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Theorem blocni 30738

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 2726	. . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
3		eqid 2726	. . . . 5 ⊢ (0_vec‘𝑈) = (0_vec‘𝑈)
4	2, 3	nvzcl 30567	. . . 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 30624	. . . . . . . . 9 ⊢ (𝑈 ∈ NrmCVec → 𝐶 ∈ (Met‘(BaseSet‘𝑈)))
8	1, 7	ax-mp 5	. . . . . . . 8 ⊢ 𝐶 ∈ (Met‘(BaseSet‘𝑈))
9		metxmet 24331	. . . . . . . 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 24436	. . . . . . 7 ⊢ (𝐶 ∈ (∞Met‘(BaseSet‘𝑈)) → 𝐽 ∈ (TopOn‘(BaseSet‘𝑈)))
13	10, 12	ax-mp 5	. . . . . 6 ⊢ 𝐽 ∈ (TopOn‘(BaseSet‘𝑈))
14	13	toponunii 22909	. . . . 5 ⊢ (BaseSet‘𝑈) = ∪ 𝐽
15	14	cncnpi 23273	. . . 4 ⊢ ((𝑇 ∈ (𝐽 Cn 𝐾) ∧ (0_vec‘𝑈) ∈ (BaseSet‘𝑈)) → 𝑇 ∈ ((𝐽 CnP 𝐾)‘(0_vec‘𝑈)))
16	5, 15	mpan2 689	. . 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 30737	. . 3 ⊢ (((0_vec‘𝑈) ∈ (BaseSet‘𝑈) ∧ 𝑇 ∈ ((𝐽 CnP 𝐾)‘(0_vec‘𝑈))) → 𝑇 ∈ 𝐵)
24	5, 16, 23	sylancr 585	. 2 ⊢ (𝑇 ∈ (𝐽 Cn 𝐾) → 𝑇 ∈ 𝐵)
25		eleq1 2814	. . 3 ⊢ (𝑇 = (𝑈 0_op 𝑊) → (𝑇 ∈ (𝐽 Cn 𝐾) ↔ (𝑈 0_op 𝑊) ∈ (𝐽 Cn 𝐾)))
26		simprr 771	. . . . . . . 8 ⊢ (((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) → 𝑦 ∈ ℝ⁺)
27		eqid 2726	. . . . . . . . . . . . 13 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
28		eqid 2726	. . . . . . . . . . . . 13 ⊢ (𝑈 normOp_OLD 𝑊) = (𝑈 normOp_OLD 𝑊)
29	2, 27, 28, 20	nmblore 30719	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → ((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ)
30	1, 21, 29	mp3an12 1448	. . . . . . . . . . 11 ⊢ (𝑇 ∈ 𝐵 → ((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ)
31		eqid 2726	. . . . . . . . . . . . . 14 ⊢ (𝑈 0_op 𝑊) = (𝑈 0_op 𝑊)
32	28, 31, 19	nmlnogt0 30730	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇 ≠ (𝑈 0_op 𝑊) ↔ 0 < ((𝑈 normOp_OLD 𝑊)‘𝑇)))
33	1, 21, 22, 32	mp3an 1458	. . . . . . . . . . . 12 ⊢ (𝑇 ≠ (𝑈 0_op 𝑊) ↔ 0 < ((𝑈 normOp_OLD 𝑊)‘𝑇))
34	33	biimpi 215	. . . . . . . . . . 11 ⊢ (𝑇 ≠ (𝑈 0_op 𝑊) → 0 < ((𝑈 normOp_OLD 𝑊)‘𝑇))
35	30, 34	anim12i 611	. . . . . . . . . 10 ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) → (((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ ∧ 0 < ((𝑈 normOp_OLD 𝑊)‘𝑇)))
36		elrp 13030	. . . . . . . . . 10 ⊢ (((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ⁺ ↔ (((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ ∧ 0 < ((𝑈 normOp_OLD 𝑊)‘𝑇)))
37	35, 36	sylibr 233	. . . . . . . . 9 ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) → ((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ⁺)
38	37	adantr 479	. . . . . . . 8 ⊢ (((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) → ((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ⁺)
39	26, 38	rpdivcld 13087	. . . . . . 7 ⊢ (((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) → (𝑦 / ((𝑈 normOp_OLD 𝑊)‘𝑇)) ∈ ℝ⁺)
40		simprl 769	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) → 𝑥 ∈ (BaseSet‘𝑈))
41		metcl 24329	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (Met‘(BaseSet‘𝑈)) ∧ 𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → (𝑥𝐶𝑤) ∈ ℝ)
42	8, 41	mp3an1 1445	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → (𝑥𝐶𝑤) ∈ ℝ)
