nmoi2 - Metamath Proof Explorer

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Theorem nmoi2 24247

Description: The operator norm is a bound on the growth of a vector under the action of the operator. (Contributed by Mario Carneiro, 19-Oct-2015.)

**Hypotheses**
Ref	Expression
nmofval.1	⊢ 𝑁 = (𝑆 normOp 𝑇)
nmoi.2	⊢ 𝑉 = (Base‘𝑆)
nmoi.3	⊢ 𝐿 = (norm‘𝑆)
nmoi.4	⊢ 𝑀 = (norm‘𝑇)
nmoi2.z	⊢ 0 = (0_g‘𝑆)

**Assertion**
Ref	Expression
nmoi2	⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → ((𝑀‘(𝐹‘𝑋)) / (𝐿‘𝑋)) ≤ (𝑁‘𝐹))

**Proof of Theorem nmoi2**
Step	Hyp	Ref	Expression
1		simpl2 1193	. . . . 5 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → 𝑇 ∈ NrmGrp)
2		simpl3 1194	. . . . . . 7 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
3		nmoi.2	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑆)
4		eqid 2733	. . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇)
5	3, 4	ghmf 19096	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑉⟶(Base‘𝑇))
6	2, 5	syl 17	. . . . . 6 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → 𝐹:𝑉⟶(Base‘𝑇))
7		simprl 770	. . . . . 6 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → 𝑋 ∈ 𝑉)
8	6, 7	ffvelcdmd 7088	. . . . 5 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → (𝐹‘𝑋) ∈ (Base‘𝑇))
9		nmoi.4	. . . . . 6 ⊢ 𝑀 = (norm‘𝑇)
10	4, 9	nmcl 24125	. . . . 5 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐹‘𝑋) ∈ (Base‘𝑇)) → (𝑀‘(𝐹‘𝑋)) ∈ ℝ)
11	1, 8, 10	syl2anc 585	. . . 4 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → (𝑀‘(𝐹‘𝑋)) ∈ ℝ)
12	11	rexrd 11264	. . 3 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → (𝑀‘(𝐹‘𝑋)) ∈ ℝ^*)
13		nmofval.1	. . . . . 6 ⊢ 𝑁 = (𝑆 normOp 𝑇)
14	13	nmocl 24237	. . . . 5 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (𝑁‘𝐹) ∈ ℝ^*)
15	14	adantr 482	. . . 4 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → (𝑁‘𝐹) ∈ ℝ^*)
16		nmoi.3	. . . . . . . 8 ⊢ 𝐿 = (norm‘𝑆)
17		nmoi2.z	. . . . . . . 8 ⊢ 0 = (0_g‘𝑆)
18	3, 16, 17	nmrpcl 24129	. . . . . . 7 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → (𝐿‘𝑋) ∈ ℝ⁺)
19	18	3expb 1121	. . . . . 6 ⊢ ((𝑆 ∈ NrmGrp ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → (𝐿‘𝑋) ∈ ℝ⁺)
20	19	3ad2antl1 1186	. . . . 5 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → (𝐿‘𝑋) ∈ ℝ⁺)
21	20	rpxrd 13017	. . . 4 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → (𝐿‘𝑋) ∈ ℝ^*)
22	15, 21	xmulcld 13281	. . 3 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → ((𝑁‘𝐹) ·_e (𝐿‘𝑋)) ∈ ℝ^*)
23	20	rpreccld 13026	. . . . 5 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → (1 / (𝐿‘𝑋)) ∈ ℝ⁺)
24	23	rpxrd 13017	. . . 4 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → (1 / (𝐿‘𝑋)) ∈ ℝ^*)
25	23	rpge0d 13020	. . . 4 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → 0 ≤ (1 / (𝐿‘𝑋)))
26	24, 25	jca 513	. . 3 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → ((1 / (𝐿‘𝑋)) ∈ ℝ^* ∧ 0 ≤ (1 / (𝐿‘𝑋))))
27	13, 3, 16, 9	nmoix 24246	. . . 4 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (𝑀‘(𝐹‘𝑋)) ≤ ((𝑁‘𝐹) ·_e (𝐿‘𝑋)))
28	27	adantrr 716	. . 3 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → (𝑀‘(𝐹‘𝑋)) ≤ ((𝑁‘𝐹) ·_e (𝐿‘𝑋)))
29		xlemul1a 13267	. . 3 ⊢ ((((𝑀‘(𝐹‘𝑋)) ∈ ℝ^* ∧ ((𝑁‘𝐹) ·_e (𝐿‘𝑋)) ∈ ℝ^* ∧ ((1 / (𝐿‘𝑋)) ∈ ℝ^* ∧ 0 ≤ (1 / (𝐿‘𝑋)))) ∧ (𝑀‘(𝐹‘𝑋)) ≤ ((𝑁‘𝐹) ·_e (𝐿‘𝑋))) → ((𝑀‘(𝐹‘𝑋)) ·_e (1 / (𝐿‘𝑋))) ≤ (((𝑁‘𝐹) ·_e (𝐿‘𝑋)) ·_e (1 / (𝐿‘𝑋))))
