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Theorem blocni 29167

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 2738	. . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
3		eqid 2738	. . . . 5 ⊢ (0_vec‘𝑈) = (0_vec‘𝑈)
4	2, 3	nvzcl 28996	. . . 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 29053	. . . . . . . . 9 ⊢ (𝑈 ∈ NrmCVec → 𝐶 ∈ (Met‘(BaseSet‘𝑈)))
8	1, 7	ax-mp 5	. . . . . . . 8 ⊢ 𝐶 ∈ (Met‘(BaseSet‘𝑈))
9		metxmet 23487	. . . . . . . 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 23592	. . . . . . 7 ⊢ (𝐶 ∈ (∞Met‘(BaseSet‘𝑈)) → 𝐽 ∈ (TopOn‘(BaseSet‘𝑈)))
13	10, 12	ax-mp 5	. . . . . 6 ⊢ 𝐽 ∈ (TopOn‘(BaseSet‘𝑈))
14	13	toponunii 22065	. . . . 5 ⊢ (BaseSet‘𝑈) = ∪ 𝐽
15	14	cncnpi 22429	. . . 4 ⊢ ((𝑇 ∈ (𝐽 Cn 𝐾) ∧ (0_vec‘𝑈) ∈ (BaseSet‘𝑈)) → 𝑇 ∈ ((𝐽 CnP 𝐾)‘(0_vec‘𝑈)))
16	5, 15	mpan2 688	. . 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 29166	. . 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 2826	. . 3 ⊢ (𝑇 = (𝑈 0_op 𝑊) → (𝑇 ∈ (𝐽 Cn 𝐾) ↔ (𝑈 0_op 𝑊) ∈ (𝐽 Cn 𝐾)))
26		simprr 770	. . . . . . . 8 ⊢ (((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) → 𝑦 ∈ ℝ⁺)
27		eqid 2738	. . . . . . . . . . . . 13 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
28		eqid 2738	. . . . . . . . . . . . 13 ⊢ (𝑈 normOp_OLD 𝑊) = (𝑈 normOp_OLD 𝑊)
29	2, 27, 28, 20	nmblore 29148	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → ((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ)
30	1, 21, 29	mp3an12 1450	. . . . . . . . . . 11 ⊢ (𝑇 ∈ 𝐵 → ((𝑈 normOp_OLD 𝑊)‘𝑇) ∈ ℝ)
31		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (𝑈 0_op 𝑊) = (𝑈 0_op 𝑊)
32	28, 31, 19	nmlnogt0 29159	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇 ≠ (𝑈 0_op 𝑊) ↔ 0 < ((𝑈 normOp_OLD 𝑊)‘𝑇)))
33	1, 21, 22, 32	mp3an 1460	. . . . . . . . . . . 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 12732	. . . . . . . . . 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 12789	. . . . . . 7 ⊢ (((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) → (𝑦 / ((𝑈 normOp_OLD 𝑊)‘𝑇)) ∈ ℝ⁺)
40		simprl 768	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) → 𝑥 ∈ (BaseSet‘𝑈))
41		metcl 23485	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (Met‘(BaseSet‘𝑈)) ∧ 𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → (𝑥𝐶𝑤) ∈ ℝ)
42	8, 41	mp3an1 1447	. . . . . . . . . . 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 775	. . . . . . . . . . 11 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → 𝑦 ∈ ℝ⁺)
45	44	rpred 12772	. . . . . . . . . 10 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → 𝑦 ∈ ℝ)
46	35	ad2antrr 723	. . . . . . . . . 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 11849	. . . . . . . . . 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 1370	. . . . . . . . 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 744	. . . . . . . . . . 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 29165	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ≤ (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)))
52	51	3expa 1117	. . . . . . . . . . 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 29117	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊))
55	1, 21, 22, 54	mp3an 1460	. . . . . . . . . . . . . 14 ⊢ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)
56	55	ffvelrni 6960	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (BaseSet‘𝑈) → (𝑇‘𝑥) ∈ (BaseSet‘𝑊))
