ℝ)
68	30, 2, 8, 67	syl3anc 1368	. . . 4 ⊢ (𝜑 → (𝐴𝐷𝑋) ∈ ℝ)
69	4, 22	nmcl 24545	. . . . . 6 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝐵 ∈ 𝑉) → (𝑁‘𝐵) ∈ ℝ)
70	21, 3, 69	syl2anc 582	. . . . 5 ⊢ (𝜑 → (𝑁‘𝐵) ∈ ℝ)
71	14	rphalfcld 13068	. . . . . . . 8 ⊢ (𝜑 → (𝑅 / 2) ∈ ℝ⁺)
72	71, 27	rpdivcld 13073	. . . . . . 7 ⊢ (𝜑 → ((𝑅 / 2) / ((𝑁‘𝐴) + 1)) ∈ ℝ⁺)
73	60, 72	eqeltrid 2833	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ ℝ⁺)
74	73	rpred 13056	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℝ)
75	70, 74	readdcld 11281	. . . 4 ⊢ (𝜑 → ((𝑁‘𝐵) + 𝑇) ∈ ℝ)
76	68, 75	remulcld 11282	. . 3 ⊢ (𝜑 → ((𝐴𝐷𝑋) · ((𝑁‘𝐵) + 𝑇)) ∈ ℝ)
77	4, 22	nmcl 24545	. . . . . 6 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑌 ∈ 𝑉) → (𝑁‘𝑌) ∈ ℝ)
78	21, 9, 77	syl2anc 582	. . . . 5 ⊢ (𝜑 → (𝑁‘𝑌) ∈ ℝ)
79	68, 78	remulcld 11282	. . . 4 ⊢ (𝜑 → ((𝐴𝐷𝑋) · (𝑁‘𝑌)) ∈ ℝ)
80	5, 4, 37	cphsubdir 25156	. . . . . . 7 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝐴(-_g‘𝑊)𝑋) , 𝑌) = ((𝐴 , 𝑌) − (𝑋 , 𝑌)))
81	1, 2, 8, 9, 80	syl13anc 1369	. . . . . 6 ⊢ (𝜑 → ((𝐴(-_g‘𝑊)𝑋) , 𝑌) = ((𝐴 , 𝑌) − (𝑋 , 𝑌)))
82	81	fveq2d 6906	. . . . 5 ⊢ (𝜑 → (abs‘((𝐴(-_g‘𝑊)𝑋) , 𝑌)) = (abs‘((𝐴 , 𝑌) − (𝑋 , 𝑌))))
83	4, 37	grpsubcl 18983	. . . . . . . 8 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝐴(-_g‘𝑊)𝑋) ∈ 𝑉)
84	42, 2, 8, 83	syl3anc 1368	. . . . . . 7 ⊢ (𝜑 → (𝐴(-_g‘𝑊)𝑋) ∈ 𝑉)
85	4, 5, 22	ipcau 25186	. . . . . . 7 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴(-_g‘𝑊)𝑋) ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (abs‘((𝐴(-_g‘𝑊)𝑋) , 𝑌)) ≤ ((𝑁‘(𝐴(-_g‘𝑊)𝑋)) · (𝑁‘𝑌)))
86	1, 84, 9, 85	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → (abs‘((𝐴(-_g‘𝑊)𝑋) , 𝑌)) ≤ ((𝑁‘(𝐴(-_g‘𝑊)𝑋)) · (𝑁‘𝑌)))
87	22, 4, 37, 31	ngpds 24533	. . . . . . . 8 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝐴𝐷𝑋) = (𝑁‘(𝐴(-_g‘𝑊)𝑋)))
88	21, 2, 8, 87	syl3anc 1368	. . . . . . 7 ⊢ (𝜑 → (𝐴𝐷𝑋) = (𝑁‘(𝐴(-_g‘𝑊)𝑋)))
89	88	oveq1d 7441	. . . . . 6 ⊢ (𝜑 → ((𝐴𝐷𝑋) · (𝑁‘𝑌)) = ((𝑁‘(𝐴(-_g‘𝑊)𝑋)) · (𝑁‘𝑌)))
90	86, 89	breqtrrd 5180	. . . . 5 ⊢ (𝜑 → (abs‘((𝐴(-_g‘𝑊)𝑋) , 𝑌)) ≤ ((𝐴𝐷𝑋) · (𝑁‘𝑌)))
