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Theorem blocni 30741

Description: A linear operator is continuous iff it is bounded. Theorem 2.7-9(a) of [Kreyszig] p. 97. (Contributed by NM, 18-Dec-2007.) (Revised by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
blocni.8	⊢ 𝐶 = (IndMet‘𝑈)
blocni.d	⊢ 𝐷 = (IndMet‘𝑊)
blocni.j	⊢ 𝐽 = (MetOpen‘𝐶)
blocni.k	⊢ 𝐾 = (MetOpen‘𝐷)
blocni.4	⊢ 𝐿 = (𝑈 LnOp 𝑊)
blocni.5	⊢ 𝐵 = (𝑈 BLnOp 𝑊)
blocni.u	⊢ 𝑈 ∈ NrmCVec
blocni.w	⊢ 𝑊 ∈ NrmCVec
blocni.l	⊢ 𝑇 ∈ 𝐿

**Assertion**
Ref	Expression
blocni	⊢ (𝑇 ∈ (𝐽 Cn 𝐾) ↔ 𝑇 ∈ 𝐵)

Proof of Theorem blocni Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		blocni.u	. . . 4 ⊢ 𝑈 ∈ NrmCVec
2		eqid 2730	. . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
3		eqid 2730	. . . . 5 ⊢ (0_vec‘𝑈) = (0_vec‘𝑈)
4	2, 3	nvzcl 30570	. . . 4 ⊢ (𝑈 ∈ NrmCVec → (0_vec‘𝑈) ∈ (BaseSet‘𝑈))
5	1, 4	ax-mp 5	. . 3 ⊢ (0_vec‘𝑈) ∈ (BaseSet‘𝑈)
6		blocni.8	. . . . . . . . . 10 ⊢ 𝐶 = (IndMet‘𝑈)
7	2, 6	imsmet 30627	. . . . . . . . 9 ⊢ (𝑈 ∈ NrmCVec → 𝐶 ∈ (Met‘(BaseSet‘𝑈)))
8	1, 7	ax-mp 5	. . . . . . . 8 ⊢ 𝐶 ∈ (Met‘(BaseSet‘𝑈))
9		metxmet 24229	. . . . . . . 8 ⊢ (𝐶 ∈ (Met‘(BaseSet‘𝑈)) → 𝐶 ∈ (∞Met‘(BaseSet‘𝑈)))
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ 𝐶 ∈ (∞Met‘(BaseSet‘𝑈))
11		blocni.j	. . . . . . . 8 ⊢ 𝐽 = (MetOpen‘𝐶)
12	11	mopntopon 24334	. . . . . . 7 ⊢ (𝐶 ∈ (∞Met‘(BaseSet‘𝑈)) → 𝐽 ∈ (TopOn‘(BaseSet‘𝑈)))
13	10, 12	ax-mp 5	. . . . . 6 ⊢ 𝐽 ∈ (TopOn‘(BaseSet‘𝑈))
14	13	toponunii 22810	. . . . 5 ⊢ (BaseSet‘𝑈) = ∪ 𝐽
15	14	cncnpi 23172	. . . 4 ⊢ ((𝑇 ∈ (𝐽 Cn 𝐾) ∧ (0_vec‘𝑈) ∈ (BaseSet‘𝑈)) → 𝑇 ∈ ((𝐽 CnP 𝐾)‘(0_vec‘𝑈)))
16	5, 15	mpan2 691	. . 3 ⊢ (𝑇 ∈ (𝐽 Cn 𝐾) → 𝑇 ∈ ((𝐽 CnP 𝐾)‘(0_vec‘𝑈)))
17		blocni.d	. . . 4 ⊢ 𝐷 = (IndMet‘𝑊)
18		blocni.k	. . . 4 ⊢ 𝐾 = (MetOpen‘𝐷)
19		blocni.4	. . . 4 ⊢ 𝐿 = (𝑈 LnOp 𝑊)
20		blocni.5	. . . 4 ⊢ 𝐵 = (𝑈 BLnOp 𝑊)
21		blocni.w	. . . 4 ⊢ 𝑊 ∈ NrmCVec
22		blocni.l	. . . 4 ⊢ 𝑇 ∈ 𝐿
23	6, 17, 11, 18, 19, 20, 1, 21, 22, 2	blocnilem 30740	. . 3 ⊢ (((0_vec‘𝑈) ∈ (BaseSet‘𝑈) ∧ 𝑇 ∈ ((𝐽 CnP 𝐾)‘(0_vec‘𝑈))) → 𝑇 ∈ 𝐵)
24	5, 16, 23	sylancr 587	. 2 ⊢ (𝑇 ∈ (𝐽 Cn 𝐾) → 𝑇 ∈ 𝐵)
25		eleq1 2817	. . 3 ⊢ (𝑇 = (𝑈 0_op 𝑊) → (𝑇 ∈ (𝐽 Cn 𝐾) ↔ (𝑈 0_op 𝑊) ∈ (𝐽 Cn 𝐾)))
26		simprr 772	. . . . . . . 8 ⊢ (((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) → 𝑦 ∈ ℝ⁺)
27		eqid 2730	. . . . . . . . . . . . 13 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
28		eqid 2730	. . . . . . . . . . . . 13 ⊢ (𝑈 normOp_OLD 𝑊) = (𝑈 normOp_OLD 𝑊)
29	2, 27, 28, 20	nmblore 30722	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → ((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ)
30	1, 21, 29	mp3an12 1453	. . . . . . . . . . 11 ⊢ (𝑇 ∈ 𝐵 → ((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ)
31		eqid 2730	. . . . . . . . . . . . . 14 ⊢ (𝑈 0_op 𝑊) = (𝑈 0_op 𝑊)
32	28, 31, 19	nmlnogt0 30733	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇 ≠ (𝑈 0_op 𝑊) ↔ 0 < ((𝑈 normOp_OLD 𝑊)‘𝑇)))
33	1, 21, 22, 32	mp3an 1463	. . . . . . . . . . . 12 ⊢ (𝑇 ≠ (𝑈 0_op 𝑊) ↔ 0 < ((𝑈 normOp_OLD 𝑊)‘𝑇))
34	33	biimpi 216	. . . . . . . . . . 11 ⊢ (𝑇 ≠ (𝑈 0_op 𝑊) → 0 < ((𝑈 normOp_OLD 𝑊)‘𝑇))
35	30, 34	anim12i 613	. . . . . . . . . 10 ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) → (((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ ∧ 0 < ((𝑈 normOp_OLD 𝑊)‘𝑇)))
