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Theorem nmblolbii 30040

Description: A lower bound for the norm of a bounded linear operator. (Contributed by NM, 7-Dec-2007.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
nmblolbi.1	⊢ 𝑋 = (BaseSet‘𝑈)
nmblolbi.4	⊢ 𝐿 = (norm_CV‘𝑈)
nmblolbi.5	⊢ 𝑀 = (norm_CV‘𝑊)
nmblolbi.6	⊢ 𝑁 = (𝑈 normOp_OLD 𝑊)
nmblolbi.7	⊢ 𝐵 = (𝑈 BLnOp 𝑊)
nmblolbi.u	⊢ 𝑈 ∈ NrmCVec
nmblolbi.w	⊢ 𝑊 ∈ NrmCVec
nmblolbii.b	⊢ 𝑇 ∈ 𝐵

**Assertion**
Ref	Expression
nmblolbii	⊢ (𝐴 ∈ 𝑋 → (𝑀‘(𝑇‘𝐴)) ≤ ((𝑁‘𝑇) · (𝐿‘𝐴)))

**Proof of Theorem nmblolbii**
Step	Hyp	Ref	Expression
1		fveq2 6889	. . . 4 ⊢ (𝐴 = (0_vec‘𝑈) → (𝑇‘𝐴) = (𝑇‘(0_vec‘𝑈)))
2	1	fveq2d 6893	. . 3 ⊢ (𝐴 = (0_vec‘𝑈) → (𝑀‘(𝑇‘𝐴)) = (𝑀‘(𝑇‘(0_vec‘𝑈))))
3		fveq2 6889	. . . 4 ⊢ (𝐴 = (0_vec‘𝑈) → (𝐿‘𝐴) = (𝐿‘(0_vec‘𝑈)))
4	3	oveq2d 7422	. . 3 ⊢ (𝐴 = (0_vec‘𝑈) → ((𝑁‘𝑇) · (𝐿‘𝐴)) = ((𝑁‘𝑇) · (𝐿‘(0_vec‘𝑈))))
5	2, 4	breq12d 5161	. 2 ⊢ (𝐴 = (0_vec‘𝑈) → ((𝑀‘(𝑇‘𝐴)) ≤ ((𝑁‘𝑇) · (𝐿‘𝐴)) ↔ (𝑀‘(𝑇‘(0_vec‘𝑈))) ≤ ((𝑁‘𝑇) · (𝐿‘(0_vec‘𝑈)))))
6		nmblolbi.u	. . . . . . . . 9 ⊢ 𝑈 ∈ NrmCVec
7		nmblolbi.1	. . . . . . . . . 10 ⊢ 𝑋 = (BaseSet‘𝑈)
8		nmblolbi.4	. . . . . . . . . 10 ⊢ 𝐿 = (norm_CV‘𝑈)
9	7, 8	nvcl 29902	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐿‘𝐴) ∈ ℝ)
10	6, 9	mpan 689	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → (𝐿‘𝐴) ∈ ℝ)
11	10	adantr 482	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝐿‘𝐴) ∈ ℝ)
12		eqid 2733	. . . . . . . . . . 11 ⊢ (0_vec‘𝑈) = (0_vec‘𝑈)
13	7, 12, 8	nvz 29910	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝐿‘𝐴) = 0 ↔ 𝐴 = (0_vec‘𝑈)))
14	6, 13	mpan 689	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑋 → ((𝐿‘𝐴) = 0 ↔ 𝐴 = (0_vec‘𝑈)))
15	14	necon3bid 2986	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → ((𝐿‘𝐴) ≠ 0 ↔ 𝐴 ≠ (0_vec‘𝑈)))
16	15	biimpar 479	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝐿‘𝐴) ≠ 0)
17	11, 16	rereccld 12038	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (1 / (𝐿‘𝐴)) ∈ ℝ)
18	7, 12, 8	nvgt0 29915	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴 ≠ (0_vec‘𝑈) ↔ 0 < (𝐿‘𝐴)))
19	6, 18	mpan 689	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ≠ (0_vec‘𝑈) ↔ 0 < (𝐿‘𝐴)))
20	19	biimpa 478	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → 0 < (𝐿‘𝐴))
21	11, 20	recgt0d 12145	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → 0 < (1 / (𝐿‘𝐴)))
22		0re 11213	. . . . . . . 8 ⊢ 0 ∈ ℝ
23		ltle 11299	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ (1 / (𝐿‘𝐴)) ∈ ℝ) → (0 < (1 / (𝐿‘𝐴)) → 0 ≤ (1 / (𝐿‘𝐴))))
24	22, 17, 23	sylancr 588	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (0 < (1 / (𝐿‘𝐴)) → 0 ≤ (1 / (𝐿‘𝐴))))
25	21, 24	mpd 15	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → 0 ≤ (1 / (𝐿‘𝐴)))
26		nmblolbi.w	. . . . . . . . 9 ⊢ 𝑊 ∈ NrmCVec
27		nmblolbii.b	. . . . . . . . 9 ⊢ 𝑇 ∈ 𝐵
28		eqid 2733	. . . . . . . . . 10 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