91	82, 90	eqbrtrrd 5176	. . . 4 ⊢ (𝜑 → (abs‘((𝐴 , 𝑌) − (𝑋 , 𝑌))) ≤ ((𝐴𝐷𝑋) · (𝑁‘𝑌)))
92	4, 31	xmsge0 24389	. . . . . 6 ⊢ ((𝑊 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → 0 ≤ (𝐴𝐷𝑋))
93	53, 2, 8, 92	syl3anc 1368	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝐴𝐷𝑋))
94	78, 70	resubcld 11680	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘𝑌) − (𝑁‘𝐵)) ∈ ℝ)
95	4, 22, 37	nm2dif 24554	. . . . . . . . 9 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑌 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝑁‘𝑌) − (𝑁‘𝐵)) ≤ (𝑁‘(𝑌(-_g‘𝑊)𝐵)))
96	21, 9, 3, 95	syl3anc 1368	. . . . . . . 8 ⊢ (𝜑 → ((𝑁‘𝑌) − (𝑁‘𝐵)) ≤ (𝑁‘(𝑌(-_g‘𝑊)𝐵)))
97	22, 4, 37, 31	ngpdsr 24534	. . . . . . . . 9 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝐵 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝐵𝐷𝑌) = (𝑁‘(𝑌(-_g‘𝑊)𝐵)))
98	21, 3, 9, 97	syl3anc 1368	. . . . . . . 8 ⊢ (𝜑 → (𝐵𝐷𝑌) = (𝑁‘(𝑌(-_g‘𝑊)𝐵)))
99	96, 98	breqtrrd 5180	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘𝑌) − (𝑁‘𝐵)) ≤ (𝐵𝐷𝑌))
100	33, 74, 59	ltled 11400	. . . . . . 7 ⊢ (𝜑 → (𝐵𝐷𝑌) ≤ 𝑇)
101	94, 33, 74, 99, 100	letrd 11409	. . . . . 6 ⊢ (𝜑 → ((𝑁‘𝑌) − (𝑁‘𝐵)) ≤ 𝑇)
102	78, 70, 74	lesubadd2d 11851	. . . . . 6 ⊢ (𝜑 → (((𝑁‘𝑌) − (𝑁‘𝐵)) ≤ 𝑇 ↔ (𝑁‘𝑌) ≤ ((𝑁‘𝐵) + 𝑇)))
103	101, 102	mpbid 231	. . . . 5 ⊢ (𝜑 → (𝑁‘𝑌) ≤ ((𝑁‘𝐵) + 𝑇))
104	78, 75, 68, 93, 103	lemul2ad 12192	. . . 4 ⊢ (𝜑 → ((𝐴𝐷𝑋) · (𝑁‘𝑌)) ≤ ((𝐴𝐷𝑋) · ((𝑁‘𝐵) + 𝑇)))
105	66, 79, 76, 91, 104	letrd 11409	. . 3 ⊢ (𝜑 → (abs‘((𝐴 , 𝑌) − (𝑋 , 𝑌))) ≤ ((𝐴𝐷𝑋) · ((𝑁‘𝐵) + 𝑇)))
106		ipcn.1	. . . . 5 ⊢ (𝜑 → (𝐴𝐷𝑋) < 𝑈)
107		ipcn.u	. . . . 5 ⊢ 𝑈 = ((𝑅 / 2) / ((𝑁‘𝐵) + 𝑇))
108	106, 107	breqtrdi 5193	. . . 4 ⊢ (𝜑 → (𝐴𝐷𝑋) < ((𝑅 / 2) / ((𝑁‘𝐵) + 𝑇)))
109		0red 11255	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ)
110	4, 22	nmge0 24546	. . . . . . 7 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝐵 ∈ 𝑉) → 0 ≤ (𝑁‘𝐵))
111	21, 3, 110	syl2anc 582	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝑁‘𝐵))
112	70, 73	ltaddrpd 13089	. . . . . 6 ⊢ (𝜑 → (𝑁‘𝐵) < ((𝑁‘𝐵) + 𝑇))