36		elrp 12960	. . . . . . . . . 10 ⊢ (((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ⁺ ↔ (((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ ∧ 0 < ((𝑈 normOp_OLD 𝑊)‘𝑇)))
37	35, 36	sylibr 234	. . . . . . . . 9 ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) → ((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ⁺)
38	37	adantr 480	. . . . . . . 8 ⊢ (((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) → ((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ⁺)
39	26, 38	rpdivcld 13019	. . . . . . 7 ⊢ (((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) → (𝑦 / ((𝑈 normOp_OLD 𝑊)‘𝑇)) ∈ ℝ⁺)
40		simprl 770	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) → 𝑥 ∈ (BaseSet‘𝑈))
41		metcl 24227	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (Met‘(BaseSet‘𝑈)) ∧ 𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → (𝑥𝐶𝑤) ∈ ℝ)
42	8, 41	mp3an1 1450	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → (𝑥𝐶𝑤) ∈ ℝ)
43	40, 42	sylan 580	. . . . . . . . . 10 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → (𝑥𝐶𝑤) ∈ ℝ)
44		simplrr 777	. . . . . . . . . . 11 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → 𝑦 ∈ ℝ⁺)
45	44	rpred 13002	. . . . . . . . . 10 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → 𝑦 ∈ ℝ)
46	35	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → (((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ ∧ 0 < ((𝑈 normOp_OLD 𝑊)‘𝑇)))
47		ltmuldiv2 12064	. . . . . . . . . 10 ⊢ (((𝑥𝐶𝑤) ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ (((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ ∧ 0 < ((𝑈 normOp_OLD 𝑊)‘𝑇))) → ((((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)) < 𝑦 ↔ (𝑥𝐶𝑤) < (𝑦 / ((𝑈 normOp_OLD 𝑊)‘𝑇))))
48	43, 45, 46, 47	syl3anc 1373	. . . . . . . . 9 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → ((((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)) < 𝑦 ↔ (𝑥𝐶𝑤) < (𝑦 / ((𝑈 normOp_OLD 𝑊)‘𝑇))))
49		id 22	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (𝑇 ∈ 𝐵 ∧ 𝑥 ∈ (BaseSet‘𝑈)))
50	49	ad2ant2r 747	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) → (𝑇 ∈ 𝐵 ∧ 𝑥 ∈ (BaseSet‘𝑈)))
51	2, 27, 6, 17, 28, 20, 1, 21	blometi 30739	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ≤ (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)))
52	51	3expa 1118	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ 𝐵 ∧ 𝑥 ∈ (BaseSet‘𝑈)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ≤ (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)))
53	50, 52	sylan 580	. . . . . . . . . 10 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ≤ (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)))
54	2, 27, 19	lnof 30691	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊))
55	1, 21, 22, 54	mp3an 1463	. . . . . . . . . . . . . 14 ⊢ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)
56	55	ffvelcdmi 7058	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (BaseSet‘𝑈) → (𝑇‘𝑥) ∈ (BaseSet‘𝑊))
57	55	ffvelcdmi 7058	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (BaseSet‘𝑈) → (𝑇‘𝑤) ∈ (BaseSet‘𝑊))
58	27, 17	imsmet 30627	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ NrmCVec → 𝐷 ∈ (Met‘(BaseSet‘𝑊)))
59	21, 58	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 𝐷 ∈ (Met‘(BaseSet‘𝑊))
60		metcl 24227	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (Met‘(BaseSet‘𝑊)) ∧ (𝑇‘𝑥) ∈ (BaseSet‘𝑊) ∧ (𝑇‘𝑤) ∈ (BaseSet‘𝑊)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ∈ ℝ)