30	12, 22, 26, 28, 29	syl31anc 1374	. 2 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → ((𝑀‘(𝐹‘𝑋)) ·_e (1 / (𝐿‘𝑋))) ≤ (((𝑁‘𝐹) ·_e (𝐿‘𝑋)) ·_e (1 / (𝐿‘𝑋))))
31	23	rpred 13016	. . . 4 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → (1 / (𝐿‘𝑋)) ∈ ℝ)
32		rexmul 13250	. . . 4 ⊢ (((𝑀‘(𝐹‘𝑋)) ∈ ℝ ∧ (1 / (𝐿‘𝑋)) ∈ ℝ) → ((𝑀‘(𝐹‘𝑋)) ·_e (1 / (𝐿‘𝑋))) = ((𝑀‘(𝐹‘𝑋)) · (1 / (𝐿‘𝑋))))
33	11, 31, 32	syl2anc 585	. . 3 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → ((𝑀‘(𝐹‘𝑋)) ·_e (1 / (𝐿‘𝑋))) = ((𝑀‘(𝐹‘𝑋)) · (1 / (𝐿‘𝑋))))
34	11	recnd 11242	. . . 4 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → (𝑀‘(𝐹‘𝑋)) ∈ ℂ)
35	20	rpcnd 13018	. . . 4 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → (𝐿‘𝑋) ∈ ℂ)
36	20	rpne0d 13021	. . . 4 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → (𝐿‘𝑋) ≠ 0)
37	34, 35, 36	divrecd 11993	. . 3 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → ((𝑀‘(𝐹‘𝑋)) / (𝐿‘𝑋)) = ((𝑀‘(𝐹‘𝑋)) · (1 / (𝐿‘𝑋))))
38	33, 37	eqtr4d 2776	. 2 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → ((𝑀‘(𝐹‘𝑋)) ·_e (1 / (𝐿‘𝑋))) = ((𝑀‘(𝐹‘𝑋)) / (𝐿‘𝑋)))
39		xmulass 13266	. . . 4 ⊢ (((𝑁‘𝐹) ∈ ℝ^* ∧ (𝐿‘𝑋) ∈ ℝ^* ∧ (1 / (𝐿‘𝑋)) ∈ ℝ^*) → (((𝑁‘𝐹) ·_e (𝐿‘𝑋)) ·_e (1 / (𝐿‘𝑋))) = ((𝑁‘𝐹) ·_e ((𝐿‘𝑋) ·_e (1 / (𝐿‘𝑋)))))
40	15, 21, 24, 39	syl3anc 1372	. . 3 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → (((𝑁‘𝐹) ·_e (𝐿‘𝑋)) ·_e (1 / (𝐿‘𝑋))) = ((𝑁‘𝐹) ·_e ((𝐿‘𝑋) ·_e (1 / (𝐿‘𝑋)))))
41	20	rpred 13016	. . . . . 6 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → (𝐿‘𝑋) ∈ ℝ)
42		rexmul 13250	. . . . . 6 ⊢ (((𝐿‘𝑋) ∈ ℝ ∧ (1 / (𝐿‘𝑋)) ∈ ℝ) → ((𝐿‘𝑋) ·_e (1 / (𝐿‘𝑋))) = ((𝐿‘𝑋) · (1 / (𝐿‘𝑋))))
43	41, 31, 42	syl2anc 585	. . . . 5 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → ((𝐿‘𝑋) ·_e (1 / (𝐿‘𝑋))) = ((𝐿‘𝑋) · (1 / (𝐿‘𝑋))))
44	35, 36	recidd 11985	. . . . 5 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → ((𝐿‘𝑋) · (1 / (𝐿‘𝑋))) = 1)
45	43, 44	eqtrd 2773	. . . 4 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → ((𝐿‘𝑋) ·_e (1 / (𝐿‘𝑋))) = 1)
46	45	oveq2d 7425	. . 3 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → ((𝑁‘𝐹) ·_e ((𝐿‘𝑋) ·_e (1 / (𝐿‘𝑋)))) = ((𝑁‘𝐹) ·_e 1))
47		xmulrid 13258	. . . 4 ⊢ ((𝑁‘𝐹) ∈ ℝ^* → ((𝑁‘𝐹) ·_e 1) = (𝑁‘𝐹))
48	15, 47	syl 17	. . 3 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → ((𝑁‘𝐹) ·_e 1) = (𝑁‘𝐹))
49	40, 46, 48	3eqtrd 2777	. 2 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → (((𝑁‘𝐹) ·_e (𝐿‘𝑋)) ·_e (1 / (𝐿‘𝑋))) = (𝑁‘𝐹))
50	30, 38, 49	3brtr3d 5180	1 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → ((𝑀‘(𝐹‘𝑋)) / (𝐿‘𝑋)) ≤ (𝑁‘𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 class class class wbr 5149 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ℝcr 11109 0cc0 11110 1c1 11111 · cmul 11115 ℝ^*cxr 11247 ≤ cle 11249 / cdiv 11871 ℝ⁺crp 12974 ·_e cxmu 13091 Basecbs 17144 0_gc0g 17385 GrpHom cghm 19089 normcnm 24085 NrmGrpcngp 24086 normOp cnmo 24222

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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

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 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-n0 12473 df-z 12559 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ico 13330 df-0g 17387 df-topgen 17389 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822 df-ghm 19090 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-xms 23826 df-ms 23827 df-nm 24091 df-ngp 24092 df-nmo 24225 df-nghm 24226

This theorem is referenced by: nmoleub 24248

W3C validator