43	40, 42	sylan 578	. . . . . . . . . 10 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → (𝑥𝐶𝑤) ∈ ℝ)
44		simplrr 776	. . . . . . . . . . 11 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → 𝑦 ∈ ℝ⁺)
45	44	rpred 13070	. . . . . . . . . 10 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → 𝑦 ∈ ℝ)
46	35	ad2antrr 724	. . . . . . . . . 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 12140	. . . . . . . . . 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 1368	. . . . . . . . 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 745	. . . . . . . . . . 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 30736	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ≤ (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)))
52	51	3expa 1115	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ 𝐵 ∧ 𝑥 ∈ (BaseSet‘𝑈)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ≤ (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)))
53	50, 52	sylan 578	. . . . . . . . . 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 30688	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊))
55	1, 21, 22, 54	mp3an 1458	. . . . . . . . . . . . . 14 ⊢ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)
56	55	ffvelcdmi 7097	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (BaseSet‘𝑈) → (𝑇‘𝑥) ∈ (BaseSet‘𝑊))
57	55	ffvelcdmi 7097	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (BaseSet‘𝑈) → (𝑇‘𝑤) ∈ (BaseSet‘𝑊))
58	27, 17	imsmet 30624	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ NrmCVec → 𝐷 ∈ (Met‘(BaseSet‘𝑊)))
59	21, 58	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 𝐷 ∈ (Met‘(BaseSet‘𝑊))
60		metcl 24329	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (Met‘(BaseSet‘𝑊)) ∧ (𝑇‘𝑥) ∈ (BaseSet‘𝑊) ∧ (𝑇‘𝑤) ∈ (BaseSet‘𝑊)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ∈ ℝ)
61	59, 60	mp3an1 1445	. . . . . . . . . . . . 13 ⊢ (((𝑇‘𝑥) ∈ (BaseSet‘𝑊) ∧ (𝑇‘𝑤) ∈ (BaseSet‘𝑊)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ∈ ℝ)
62	56, 57, 61	syl2an 594	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ∈ ℝ)
63	40, 62	sylan 578	. . . . . . . . . . 11 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ∈ ℝ)
64		remulcl 11243	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ ∧ (𝑥𝐶𝑤) ∈ ℝ) → (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)) ∈ ℝ)
65	30, 42, 64	syl2an 594	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ 𝐵 ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑤 ∈ (BaseSet‘𝑈))) → (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)) ∈ ℝ)
66	65	anassrs 466	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ 𝐵 ∧ 𝑥 ∈ (BaseSet‘𝑈)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)) ∈ ℝ)
67	66	adantllr 717	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ 𝑥 ∈ (BaseSet‘𝑈)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)) ∈ ℝ)
