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Theorem nmblolbii 30319

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 6890	. . . 4 ⊢ (𝐴 = (0_vec‘𝑈) → (𝑇‘𝐴) = (𝑇‘(0_vec‘𝑈)))
2	1	fveq2d 6894	. . 3 ⊢ (𝐴 = (0_vec‘𝑈) → (𝑀‘(𝑇‘𝐴)) = (𝑀‘(𝑇‘(0_vec‘𝑈))))
3		fveq2 6890	. . . 4 ⊢ (𝐴 = (0_vec‘𝑈) → (𝐿‘𝐴) = (𝐿‘(0_vec‘𝑈)))
4	3	oveq2d 7427	. . 3 ⊢ (𝐴 = (0_vec‘𝑈) → ((𝑁‘𝑇) · (𝐿‘𝐴)) = ((𝑁‘𝑇) · (𝐿‘(0_vec‘𝑈))))
5	2, 4	breq12d 5160	. 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 30181	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐿‘𝐴) ∈ ℝ)
10	6, 9	mpan 686	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → (𝐿‘𝐴) ∈ ℝ)
11	10	adantr 479	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝐿‘𝐴) ∈ ℝ)
12		eqid 2730	. . . . . . . . . . 11 ⊢ (0_vec‘𝑈) = (0_vec‘𝑈)
13	7, 12, 8	nvz 30189	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝐿‘𝐴) = 0 ↔ 𝐴 = (0_vec‘𝑈)))
14	6, 13	mpan 686	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑋 → ((𝐿‘𝐴) = 0 ↔ 𝐴 = (0_vec‘𝑈)))
15	14	necon3bid 2983	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → ((𝐿‘𝐴) ≠ 0 ↔ 𝐴 ≠ (0_vec‘𝑈)))
16	15	biimpar 476	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝐿‘𝐴) ≠ 0)
17	11, 16	rereccld 12045	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (1 / (𝐿‘𝐴)) ∈ ℝ)
18	7, 12, 8	nvgt0 30194	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴 ≠ (0_vec‘𝑈) ↔ 0 < (𝐿‘𝐴)))
19	6, 18	mpan 686	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ≠ (0_vec‘𝑈) ↔ 0 < (𝐿‘𝐴)))
20	19	biimpa 475	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → 0 < (𝐿‘𝐴))
21	11, 20	recgt0d 12152	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → 0 < (1 / (𝐿‘𝐴)))
22		0re 11220	. . . . . . . 8 ⊢ 0 ∈ ℝ
23		ltle 11306	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ (1 / (𝐿‘𝐴)) ∈ ℝ) → (0 < (1 / (𝐿‘𝐴)) → 0 ≤ (1 / (𝐿‘𝐴))))
24	22, 17, 23	sylancr 585	. . . . . . 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 2730	. . . . . . . . . 10 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
29		nmblolbi.7	. . . . . . . . . 10 ⊢ 𝐵 = (𝑈 BLnOp 𝑊)
30	7, 28, 29	blof 30305	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → 𝑇:𝑋⟶(BaseSet‘𝑊))
31	6, 26, 27, 30	mp3an 1459	. . . . . . . 8 ⊢ 𝑇:𝑋⟶(BaseSet‘𝑊)
32	31	ffvelcdmi 7084	. . . . . . 7 ⊢ (𝐴 ∈ 𝑋 → (𝑇‘𝐴) ∈ (BaseSet‘𝑊))
33	32	adantr 479	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝑇‘𝐴) ∈ (BaseSet‘𝑊))
34		eqid 2730	. . . . . . . 8 ⊢ ( ·_𝑠OLD ‘𝑊) = ( ·_𝑠OLD ‘𝑊)
35		nmblolbi.5	. . . . . . . 8 ⊢ 𝑀 = (norm_CV‘𝑊)
36	28, 34, 35	nvsge0 30184	. . . . . . 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 1446	. . . . . 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 834	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝑀‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑊)(𝑇‘𝐴))) = ((1 / (𝐿‘𝐴)) · (𝑀‘(𝑇‘𝐴))))
39	17	recnd 11246	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (1 / (𝐿‘𝐴)) ∈ ℂ)
40		simpl 481	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → 𝐴 ∈ 𝑋)
41		eqid 2730	. . . . . . . . . . 11 ⊢ (𝑈 LnOp 𝑊) = (𝑈 LnOp 𝑊)
42	41, 29	bloln 30304	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → 𝑇 ∈ (𝑈 LnOp 𝑊))
43	6, 26, 27, 42	mp3an 1459	. . . . . . . . 9 ⊢ 𝑇 ∈ (𝑈 LnOp 𝑊)
44	6, 26, 43	3pm3.2i 1337	. . . . . . . 8 ⊢ (𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ (𝑈 LnOp 𝑊))
45		eqid 2730	. . . . . . . . 9 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
46	7, 45, 34, 41	lnomul 30280	. . . . . . . 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 686	. . . . . . 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 582	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝑇‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴)) = ((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑊)(𝑇‘𝐴)))
49	48	fveq2d 6894	. . . . 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 30181	. . . . . . . . 9 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑇‘𝐴) ∈ (BaseSet‘𝑊)) → (𝑀‘(𝑇‘𝐴)) ∈ ℝ)
51	26, 32, 50	sylancr 585	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → (𝑀‘(𝑇‘𝐴)) ∈ ℝ)
52	51	adantr 479	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝑀‘(𝑇‘𝐴)) ∈ ℝ)
53	52	recnd 11246	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝑀‘(𝑇‘𝐴)) ∈ ℂ)
54	11	recnd 11246	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝐿‘𝐴) ∈ ℂ)
