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Theorem nmoub3i 30793

Description: An upper bound for an operator norm. (Contributed by NM, 12-Dec-2007.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
nmoubi.1	⊢ 𝑋 = (BaseSet‘𝑈)
nmoubi.y	⊢ 𝑌 = (BaseSet‘𝑊)
nmoubi.l	⊢ 𝐿 = (norm_CV‘𝑈)
nmoubi.m	⊢ 𝑀 = (norm_CV‘𝑊)
nmoubi.3	⊢ 𝑁 = (𝑈 normOp_OLD 𝑊)
nmoubi.u	⊢ 𝑈 ∈ NrmCVec
nmoubi.w	⊢ 𝑊 ∈ NrmCVec

**Assertion**
Ref	Expression
nmoub3i	⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ ∧ ∀𝑥 ∈ 𝑋 (𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) → (𝑁‘𝑇) ≤ (abs‘𝐴))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐿 𝑥,𝑈 𝑥,𝑊 𝑥,𝑌 𝑥,𝑀 𝑥,𝑇 𝑥,𝑋 Allowed substitution hint: 𝑁(𝑥)

**Proof of Theorem nmoub3i**
Step	Hyp	Ref	Expression
1		nmoubi.u	. . . . . . . . . . . . 13 ⊢ 𝑈 ∈ NrmCVec
2		nmoubi.1	. . . . . . . . . . . . . 14 ⊢ 𝑋 = (BaseSet‘𝑈)
3		nmoubi.l	. . . . . . . . . . . . . 14 ⊢ 𝐿 = (norm_CV‘𝑈)
4	2, 3	nvcl 30681	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → (𝐿‘𝑥) ∈ ℝ)
5	1, 4	mpan 690	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 → (𝐿‘𝑥) ∈ ℝ)
6		remulcl 11241	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ (𝐿‘𝑥) ∈ ℝ) → (𝐴 · (𝐿‘𝑥)) ∈ ℝ)
7	5, 6	sylan2 593	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → (𝐴 · (𝐿‘𝑥)) ∈ ℝ)
8	7	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → (𝐴 · (𝐿‘𝑥)) ∈ ℝ)
9		recn 11246	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
10	9	abscld 15476	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → (abs‘𝐴) ∈ ℝ)
11		remulcl 11241	. . . . . . . . . . . 12 ⊢ (((abs‘𝐴) ∈ ℝ ∧ (𝐿‘𝑥) ∈ ℝ) → ((abs‘𝐴) · (𝐿‘𝑥)) ∈ ℝ)
12	10, 5, 11	syl2an 596	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → ((abs‘𝐴) · (𝐿‘𝑥)) ∈ ℝ)
13	12	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → ((abs‘𝐴) · (𝐿‘𝑥)) ∈ ℝ)
14	10	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → (abs‘𝐴) ∈ ℝ)
15		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℝ)
16	10	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → (abs‘𝐴) ∈ ℝ)
17	2, 3	nvge0 30693	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → 0 ≤ (𝐿‘𝑥))
18	1, 17	mpan 690	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 → 0 ≤ (𝐿‘𝑥))
19	5, 18	jca 511	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 → ((𝐿‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐿‘𝑥)))
20	19	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → ((𝐿‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐿‘𝑥)))
21		leabs 15339	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ (abs‘𝐴))
22	21	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → 𝐴 ≤ (abs‘𝐴))
23		lemul1a 12122	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ (abs‘𝐴) ∈ ℝ ∧ ((𝐿‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐿‘𝑥))) ∧ 𝐴 ≤ (abs‘𝐴)) → (𝐴 · (𝐿‘𝑥)) ≤ ((abs‘𝐴) · (𝐿‘𝑥)))
24	15, 16, 20, 22, 23	syl31anc 1374	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → (𝐴 · (𝐿‘𝑥)) ≤ ((abs‘𝐴) · (𝐿‘𝑥)))
25	24	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → (𝐴 · (𝐿‘𝑥)) ≤ ((abs‘𝐴) · (𝐿‘𝑥)))
26	5	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → (𝐿‘𝑥) ∈ ℝ)
27		1red 11263	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → 1 ∈ ℝ)
28	9	absge0d 15484	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ → 0 ≤ (abs‘𝐴))
29	28	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘𝐴))
30	16, 29	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)))
31	26, 27, 30	3jca 1128	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → ((𝐿‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))))
32		lemul2a 12123	. . . . . . . . . . . 12 ⊢ ((((𝐿‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))) ∧ (𝐿‘𝑥) ≤ 1) → ((abs‘𝐴) · (𝐿‘𝑥)) ≤ ((abs‘𝐴) · 1))
33	31, 32	sylan 580	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → ((abs‘𝐴) · (𝐿‘𝑥)) ≤ ((abs‘𝐴) · 1))
34	10	recnd 11290	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ → (abs‘𝐴) ∈ ℂ)
35	34	mulridd 11279	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → ((abs‘𝐴) · 1) = (abs‘𝐴))
36	35	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → ((abs‘𝐴) · 1) = (abs‘𝐴))
37	33, 36	breqtrd 5168	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → ((abs‘𝐴) · (𝐿‘𝑥)) ≤ (abs‘𝐴))
38	8, 13, 14, 25, 37	letrd 11419	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → (𝐴 · (𝐿‘𝑥)) ≤ (abs‘𝐴))