68	67	adantlrr 719	. . . . . . . . . . 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 11354	. . . . . . . . . . 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 1368	. . . . . . . . . 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 693	. . . . . . . . 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 259	. . . . . . . 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 3136	. . . . . . 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 5157	. . . . . . . 8 ⊢ (𝑧 = (𝑦 / ((𝑈 normOp_OLD 𝑊)‘𝑇)) → ((𝑥𝐶𝑤) < 𝑧 ↔ (𝑥𝐶𝑤) < (𝑦 / ((𝑈 normOp_OLD 𝑊)‘𝑇))))
75	74	rspceaimv 3614	. . . . . . 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 582	. . . . . 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 3191	. . . . 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 518	. . . 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 24331	. . . . . 6 ⊢ (𝐷 ∈ (Met‘(BaseSet‘𝑊)) → 𝐷 ∈ (∞Met‘(BaseSet‘𝑊)))
80	59, 79	ax-mp 5	. . . . 5 ⊢ 𝐷 ∈ (∞Met‘(BaseSet‘𝑊))
81	11, 18	metcn 24543	. . . . 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 690	. . . 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 233	. . 3 ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) → 𝑇 ∈ (𝐽 Cn 𝐾))
84		eqid 2726	. . . . . . 7 ⊢ (0_vec‘𝑊) = (0_vec‘𝑊)
85	2, 84, 31	0ofval 30720	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 0_op 𝑊) = ((BaseSet‘𝑈) × {(0_vec‘𝑊)}))
86	1, 21, 85	mp2an 690	. . . . 5 ⊢ (𝑈 0_op 𝑊) = ((BaseSet‘𝑈) × {(0_vec‘𝑊)})
87	18	mopntopon 24436	. . . . . . 7 ⊢ (𝐷 ∈ (∞Met‘(BaseSet‘𝑊)) → 𝐾 ∈ (TopOn‘(BaseSet‘𝑊)))
88	80, 87	ax-mp 5	. . . . . 6 ⊢ 𝐾 ∈ (TopOn‘(BaseSet‘𝑊))
89	27, 84	nvzcl 30567	. . . . . . 7 ⊢ (𝑊 ∈ NrmCVec → (0_vec‘𝑊) ∈ (BaseSet‘𝑊))
90	21, 89	ax-mp 5	. . . . . 6 ⊢ (0_vec‘𝑊) ∈ (BaseSet‘𝑊)
91		cnconst2 23278	. . . . . 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 1458	. . . . 5 ⊢ ((BaseSet‘𝑈) × {(0_vec‘𝑊)}) ∈ (𝐽 Cn 𝐾)
93	86, 92	eqeltri 2822	. . . 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 208	1 ⊢ (𝑇 ∈ (𝐽 Cn 𝐾) ↔ 𝑇 ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 {csn 4633 class class class wbr 5153 × cxp 5680 ⟶wf 6550 ‘cfv 6554 (class class class)co 7424 ℝcr 11157 0cc0 11158 · cmul 11163 < clt 11298 ≤ cle 11299 / cdiv 11921 ℝ⁺crp 13028 ∞Metcxmet 21328 Metcmet 21329 MetOpencmopn 21333 TopOnctopon 22903 Cn ccn 23219 CnP ccnp 23220 NrmCVeccnv 30517 BaseSetcba 30519 0_veccn0v 30521 IndMetcims 30524 LnOp clno 30673 normOp_OLD cnmoo 30674 BLnOp cblo 30675 0_op c0o 30676

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 ax-addf 11237 ax-mulf 11238

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 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 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-sup 9485 df-inf 9486 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12611 df-uz 12875 df-q 12985 df-rp 13029 df-xneg 13146 df-xadd 13147 df-xmul 13148 df-seq 14022 df-exp 14082 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-topgen 17458 df-psmet 21335 df-xmet 21336 df-met 21337 df-bl 21338 df-mopn 21339 df-top 22887 df-topon 22904 df-bases 22940 df-cn 23222 df-cnp 23223 df-grpo 30426 df-gid 30427 df-ginv 30428 df-gdiv 30429 df-ablo 30478 df-vc 30492 df-nv 30525 df-va 30528 df-ba 30529 df-sm 30530 df-0v 30531 df-vs 30532 df-nmcv 30533 df-ims 30534 df-lno 30677 df-nmoo 30678 df-blo 30679 df-0o 30680

This theorem is referenced by: lnocni 30739 blocn 30740

W3C validator