113	109, 70, 75, 111, 112	lelttrd 11410	. . . . 5 ⊢ (𝜑 → 0 < ((𝑁‘𝐵) + 𝑇))
114		ltmuldiv 12125	. . . . 5 ⊢ (((𝐴𝐷𝑋) ∈ ℝ ∧ (𝑅 / 2) ∈ ℝ ∧ (((𝑁‘𝐵) + 𝑇) ∈ ℝ ∧ 0 < ((𝑁‘𝐵) + 𝑇))) → (((𝐴𝐷𝑋) · ((𝑁‘𝐵) + 𝑇)) < (𝑅 / 2) ↔ (𝐴𝐷𝑋) < ((𝑅 / 2) / ((𝑁‘𝐵) + 𝑇))))
115	68, 35, 75, 113, 114	syl112anc 1371	. . . 4 ⊢ (𝜑 → (((𝐴𝐷𝑋) · ((𝑁‘𝐵) + 𝑇)) < (𝑅 / 2) ↔ (𝐴𝐷𝑋) < ((𝑅 / 2) / ((𝑁‘𝐵) + 𝑇))))
116	108, 115	mpbird 256	. . 3 ⊢ (𝜑 → ((𝐴𝐷𝑋) · ((𝑁‘𝐵) + 𝑇)) < (𝑅 / 2))
117	66, 76, 35, 105, 116	lelttrd 11410	. 2 ⊢ (𝜑 → (abs‘((𝐴 , 𝑌) − (𝑋 , 𝑌))) < (𝑅 / 2))
118	7, 11, 13, 15, 64, 117	abs3lemd 15448	1 ⊢ (𝜑 → (abs‘((𝐴 , 𝐵) − (𝑋 , 𝑌))) < 𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 class class class wbr 5152 ‘cfv 6553 (class class class)co 7426 ℂcc 11144 ℝcr 11145 0cc0 11146 1c1 11147 + caddc 11149 · cmul 11151 < clt 11286 ≤ cle 11287 − cmin 11482 / cdiv 11909 2c2 12305 ℝ⁺crp 13014 abscabs 15221 Basecbs 17187 ·_𝑖cip 17245 distcds 17249 Grpcgrp 18897 -_gcsg 18899 ∞MetSpcxms 24243 MetSpcms 24244 normcnm 24505 NrmGrpcngp 24506 NrmModcnlm 24509 ℂPreHilccph 25114

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 ax-mulf 11226

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ico 13370 df-fz 13525 df-seq 14007 df-exp 14067 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-0g 17430 df-topgen 17432 df-xrs 17491 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-grp 18900 df-minusg 18901 df-sbg 18902 df-subg 19085 df-ghm 19175 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-invr 20334 df-dvr 20347 df-rhm 20418 df-subrng 20490 df-subrg 20515 df-drng 20633 df-staf 20732 df-srng 20733 df-lmod 20752 df-lmhm 20914 df-lvec 20995 df-sra 21065 df-rgmod 21066 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-cnfld 21287 df-phl 21565 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-xms 24246 df-ms 24247 df-nm 24511 df-ngp 24512 df-tng 24513 df-nlm 24515 df-clm 25010 df-cph 25116 df-tcph 25117

This theorem is referenced by: ipcnlem1 25193

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