29		nmblolbi.7	. . . . . . . . . 10 ⊢ 𝐵 = (𝑈 BLnOp 𝑊)
30	7, 28, 29	blof 30026	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → 𝑇:𝑋⟶(BaseSet‘𝑊))
31	6, 26, 27, 30	mp3an 1462	. . . . . . . 8 ⊢ 𝑇:𝑋⟶(BaseSet‘𝑊)
32	31	ffvelcdmi 7083	. . . . . . 7 ⊢ (𝐴 ∈ 𝑋 → (𝑇‘𝐴) ∈ (BaseSet‘𝑊))
33	32	adantr 482	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝑇‘𝐴) ∈ (BaseSet‘𝑊))
34		eqid 2733	. . . . . . . 8 ⊢ ( ·_𝑠OLD ‘𝑊) = ( ·_𝑠OLD ‘𝑊)
35		nmblolbi.5	. . . . . . . 8 ⊢ 𝑀 = (norm_CV‘𝑊)
36	28, 34, 35	nvsge0 29905	. . . . . . 7 ⊢ ((𝑊 ∈ NrmCVec ∧ ((1 / (𝐿‘𝐴)) ∈ ℝ ∧ 0 ≤ (1 / (𝐿‘𝐴))) ∧ (𝑇‘𝐴) ∈ (BaseSet‘𝑊)) → (𝑀‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑊)(𝑇‘𝐴))) = ((1 / (𝐿‘𝐴)) · (𝑀‘(𝑇‘𝐴))))
37	26, 36	mp3an1 1449	. . . . . 6 ⊢ ((((1 / (𝐿‘𝐴)) ∈ ℝ ∧ 0 ≤ (1 / (𝐿‘𝐴))) ∧ (𝑇‘𝐴) ∈ (BaseSet‘𝑊)) → (𝑀‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑊)(𝑇‘𝐴))) = ((1 / (𝐿‘𝐴)) · (𝑀‘(𝑇‘𝐴))))
38	17, 25, 33, 37	syl21anc 837	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝑀‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑊)(𝑇‘𝐴))) = ((1 / (𝐿‘𝐴)) · (𝑀‘(𝑇‘𝐴))))
39	17	recnd 11239	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (1 / (𝐿‘𝐴)) ∈ ℂ)
40		simpl 484	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → 𝐴 ∈ 𝑋)
41		eqid 2733	. . . . . . . . . . 11 ⊢ (𝑈 LnOp 𝑊) = (𝑈 LnOp 𝑊)
42	41, 29	bloln 30025	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → 𝑇 ∈ (𝑈 LnOp 𝑊))
43	6, 26, 27, 42	mp3an 1462	. . . . . . . . 9 ⊢ 𝑇 ∈ (𝑈 LnOp 𝑊)
44	6, 26, 43	3pm3.2i 1340	. . . . . . . 8 ⊢ (𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ (𝑈 LnOp 𝑊))
45		eqid 2733	. . . . . . . . 9 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
46	7, 45, 34, 41	lnomul 30001	. . . . . . . 8 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ (𝑈 LnOp 𝑊)) ∧ ((1 / (𝐿‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → (𝑇‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴)) = ((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑊)(𝑇‘𝐴)))
47	44, 46	mpan 689	. . . . . . 7 ⊢ (((1 / (𝐿‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (𝑇‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴)) = ((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑊)(𝑇‘𝐴)))
48	39, 40, 47	syl2anc 585	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝑇‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴)) = ((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑊)(𝑇‘𝐴)))
49	48	fveq2d 6893	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝑀‘(𝑇‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴))) = (𝑀‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑊)(𝑇‘𝐴))))
50	28, 35	nvcl 29902	. . . . . . . . 9 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑇‘𝐴) ∈ (BaseSet‘𝑊)) → (𝑀‘(𝑇‘𝐴)) ∈ ℝ)
51	26, 32, 50	sylancr 588	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → (𝑀‘(𝑇‘𝐴)) ∈ ℝ)
52	51	adantr 482	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝑀‘(𝑇‘𝐴)) ∈ ℝ)