39	38	adantlll 718	. . . . . . . 8 ⊢ ((((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → (𝐴 · (𝐿‘𝑥)) ≤ (abs‘𝐴))
40		nmoubi.w	. . . . . . . . . . . 12 ⊢ 𝑊 ∈ NrmCVec
41		ffvelcdm 7100	. . . . . . . . . . . 12 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑥 ∈ 𝑋) → (𝑇‘𝑥) ∈ 𝑌)
42		nmoubi.y	. . . . . . . . . . . . 13 ⊢ 𝑌 = (BaseSet‘𝑊)
43		nmoubi.m	. . . . . . . . . . . . 13 ⊢ 𝑀 = (norm_CV‘𝑊)
44	42, 43	nvcl 30681	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑇‘𝑥) ∈ 𝑌) → (𝑀‘(𝑇‘𝑥)) ∈ ℝ)
45	40, 41, 44	sylancr 587	. . . . . . . . . . 11 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑥 ∈ 𝑋) → (𝑀‘(𝑇‘𝑥)) ∈ ℝ)
46	45	adantlr 715	. . . . . . . . . 10 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) → (𝑀‘(𝑇‘𝑥)) ∈ ℝ)
47	7	adantll 714	. . . . . . . . . 10 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) → (𝐴 · (𝐿‘𝑥)) ∈ ℝ)
48	10	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) → (abs‘𝐴) ∈ ℝ)
49		letr 11356	. . . . . . . . . 10 ⊢ (((𝑀‘(𝑇‘𝑥)) ∈ ℝ ∧ (𝐴 · (𝐿‘𝑥)) ∈ ℝ ∧ (abs‘𝐴) ∈ ℝ) → (((𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)) ∧ (𝐴 · (𝐿‘𝑥)) ≤ (abs‘𝐴)) → (𝑀‘(𝑇‘𝑥)) ≤ (abs‘𝐴)))
50	46, 47, 48, 49	syl3anc 1372	. . . . . . . . 9 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) → (((𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)) ∧ (𝐴 · (𝐿‘𝑥)) ≤ (abs‘𝐴)) → (𝑀‘(𝑇‘𝑥)) ≤ (abs‘𝐴)))
51	50	adantr 480	. . . . . . . 8 ⊢ ((((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → (((𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)) ∧ (𝐴 · (𝐿‘𝑥)) ≤ (abs‘𝐴)) → (𝑀‘(𝑇‘𝑥)) ≤ (abs‘𝐴)))
52	39, 51	mpan2d 694	. . . . . . 7 ⊢ ((((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → ((𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)) → (𝑀‘(𝑇‘𝑥)) ≤ (abs‘𝐴)))
53	52	ex 412	. . . . . 6 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) → ((𝐿‘𝑥) ≤ 1 → ((𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)) → (𝑀‘(𝑇‘𝑥)) ≤ (abs‘𝐴))))
54	53	com23 86	. . . . 5 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) → ((𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)) → ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ (abs‘𝐴))))
55	54	ralimdva 3166	. . . 4 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) → (∀𝑥 ∈ 𝑋 (𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)) → ∀𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ (abs‘𝐴))))
56	55	imp 406	. . 3 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ ∀𝑥 ∈ 𝑋 (𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) → ∀𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ (abs‘𝐴)))
57	10	rexrd 11312	. . . . 5 ⊢ (𝐴 ∈ ℝ → (abs‘𝐴) ∈ ℝ^*)
58		nmoubi.3	. . . . . 6 ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊)
59	2, 42, 3, 43, 58, 1, 40	nmoubi 30792	. . . . 5 ⊢ ((𝑇:𝑋⟶𝑌 ∧ (abs‘𝐴) ∈ ℝ^*) → ((𝑁‘𝑇) ≤ (abs‘𝐴) ↔ ∀𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ (abs‘𝐴))))
60	57, 59	sylan2 593	. . . 4 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) → ((𝑁‘𝑇) ≤ (abs‘𝐴) ↔ ∀𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ (abs‘𝐴))))
61	60	biimpar 477	. . 3 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ ∀𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ (abs‘𝐴))) → (𝑁‘𝑇) ≤ (abs‘𝐴))
62	56, 61	syldan 591	. 2 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ ∀𝑥 ∈ 𝑋 (𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) → (𝑁‘𝑇) ≤ (abs‘𝐴))
63	62	3impa 1109	1 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ ∧ ∀𝑥 ∈ 𝑋 (𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) → (𝑁‘𝑇) ≤ (abs‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3060 class class class wbr 5142 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ℝcr 11155 0cc0 11156 1c1 11157 · cmul 11161 ℝ^*cxr 11295 ≤ cle 11297 abscabs 15274 NrmCVeccnv 30604 BaseSetcba 30606 norm_CVcnmcv 30610 normOp_OLD cnmoo 30761

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-sup 9483 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-n0 12529 df-z 12616 df-uz 12880 df-rp 13036 df-seq 14044 df-exp 14104 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-grpo 30513 df-gid 30514 df-ginv 30515 df-ablo 30565 df-vc 30579 df-nv 30612 df-va 30615 df-ba 30616 df-sm 30617 df-0v 30618 df-nmcv 30620 df-nmoo 30765

This theorem is referenced by: nmoub2i 30794 isblo3i 30821

W3C validator