61	59, 60	mp3an1 1450	. . . . . . . . . . . . 13 ⊢ (((𝑇‘𝑥) ∈ (BaseSet‘𝑊) ∧ (𝑇‘𝑤) ∈ (BaseSet‘𝑊)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ∈ ℝ)
62	56, 57, 61	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ∈ ℝ)
63	40, 62	sylan 580	. . . . . . . . . . 11 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ∈ ℝ)
64		remulcl 11160	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ ∧ (𝑥𝐶𝑤) ∈ ℝ) → (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)) ∈ ℝ)
65	30, 42, 64	syl2an 596	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ 𝐵 ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑤 ∈ (BaseSet‘𝑈))) → (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)) ∈ ℝ)
66	65	anassrs 467	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ 𝐵 ∧ 𝑥 ∈ (BaseSet‘𝑈)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)) ∈ ℝ)
67	66	adantllr 719	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ 𝑥 ∈ (BaseSet‘𝑈)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)) ∈ ℝ)
68	67	adantlrr 721	. . . . . . . . . . 11 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)) ∈ ℝ)
69		lelttr 11271	. . . . . . . . . . 11 ⊢ ((((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ∈ ℝ ∧ (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ≤ (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)) ∧ (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)) < 𝑦) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) < 𝑦))
70	63, 68, 45, 69	syl3anc 1373	. . . . . . . . . 10 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → ((((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ≤ (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)) ∧ (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)) < 𝑦) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) < 𝑦))
71	53, 70	mpand 695	. . . . . . . . 9 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → ((((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)) < 𝑦 → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) < 𝑦))
72	48, 71	sylbird 260	. . . . . . . 8 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → ((𝑥𝐶𝑤) < (𝑦 / ((𝑈 normOp_OLD 𝑊)‘𝑇)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) < 𝑦))
73	72	ralrimiva 3126	. . . . . . 7 ⊢ (((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) → ∀𝑤 ∈ (BaseSet‘𝑈)((𝑥𝐶𝑤) < (𝑦 / ((𝑈 normOp_OLD 𝑊)‘𝑇)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) < 𝑦))
74		breq2 5114	. . . . . . . 8 ⊢ (𝑧 = (𝑦 / ((𝑈 normOp_OLD 𝑊)‘𝑇)) → ((𝑥𝐶𝑤) < 𝑧 ↔ (𝑥𝐶𝑤) < (𝑦 / ((𝑈 normOp_OLD 𝑊)‘𝑇))))
75	74	rspceaimv 3597	. . . . . . 7 ⊢ (((𝑦 / ((𝑈 normOp_OLD 𝑊)‘𝑇)) ∈ ℝ⁺ ∧ ∀𝑤 ∈ (BaseSet‘𝑈)((𝑥𝐶𝑤) < (𝑦 / ((𝑈 normOp_OLD 𝑊)‘𝑇)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) < 𝑦)) → ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ (BaseSet‘𝑈)((𝑥𝐶𝑤) < 𝑧 → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) < 𝑦))
76	39, 73, 75	syl2anc 584	. . . . . 6 ⊢ (((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) → ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ (BaseSet‘𝑈)((𝑥𝐶𝑤) < 𝑧 → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) < 𝑦))
77	76	ralrimivva 3181	. . . . 5 ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) → ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ (BaseSet‘𝑈)((𝑥𝐶𝑤) < 𝑧 → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) < 𝑦))