53	52	recnd 11239	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝑀‘(𝑇‘𝐴)) ∈ ℂ)
54	11	recnd 11239	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝐿‘𝐴) ∈ ℂ)
55	53, 54, 16	divrec2d 11991	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → ((𝑀‘(𝑇‘𝐴)) / (𝐿‘𝐴)) = ((1 / (𝐿‘𝐴)) · (𝑀‘(𝑇‘𝐴))))
56	38, 49, 55	3eqtr4rd 2784	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → ((𝑀‘(𝑇‘𝐴)) / (𝐿‘𝐴)) = (𝑀‘(𝑇‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴))))
57	7, 45	nvscl 29867	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ (1 / (𝐿‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ 𝑋) → ((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴) ∈ 𝑋)
58	6, 57	mp3an1 1449	. . . . . . 7 ⊢ (((1 / (𝐿‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ 𝑋) → ((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴) ∈ 𝑋)
59	58	ancoms 460	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ (1 / (𝐿‘𝐴)) ∈ ℂ) → ((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴) ∈ 𝑋)
60	39, 59	syldan 592	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → ((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴) ∈ 𝑋)
61	7, 8	nvcl 29902	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ ((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴) ∈ 𝑋) → (𝐿‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴)) ∈ ℝ)
62	6, 60, 61	sylancr 588	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝐿‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴)) ∈ ℝ)
63	7, 45, 12, 8	nv1 29916	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝐿‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴)) = 1)
64	6, 63	mp3an1 1449	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝐿‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴)) = 1)
65		eqle 11313	. . . . . 6 ⊢ (((𝐿‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴)) ∈ ℝ ∧ (𝐿‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴)) = 1) → (𝐿‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴)) ≤ 1)
66	62, 64, 65	syl2anc 585	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝐿‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴)) ≤ 1)
67	6, 26, 31	3pm3.2i 1340	. . . . . 6 ⊢ (𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶(BaseSet‘𝑊))
68		nmblolbi.6	. . . . . . 7 ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊)
69	7, 28, 8, 35, 68	nmoolb 30012	. . . . . 6 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶(BaseSet‘𝑊)) ∧ (((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴) ∈ 𝑋 ∧ (𝐿‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴)) ≤ 1)) → (𝑀‘(𝑇‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴))) ≤ (𝑁‘𝑇))
70	67, 69	mpan 689	. . . . 5 ⊢ ((((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴) ∈ 𝑋 ∧ (𝐿‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴)) ≤ 1) → (𝑀‘(𝑇‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴))) ≤ (𝑁‘𝑇))
