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Theorem blocni 30876

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 2736	. . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
3		eqid 2736	. . . . 5 ⊢ (0_vec‘𝑈) = (0_vec‘𝑈)
4	2, 3	nvzcl 30705	. . . 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 30762	. . . . . . . . 9 ⊢ (𝑈 ∈ NrmCVec → 𝐶 ∈ (Met‘(BaseSet‘𝑈)))
8	1, 7	ax-mp 5	. . . . . . . 8 ⊢ 𝐶 ∈ (Met‘(BaseSet‘𝑈))
9		metxmet 24299	. . . . . . . 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 24404	. . . . . . 7 ⊢ (𝐶 ∈ (∞Met‘(BaseSet‘𝑈)) → 𝐽 ∈ (TopOn‘(BaseSet‘𝑈)))
13	10, 12	ax-mp 5	. . . . . 6 ⊢ 𝐽 ∈ (TopOn‘(BaseSet‘𝑈))
14	13	toponunii 22881	. . . . 5 ⊢ (BaseSet‘𝑈) = ∪ 𝐽
15	14	cncnpi 23243	. . . 4 ⊢ ((𝑇 ∈ (𝐽 Cn 𝐾) ∧ (0_vec‘𝑈) ∈ (BaseSet‘𝑈)) → 𝑇 ∈ ((𝐽 CnP 𝐾)‘(0_vec‘𝑈)))
16	5, 15	mpan2 692	. . 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 30875	. . 3 ⊢ (((0_vec‘𝑈) ∈ (BaseSet‘𝑈) ∧ 𝑇 ∈ ((𝐽 CnP 𝐾)‘(0_vec‘𝑈))) → 𝑇 ∈ 𝐵)
24	5, 16, 23	sylancr 588	. 2 ⊢ (𝑇 ∈ (𝐽 Cn 𝐾) → 𝑇 ∈ 𝐵)
25		eleq1 2824	. . 3 ⊢ (𝑇 = (𝑈 0_op 𝑊) → (𝑇 ∈ (𝐽 Cn 𝐾) ↔ (𝑈 0_op 𝑊) ∈ (𝐽 Cn 𝐾)))
26		simprr 773	. . . . . . . 8 ⊢ (((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) → 𝑦 ∈ ℝ⁺)
27		eqid 2736	. . . . . . . . . . . . 13 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
28		eqid 2736	. . . . . . . . . . . . 13 ⊢ (𝑈 normOp_OLD 𝑊) = (𝑈 normOp_OLD 𝑊)
29	2, 27, 28, 20	nmblore 30857	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → ((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ)
30	1, 21, 29	mp3an12 1454	. . . . . . . . . . 11 ⊢ (𝑇 ∈ 𝐵 → ((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ)
31		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (𝑈 0_op 𝑊) = (𝑈 0_op 𝑊)
32	28, 31, 19	nmlnogt0 30868	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇 ≠ (𝑈 0_op 𝑊) ↔ 0 < ((𝑈 normOp_OLD 𝑊)‘𝑇)))
33	1, 21, 22, 32	mp3an 1464	. . . . . . . . . . . 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 614	. . . . . . . . . 10 ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) → (((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ ∧ 0 < ((𝑈 normOp_OLD 𝑊)‘𝑇)))
36		elrp 12944	. . . . . . . . . 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 13003	. . . . . . 7 ⊢ (((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) → (𝑦 / ((𝑈 normOp_OLD 𝑊)‘𝑇)) ∈ ℝ⁺)
40		simprl 771	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) → 𝑥 ∈ (BaseSet‘𝑈))
41		metcl 24297	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (Met‘(BaseSet‘𝑈)) ∧ 𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → (𝑥𝐶𝑤) ∈ ℝ)
42	8, 41	mp3an1 1451	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → (𝑥𝐶𝑤) ∈ ℝ)
43	40, 42	sylan 581	. . . . . . . . . 10 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → (𝑥𝐶𝑤) ∈ ℝ)
44		simplrr 778	. . . . . . . . . . 11 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → 𝑦 ∈ ℝ⁺)
45	44	rpred 12986	. . . . . . . . . 10 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → 𝑦 ∈ ℝ)
46	35	ad2antrr 727	. . . . . . . . . 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 12030	. . . . . . . . . 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 1374	. . . . . . . . 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 748	. . . . . . . . . . 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 30874	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ≤ (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)))
52	51	3expa 1119	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ 𝐵 ∧ 𝑥 ∈ (BaseSet‘𝑈)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ≤ (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)))
53	50, 52	sylan 581	. . . . . . . . . 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 30826	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊))
55	1, 21, 22, 54	mp3an 1464	. . . . . . . . . . . . . 14 ⊢ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)
56	55	ffvelcdmi 7035	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (BaseSet‘𝑈) → (𝑇‘𝑥) ∈ (BaseSet‘𝑊))
57	55	ffvelcdmi 7035	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (BaseSet‘𝑈) → (𝑇‘𝑤) ∈ (BaseSet‘𝑊))