55	53, 54, 16	divrec2d 11998	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → ((𝑀‘(𝑇‘𝐴)) / (𝐿‘𝐴)) = ((1 / (𝐿‘𝐴)) · (𝑀‘(𝑇‘𝐴))))
56	38, 49, 55	3eqtr4rd 2781	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → ((𝑀‘(𝑇‘𝐴)) / (𝐿‘𝐴)) = (𝑀‘(𝑇‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴))))
57	7, 45	nvscl 30146	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ (1 / (𝐿‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ 𝑋) → ((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴) ∈ 𝑋)
58	6, 57	mp3an1 1446	. . . . . . 7 ⊢ (((1 / (𝐿‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ 𝑋) → ((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴) ∈ 𝑋)
59	58	ancoms 457	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ (1 / (𝐿‘𝐴)) ∈ ℂ) → ((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴) ∈ 𝑋)
60	39, 59	syldan 589	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → ((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴) ∈ 𝑋)
61	7, 8	nvcl 30181	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ ((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴) ∈ 𝑋) → (𝐿‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴)) ∈ ℝ)
62	6, 60, 61	sylancr 585	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝐿‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴)) ∈ ℝ)
63	7, 45, 12, 8	nv1 30195	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝐿‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴)) = 1)
64	6, 63	mp3an1 1446	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝐿‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴)) = 1)
65		eqle 11320	. . . . . 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 582	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝐿‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴)) ≤ 1)
67	6, 26, 31	3pm3.2i 1337	. . . . . 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 30291	. . . . . 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 686	. . . . 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 582	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝑀‘(𝑇‘((1 / (𝐿‘𝐴))( ·_𝑠OLD ‘𝑈)𝐴))) ≤ (𝑁‘𝑇))
72	56, 71	eqbrtrd 5169	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → ((𝑀‘(𝑇‘𝐴)) / (𝐿‘𝐴)) ≤ (𝑁‘𝑇))
73	7, 28, 68, 29	nmblore 30306	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → (𝑁‘𝑇) ∈ ℝ)
74	6, 26, 27, 73	mp3an 1459	. . . . 5 ⊢ (𝑁‘𝑇) ∈ ℝ
75	74	a1i 11	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ≠ (0_vec‘𝑈)) → (𝑁‘𝑇) ∈ ℝ)
76		ledivmul2 12097	. . . 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 1372	. . 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 12317	. . . 4 ⊢ 0 ≤ 0
80		eqid 2730	. . . . . . . 8 ⊢ (0_vec‘𝑊) = (0_vec‘𝑊)
81	7, 28, 12, 80, 41	lno0 30276	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ (𝑈 LnOp 𝑊)) → (𝑇‘(0_vec‘𝑈)) = (0_vec‘𝑊))
82	6, 26, 43, 81	mp3an 1459	. . . . . 6 ⊢ (𝑇‘(0_vec‘𝑈)) = (0_vec‘𝑊)
83	82	fveq2i 6893	. . . . 5 ⊢ (𝑀‘(𝑇‘(0_vec‘𝑈))) = (𝑀‘(0_vec‘𝑊))
84	80, 35	nvz0 30188	. . . . . 6 ⊢ (𝑊 ∈ NrmCVec → (𝑀‘(0_vec‘𝑊)) = 0)
85	26, 84	ax-mp 5	. . . . 5 ⊢ (𝑀‘(0_vec‘𝑊)) = 0
86	83, 85	eqtri 2758	. . . 4 ⊢ (𝑀‘(𝑇‘(0_vec‘𝑈))) = 0
87	12, 8	nvz0 30188	. . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → (𝐿‘(0_vec‘𝑈)) = 0)
88	6, 87	ax-mp 5	. . . . . 6 ⊢ (𝐿‘(0_vec‘𝑈)) = 0
89	88	oveq2i 7422	. . . . 5 ⊢ ((𝑁‘𝑇) · (𝐿‘(0_vec‘𝑈))) = ((𝑁‘𝑇) · 0)
90	74	recni 11232	. . . . . 6 ⊢ (𝑁‘𝑇) ∈ ℂ
91	90	mul01i 11408	. . . . 5 ⊢ ((𝑁‘𝑇) · 0) = 0
92	89, 91	eqtri 2758	. . . 4 ⊢ ((𝑁‘𝑇) · (𝐿‘(0_vec‘𝑈))) = 0
93	79, 86, 92	3brtr4i 5177	. . 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 3025	1 ⊢ (𝐴 ∈ 𝑋 → (𝑀‘(𝑇‘𝐴)) ≤ ((𝑁‘𝑇) · (𝐿‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 class class class wbr 5147 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 · cmul 11117 < clt 11252 ≤ cle 11253 / cdiv 11875 NrmCVeccnv 30104 BaseSetcba 30106 ·_𝑠OLD cns 30107 0_veccn0v 30108 norm_CVcnmcv 30110 LnOp clno 30260 normOp_OLD cnmoo 30261 BLnOp cblo 30262

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-seq 13971 df-exp 14032 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-grpo 30013 df-gid 30014 df-ginv 30015 df-ablo 30065 df-vc 30079 df-nv 30112 df-va 30115 df-ba 30116 df-sm 30117 df-0v 30118 df-nmcv 30120 df-lno 30264 df-nmoo 30265 df-blo 30266

This theorem is referenced by: nmblolbi 30320

W3C validator