71	60, 66, 70	syl2anc 585	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝑀‘(𝑇‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴))) ≤ (𝑁‘𝑇))
72	56, 71	eqbrtrd 5170	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → ((𝑀‘(𝑇‘𝐴)) / (𝐿‘𝐴)) ≤ (𝑁‘𝑇))
73	7, 28, 68, 29	nmblore 30027	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → (𝑁‘𝑇) ∈ ℝ)
74	6, 26, 27, 73	mp3an 1462	. . . . 5 ⊢ (𝑁‘𝑇) ∈ ℝ
75	74	a1i 11	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝑁‘𝑇) ∈ ℝ)
76		ledivmul2 12090	. . . 4 ⊢ (((𝑀‘(𝑇‘𝐴)) ∈ ℝ ∧ (𝑁‘𝑇) ∈ ℝ ∧ ((𝐿‘𝐴) ∈ ℝ ∧ 0 < (𝐿‘𝐴))) → (((𝑀‘(𝑇‘𝐴)) / (𝐿‘𝐴)) ≤ (𝑁‘𝑇) ↔ (𝑀‘(𝑇‘𝐴)) ≤ ((𝑁‘𝑇) · (𝐿‘𝐴))))
77	52, 75, 11, 20, 76	syl112anc 1375	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (((𝑀‘(𝑇‘𝐴)) / (𝐿‘𝐴)) ≤ (𝑁‘𝑇) ↔ (𝑀‘(𝑇‘𝐴)) ≤ ((𝑁‘𝑇) · (𝐿‘𝐴))))
78	72, 77	mpbid 231	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝑀‘(𝑇‘𝐴)) ≤ ((𝑁‘𝑇) · (𝐿‘𝐴)))
79		0le0 12310	. . . 4 ⊢ 0 ≤ 0
80		eqid 2733	. . . . . . . 8 ⊢ (0_vec‘𝑊) = (0_vec‘𝑊)
81	7, 28, 12, 80, 41	lno0 29997	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ (𝑈 LnOp 𝑊)) → (𝑇‘(0_vec‘𝑈)) = (0_vec‘𝑊))
82	6, 26, 43, 81	mp3an 1462	. . . . . 6 ⊢ (𝑇‘(0_vec‘𝑈)) = (0_vec‘𝑊)
83	82	fveq2i 6892	. . . . 5 ⊢ (𝑀‘(𝑇‘(0_vec‘𝑈))) = (𝑀‘(0_vec‘𝑊))
84	80, 35	nvz0 29909	. . . . . 6 ⊢ (𝑊 ∈ NrmCVec → (𝑀‘(0_vec‘𝑊)) = 0)
85	26, 84	ax-mp 5	. . . . 5 ⊢ (𝑀‘(0_vec‘𝑊)) = 0
86	83, 85	eqtri 2761	. . . 4 ⊢ (𝑀‘(𝑇‘(0_vec‘𝑈))) = 0
87	12, 8	nvz0 29909	. . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → (𝐿‘(0_vec‘𝑈)) = 0)
88	6, 87	ax-mp 5	. . . . . 6 ⊢ (𝐿‘(0_vec‘𝑈)) = 0
89	88	oveq2i 7417	. . . . 5 ⊢ ((𝑁‘𝑇) · (𝐿‘(0_vec‘𝑈))) = ((𝑁‘𝑇) · 0)
90	74	recni 11225	. . . . . 6 ⊢ (𝑁‘𝑇) ∈ ℂ
91	90	mul01i 11401	. . . . 5 ⊢ ((𝑁‘𝑇) · 0) = 0
92	89, 91	eqtri 2761	. . . 4 ⊢ ((𝑁‘𝑇) · (𝐿‘(0_vec‘𝑈))) = 0
93	79, 86, 92	3brtr4i 5178	. . 3 ⊢ (𝑀‘(𝑇‘(0_vec‘𝑈))) ≤ ((𝑁‘𝑇) · (𝐿‘(0_vec‘𝑈)))
94	93	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑋 → (𝑀‘(𝑇‘(0_vec‘𝑈))) ≤ ((𝑁‘𝑇) · (𝐿‘(0_vec‘𝑈))))
95	5, 78, 94	pm2.61ne 3028	1 ⊢ (𝐴 ∈ 𝑋 → (𝑀‘(𝑇‘𝐴)) ≤ ((𝑁‘𝑇) · (𝐿‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 class class class wbr 5148 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 ℂcc 11105 ℝcr 11106 0cc0 11107 1c1 11108 · cmul 11112 < clt 11245 ≤ cle 11246 / cdiv 11868 NrmCVeccnv 29825 BaseSetcba 29827 ·_𝑠OLD cns 29828 0_veccn0v 29829 norm_CVcnmcv 29831 LnOp clno 29981 normOp_OLD cnmoo 29982 BLnOp cblo 29983

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-sup 9434 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-rp 12972 df-seq 13964 df-exp 14025 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-grpo 29734 df-gid 29735 df-ginv 29736 df-ablo 29786 df-vc 29800 df-nv 29833 df-va 29836 df-ba 29837 df-sm 29838 df-0v 29839 df-nmcv 29841 df-lno 29985 df-nmoo 29986 df-blo 29987

This theorem is referenced by: nmblolbi 30041

W3C validator