58	27, 17	imsmet 30762	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ NrmCVec → 𝐷 ∈ (Met‘(BaseSet‘𝑊)))
59	21, 58	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 𝐷 ∈ (Met‘(BaseSet‘𝑊))
60		metcl 24297	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (Met‘(BaseSet‘𝑊)) ∧ (𝑇‘𝑥) ∈ (BaseSet‘𝑊) ∧ (𝑇‘𝑤) ∈ (BaseSet‘𝑊)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ∈ ℝ)
61	59, 60	mp3an1 1451	. . . . . . . . . . . . 13 ⊢ (((𝑇‘𝑥) ∈ (BaseSet‘𝑊) ∧ (𝑇‘𝑤) ∈ (BaseSet‘𝑊)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ∈ ℝ)
62	56, 57, 61	syl2an 597	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ∈ ℝ)
63	40, 62	sylan 581	. . . . . . . . . . 11 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ∈ ℝ)
64		remulcl 11123	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ ∧ (𝑥𝐶𝑤) ∈ ℝ) → (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)) ∈ ℝ)
65	30, 42, 64	syl2an 597	. . . . . . . . . . . . . 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 720	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ 𝑥 ∈ (BaseSet‘𝑈)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)) ∈ ℝ)
68	67	adantlrr 722	. . . . . . . . . . 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 11236	. . . . . . . . . . 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 1374	. . . . . . . . . 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 696	. . . . . . . . 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 3129	. . . . . . 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 5089	. . . . . . . 8 ⊢ (𝑧 = (𝑦 / ((𝑈 normOp_OLD 𝑊)‘𝑇)) → ((𝑥𝐶𝑤) < 𝑧 ↔ (𝑥𝐶𝑤) < (𝑦 / ((𝑈 normOp_OLD 𝑊)‘𝑇))))
75	74	rspceaimv 3570	. . . . . . 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 585	. . . . . 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 3180	. . . . 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 24299	. . . . . 6 ⊢ (𝐷 ∈ (Met‘(BaseSet‘𝑊)) → 𝐷 ∈ (∞Met‘(BaseSet‘𝑊)))
80	59, 79	ax-mp 5	. . . . 5 ⊢ 𝐷 ∈ (∞Met‘(BaseSet‘𝑊))
81	11, 18	metcn 24508	. . . . 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 693	. . . 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 2736	. . . . . . 7 ⊢ (0_vec‘𝑊) = (0_vec‘𝑊)
85	2, 84, 31	0ofval 30858	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 0_op 𝑊) = ((BaseSet‘𝑈) × {(0_vec‘𝑊)}))
86	1, 21, 85	mp2an 693	. . . . 5 ⊢ (𝑈 0_op 𝑊) = ((BaseSet‘𝑈) × {(0_vec‘𝑊)})
87	18	mopntopon 24404	. . . . . . 7 ⊢ (𝐷 ∈ (∞Met‘(BaseSet‘𝑊)) → 𝐾 ∈ (TopOn‘(BaseSet‘𝑊)))
88	80, 87	ax-mp 5	. . . . . 6 ⊢ 𝐾 ∈ (TopOn‘(BaseSet‘𝑊))
89	27, 84	nvzcl 30705	. . . . . . 7 ⊢ (𝑊 ∈ NrmCVec → (0_vec‘𝑊) ∈ (BaseSet‘𝑊))
90	21, 89	ax-mp 5	. . . . . 6 ⊢ (0_vec‘𝑊) ∈ (BaseSet‘𝑊)
91		cnconst2 23248	. . . . . 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 1464	. . . . 5 ⊢ ((BaseSet‘𝑈) × {(0_vec‘𝑊)}) ∈ (𝐽 Cn 𝐾)
93	86, 92	eqeltri 2832	. . . 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 3017	. 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 1542 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 ∃wrex 3061 {csn 4567 class class class wbr 5085 × cxp 5629 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ℝcr 11037 0cc0 11038 · cmul 11043 < clt 11179 ≤ cle 11180 / cdiv 11807 ℝ⁺crp 12942 ∞Metcxmet 21337 Metcmet 21338 MetOpencmopn 21342 TopOnctopon 22875 Cn ccn 23189 CnP ccnp 23190 NrmCVeccnv 30655 BaseSetcba 30657 0_veccn0v 30659 IndMetcims 30662 LnOp clno 30811 normOp_OLD cnmoo 30812 BLnOp cblo 30813 0_op c0o 30814

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-sup 9355 df-inf 9356 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-z 12525 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-seq 13964 df-exp 14024 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-topgen 17406 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-top 22859 df-topon 22876 df-bases 22911 df-cn 23192 df-cnp 23193 df-grpo 30564 df-gid 30565 df-ginv 30566 df-gdiv 30567 df-ablo 30616 df-vc 30630 df-nv 30663 df-va 30666 df-ba 30667 df-sm 30668 df-0v 30669 df-vs 30670 df-nmcv 30671 df-ims 30672 df-lno 30815 df-nmoo 30816 df-blo 30817 df-0o 30818

This theorem is referenced by: lnocni 30877 blocn 30878

W3C validator