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Theorem blocni 29747

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 29576	. . . 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 29633	. . . . . . . . 9 ⊢ (𝑈 ∈ NrmCVec → 𝐶 ∈ (Met‘(BaseSet‘𝑈)))
8	1, 7	ax-mp 5	. . . . . . . 8 ⊢ 𝐶 ∈ (Met‘(BaseSet‘𝑈))
9		metxmet 23687	. . . . . . . 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 23792	. . . . . . 7 ⊢ (𝐶 ∈ (∞Met‘(BaseSet‘𝑈)) → 𝐽 ∈ (TopOn‘(BaseSet‘𝑈)))
13	10, 12	ax-mp 5	. . . . . 6 ⊢ 𝐽 ∈ (TopOn‘(BaseSet‘𝑈))
14	13	toponunii 22265	. . . . 5 ⊢ (BaseSet‘𝑈) = ∪ 𝐽
15	14	cncnpi 22629	. . . 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 29746	. . 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 2825	. . 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 2736	. . . . . . . . . . . . 13 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
28		eqid 2736	. . . . . . . . . . . . 13 ⊢ (𝑈 normOp_OLD 𝑊) = (𝑈 normOp_OLD 𝑊)
29	2, 27, 28, 20	nmblore 29728	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → ((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ)
30	1, 21, 29	mp3an12 1451	. . . . . . . . . . 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 29739	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇 ≠ (𝑈 0_op 𝑊) ↔ 0 < ((𝑈 normOp_OLD 𝑊)‘𝑇)))
33	1, 21, 22, 32	mp3an 1461	. . . . . . . . . . . 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 613	. . . . . . . . . 10 ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) → (((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ ∧ 0 < ((𝑈 normOp_OLD 𝑊)‘𝑇)))
36		elrp 12917	. . . . . . . . . 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 481	. . . . . . . 8 ⊢ (((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) → ((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ⁺)
39	26, 38	rpdivcld 12974	. . . . . . 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 23685	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (Met‘(BaseSet‘𝑈)) ∧ 𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → (𝑥𝐶𝑤) ∈ ℝ)
42	8, 41	mp3an1 1448	. . . . . . . . . . 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 776	. . . . . . . . . . 11 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → 𝑦 ∈ ℝ⁺)
45	44	rpred 12957	. . . . . . . . . 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 12029	. . . . . . . . . 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 1371	. . . . . . . . 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 29745	. . . . . . . . . . . 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 29697	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊))
55	1, 21, 22, 54	mp3an 1461	. . . . . . . . . . . . . 14 ⊢ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)
56	55	ffvelcdmi 7034	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (BaseSet‘𝑈) → (𝑇‘𝑥) ∈ (BaseSet‘𝑊))
57	55	ffvelcdmi 7034	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (BaseSet‘𝑈) → (𝑇‘𝑤) ∈ (BaseSet‘𝑊))
58	27, 17	imsmet 29633	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ NrmCVec → 𝐷 ∈ (Met‘(BaseSet‘𝑊)))
59	21, 58	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 𝐷 ∈ (Met‘(BaseSet‘𝑊))
60		metcl 23685	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (Met‘(BaseSet‘𝑊)) ∧ (𝑇‘𝑥) ∈ (BaseSet‘𝑊) ∧ (𝑇‘𝑤) ∈ (BaseSet‘𝑊)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ∈ ℝ)
61	59, 60	mp3an1 1448	. . . . . . . . . . . . 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 11136	. . . . . . . . . . . . . . 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 468	. . . . . . . . . . . . 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 11245	. . . . . . . . . . 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 1371	. . . . . . . . . 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 3143	. . . . . . 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 5109	. . . . . . . 8 ⊢ (𝑧 = (𝑦 / ((𝑈 normOp_OLD 𝑊)‘𝑇)) → ((𝑥𝐶𝑤) < 𝑧 ↔ (𝑥𝐶𝑤) < (𝑦 / ((𝑈 normOp_OLD 𝑊)‘𝑇))))
75	74	rspceaimv 3585	. . . . . . 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 3197	. . . . 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 520	. . . 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 23687	. . . . . 6 ⊢ (𝐷 ∈ (Met‘(BaseSet‘𝑊)) → 𝐷 ∈ (∞Met‘(BaseSet‘𝑊)))
80	59, 79	ax-mp 5	. . . . 5 ⊢ 𝐷 ∈ (∞Met‘(BaseSet‘𝑊))
81	11, 18	metcn 23899	. . . . 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 2736	. . . . . . 7 ⊢ (0_vec‘𝑊) = (0_vec‘𝑊)
85	2, 84, 31	0ofval 29729	. . . . . 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 23792	. . . . . . 7 ⊢ (𝐷 ∈ (∞Met‘(BaseSet‘𝑊)) → 𝐾 ∈ (TopOn‘(BaseSet‘𝑊)))
88	80, 87	ax-mp 5	. . . . . 6 ⊢ 𝐾 ∈ (TopOn‘(BaseSet‘𝑊))
89	27, 84	nvzcl 29576	. . . . . . 7 ⊢ (𝑊 ∈ NrmCVec → (0_vec‘𝑊) ∈ (BaseSet‘𝑊))
90	21, 89	ax-mp 5	. . . . . 6 ⊢ (0_vec‘𝑊) ∈ (BaseSet‘𝑊)
91		cnconst2 22634	. . . . . 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 1461	. . . . 5 ⊢ ((BaseSet‘𝑈) × {(0_vec‘𝑊)}) ∈ (𝐽 Cn 𝐾)
93	86, 92	eqeltri 2834	. . . 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 3030	. 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 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 {csn 4586 class class class wbr 5105 × cxp 5631 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ℝcr 11050 0cc0 11051 · cmul 11056 < clt 11189 ≤ cle 11190 / cdiv 11812 ℝ⁺crp 12915 ∞Metcxmet 20781 Metcmet 20782 MetOpencmopn 20786 TopOnctopon 22259 Cn ccn 22575 CnP ccnp 22576 NrmCVeccnv 29526 BaseSetcba 29528 0_veccn0v 29530 IndMetcims 29533 LnOp clno 29682 normOp_OLD cnmoo 29683 BLnOp cblo 29684 0_op c0o 29685

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9378 df-inf 9379 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-z 12500 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-seq 13907 df-exp 13968 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-topgen 17325 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-top 22243 df-topon 22260 df-bases 22296 df-cn 22578 df-cnp 22579 df-grpo 29435 df-gid 29436 df-ginv 29437 df-gdiv 29438 df-ablo 29487 df-vc 29501 df-nv 29534 df-va 29537 df-ba 29538 df-sm 29539 df-0v 29540 df-vs 29541 df-nmcv 29542 df-ims 29543 df-lno 29686 df-nmoo 29687 df-blo 29688 df-0o 29689

This theorem is referenced by: lnocni 29748 blocn 29749

W3C validator