57	55	ffvelrni 6960	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (BaseSet‘𝑈) → (𝑇‘𝑤) ∈ (BaseSet‘𝑊))
58	27, 17	imsmet 29053	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ NrmCVec → 𝐷 ∈ (Met‘(BaseSet‘𝑊)))
59	21, 58	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 𝐷 ∈ (Met‘(BaseSet‘𝑊))
60		metcl 23485	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (Met‘(BaseSet‘𝑊)) ∧ (𝑇‘𝑥) ∈ (BaseSet‘𝑊) ∧ (𝑇‘𝑤) ∈ (BaseSet‘𝑊)) → ((𝑇‘𝑥)𝐷(𝑇‘𝑤)) ∈ ℝ)
61	59, 60	mp3an1 1447	. . . . . . . . . . . . 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 10956	. . . . . . . . . . . . . . 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 716	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ 𝐵 ∧ 𝑇 ≠ (𝑈 0_op 𝑊)) ∧ 𝑥 ∈ (BaseSet‘𝑈)) ∧ 𝑤 ∈ (BaseSet‘𝑈)) → (((𝑈 normOp_OLD 𝑊)‘𝑇) · (𝑥𝐶𝑤)) ∈ ℝ)
68	67	adantlrr 718	. . . . . . . . . . 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 11065	. . . . . . . . . . 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 1370	. . . . . . . . . 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 692	. . . . . . . . 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 3103	. . . . . . 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 5078	. . . . . . . 8 ⊢ (𝑧 = (𝑦 / ((𝑈 normOp_OLD 𝑊)‘𝑇)) → ((𝑥𝐶𝑤) < 𝑧 ↔ (𝑥𝐶𝑤) < (𝑦 / ((𝑈 normOp_OLD 𝑊)‘𝑇))))
75	74	rspceaimv 3565	. . . . . . 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 3123	. . . . 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 23487	. . . . . 6 ⊢ (𝐷 ∈ (Met‘(BaseSet‘𝑊)) → 𝐷 ∈ (∞Met‘(BaseSet‘𝑊)))
80	59, 79	ax-mp 5	. . . . 5 ⊢ 𝐷 ∈ (∞Met‘(BaseSet‘𝑊))
81	11, 18	metcn 23699	. . . . 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 689	. . . 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 2738	. . . . . . 7 ⊢ (0_vec‘𝑊) = (0_vec‘𝑊)
85	2, 84, 31	0ofval 29149	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 0_op 𝑊) = ((BaseSet‘𝑈) × {(0_vec‘𝑊)}))
86	1, 21, 85	mp2an 689	. . . . 5 ⊢ (𝑈 0_op 𝑊) = ((BaseSet‘𝑈) × {(0_vec‘𝑊)})
87	18	mopntopon 23592	. . . . . . 7 ⊢ (𝐷 ∈ (∞Met‘(BaseSet‘𝑊)) → 𝐾 ∈ (TopOn‘(BaseSet‘𝑊)))
88	80, 87	ax-mp 5	. . . . . 6 ⊢ 𝐾 ∈ (TopOn‘(BaseSet‘𝑊))
89	27, 84	nvzcl 28996	. . . . . . 7 ⊢ (𝑊 ∈ NrmCVec → (0_vec‘𝑊) ∈ (BaseSet‘𝑊))
90	21, 89	ax-mp 5	. . . . . 6 ⊢ (0_vec‘𝑊) ∈ (BaseSet‘𝑊)
91		cnconst2 22434	. . . . . 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 1460	. . . . 5 ⊢ ((BaseSet‘𝑈) × {(0_vec‘𝑊)}) ∈ (𝐽 Cn 𝐾)
93	86, 92	eqeltri 2835	. . . 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 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 {csn 4561 class class class wbr 5074 × cxp 5587 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ℝcr 10870 0cc0 10871 · cmul 10876 < clt 11009 ≤ cle 11010 / cdiv 11632 ℝ⁺crp 12730 ∞Metcxmet 20582 Metcmet 20583 MetOpencmopn 20587 TopOnctopon 22059 Cn ccn 22375 CnP ccnp 22376 NrmCVeccnv 28946 BaseSetcba 28948 0_veccn0v 28950 IndMetcims 28953 LnOp clno 29102 normOp_OLD cnmoo 29103 BLnOp cblo 29104 0_op c0o 29105

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-topgen 17154 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-top 22043 df-topon 22060 df-bases 22096 df-cn 22378 df-cnp 22379 df-grpo 28855 df-gid 28856 df-ginv 28857 df-gdiv 28858 df-ablo 28907 df-vc 28921 df-nv 28954 df-va 28957 df-ba 28958 df-sm 28959 df-0v 28960 df-vs 28961 df-nmcv 28962 df-ims 28963 df-lno 29106 df-nmoo 29107 df-blo 29108 df-0o 29109

This theorem is referenced by: lnocni 29168 blocn 29169

W3C validator