78	77, 55	jctil 519	. . . 4 ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) → (𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ (BaseSet‘𝑈)((𝑥𝐶𝑤) < 𝑧 → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) < 𝑦)))
79		metxmet 24229	. . . . . 6 ⊢ (𝐷 ∈ (Met‘(BaseSet‘𝑊)) → 𝐷 ∈ (∞Met‘(BaseSet‘𝑊)))
80	59, 79	ax-mp 5	. . . . 5 ⊢ 𝐷 ∈ (∞Met‘(BaseSet‘𝑊))
81	11, 18	metcn 24438	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘(BaseSet‘𝑈)) ∧ 𝐷 ∈ (∞Met‘(BaseSet‘𝑊))) → (𝑇 ∈ (𝐽 Cn 𝐾) ↔ (𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ (BaseSet‘𝑈)((𝑥𝐶𝑤) < 𝑧 → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) < 𝑦))))
82	10, 80, 81	mp2an 692	. . . 4 ⊢ (𝑇 ∈ (𝐽 Cn 𝐾) ↔ (𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ (BaseSet‘𝑈)((𝑥𝐶𝑤) < 𝑧 → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) < 𝑦)))
83	78, 82	sylibr 234	. . 3 ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) → 𝑇 ∈ (𝐽 Cn 𝐾))
84		eqid 2730	. . . . . . 7 ⊢ (0_vec‘𝑊) = (0_vec‘𝑊)
85	2, 84, 31	0ofval 30723	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 0_op 𝑊) = ((BaseSet‘𝑈) × {(0_vec‘𝑊)}))
86	1, 21, 85	mp2an 692	. . . . 5 ⊢ (𝑈 0_op 𝑊) = ((BaseSet‘𝑈) × {(0_vec‘𝑊)})
87	18	mopntopon 24334	. . . . . . 7 ⊢ (𝐷 ∈ (∞Met‘(BaseSet‘𝑊)) → 𝐾 ∈ (TopOn‘(BaseSet‘𝑊)))
88	80, 87	ax-mp 5	. . . . . 6 ⊢ 𝐾 ∈ (TopOn‘(BaseSet‘𝑊))
89	27, 84	nvzcl 30570	. . . . . . 7 ⊢ (𝑊 ∈ NrmCVec → (0_vec‘𝑊) ∈ (BaseSet‘𝑊))
90	21, 89	ax-mp 5	. . . . . 6 ⊢ (0_vec‘𝑊) ∈ (BaseSet‘𝑊)
91		cnconst2 23177	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘(BaseSet‘𝑈)) ∧ 𝐾 ∈ (TopOn‘(BaseSet‘𝑊)) ∧ (0_vec‘𝑊) ∈ (BaseSet‘𝑊)) → ((BaseSet‘𝑈) × {(0_vec‘𝑊)}) ∈ (𝐽 Cn 𝐾))
92	13, 88, 90, 91	mp3an 1463	. . . . 5 ⊢ ((BaseSet‘𝑈) × {(0_vec‘𝑊)}) ∈ (𝐽 Cn 𝐾)
93	86, 92	eqeltri 2825	. . . 4 ⊢ (𝑈 0_op 𝑊) ∈ (𝐽 Cn 𝐾)
94	93	a1i 11	. . 3 ⊢ (𝑇 ∈ 𝐵 → (𝑈 0_op 𝑊) ∈ (𝐽 Cn 𝐾))
95	25, 83, 94	pm2.61ne 3011	. 2 ⊢ (𝑇 ∈ 𝐵 → 𝑇 ∈ (𝐽 Cn 𝐾))
96	24, 95	impbii 209	1 ⊢ (𝑇 ∈ (𝐽 Cn 𝐾) ↔ 𝑇 ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∀wral 3045 ∃wrex 3054 {csn 4592 class class class wbr 5110 × cxp 5639 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ℝcr 11074 0cc0 11075 · cmul 11080 < clt 11215 ≤ cle 11216 / cdiv 11842 ℝ⁺crp 12958 ∞Metcxmet 21256 Metcmet 21257 MetOpencmopn 21261 TopOnctopon 22804 Cn ccn 23118 CnP ccnp 23119 NrmCVeccnv 30520 BaseSetcba 30522 0_veccn0v 30524 IndMetcims 30527 LnOp clno 30676 normOp_OLD cnmoo 30677 BLnOp cblo 30678 0_op c0o 30679

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154 ax-mulf 11155

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-sup 9400 df-inf 9401 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-z 12537 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-seq 13974 df-exp 14034 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-topgen 17413 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-top 22788 df-topon 22805 df-bases 22840 df-cn 23121 df-cnp 23122 df-grpo 30429 df-gid 30430 df-ginv 30431 df-gdiv 30432 df-ablo 30481 df-vc 30495 df-nv 30528 df-va 30531 df-ba 30532 df-sm 30533 df-0v 30534 df-vs 30535 df-nmcv 30536 df-ims 30537 df-lno 30680 df-nmoo 30681 df-blo 30682 df-0o 30683

This theorem is referenced by: lnocni 30742 blocn 30743

W3C validator