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Theorem nmoub3i 30026

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 29914	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → (𝐿‘𝑥) ∈ ℝ)
5	1, 4	mpan 689	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 → (𝐿‘𝑥) ∈ ℝ)
6		remulcl 11195	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ (𝐿‘𝑥) ∈ ℝ) → (𝐴 · (𝐿‘𝑥)) ∈ ℝ)
7	5, 6	sylan2 594	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → (𝐴 · (𝐿‘𝑥)) ∈ ℝ)
8	7	adantr 482	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → (𝐴 · (𝐿‘𝑥)) ∈ ℝ)
9		recn 11200	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
10	9	abscld 15383	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → (abs‘𝐴) ∈ ℝ)
11		remulcl 11195	. . . . . . . . . . . 12 ⊢ (((abs‘𝐴) ∈ ℝ ∧ (𝐿‘𝑥) ∈ ℝ) → ((abs‘𝐴) · (𝐿‘𝑥)) ∈ ℝ)
12	10, 5, 11	syl2an 597	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → ((abs‘𝐴) · (𝐿‘𝑥)) ∈ ℝ)
13	12	adantr 482	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → ((abs‘𝐴) · (𝐿‘𝑥)) ∈ ℝ)
14	10	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → (abs‘𝐴) ∈ ℝ)
15		simpl 484	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℝ)
16	10	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → (abs‘𝐴) ∈ ℝ)
17	2, 3	nvge0 29926	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → 0 ≤ (𝐿‘𝑥))
18	1, 17	mpan 689	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 → 0 ≤ (𝐿‘𝑥))
19	5, 18	jca 513	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 → ((𝐿‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐿‘𝑥)))
20	19	adantl 483	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → ((𝐿‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐿‘𝑥)))
21		leabs 15246	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ (abs‘𝐴))
22	21	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → 𝐴 ≤ (abs‘𝐴))
23		lemul1a 12068	. . . . . . . . . . . 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 482	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → (𝐴 · (𝐿‘𝑥)) ≤ ((abs‘𝐴) · (𝐿‘𝑥)))
26	5	adantl 483	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → (𝐿‘𝑥) ∈ ℝ)
27		1red 11215	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → 1 ∈ ℝ)
28	9	absge0d 15391	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ → 0 ≤ (abs‘𝐴))
29	28	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘𝐴))
30	16, 29	jca 513	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)))
31	26, 27, 30	3jca 1129	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → ((𝐿‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))))
32		lemul2a 12069	. . . . . . . . . . . 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 581	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → ((abs‘𝐴) · (𝐿‘𝑥)) ≤ ((abs‘𝐴) · 1))
34	10	recnd 11242	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ → (abs‘𝐴) ∈ ℂ)
35	34	mulridd 11231	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → ((abs‘𝐴) · 1) = (abs‘𝐴))
36	35	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → ((abs‘𝐴) · 1) = (abs‘𝐴))
37	33, 36	breqtrd 5175	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → ((abs‘𝐴) · (𝐿‘𝑥)) ≤ (abs‘𝐴))
38	8, 13, 14, 25, 37	letrd 11371	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → (𝐴 · (𝐿‘𝑥)) ≤ (abs‘𝐴))
39	38	adantlll 717	. . . . . . . 8 ⊢ ((((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → (𝐴 · (𝐿‘𝑥)) ≤ (abs‘𝐴))
40		nmoubi.w	. . . . . . . . . . . 12 ⊢ 𝑊 ∈ NrmCVec
41		ffvelcdm 7084	. . . . . . . . . . . 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 29914	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑇‘𝑥) ∈ 𝑌) → (𝑀‘(𝑇‘𝑥)) ∈ ℝ)
45	40, 41, 44	sylancr 588	. . . . . . . . . . 11 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑥 ∈ 𝑋) → (𝑀‘(𝑇‘𝑥)) ∈ ℝ)
46	45	adantlr 714	. . . . . . . . . 10 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) → (𝑀‘(𝑇‘𝑥)) ∈ ℝ)
47	7	adantll 713	. . . . . . . . . 10 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) → (𝐴 · (𝐿‘𝑥)) ∈ ℝ)
48	10	ad2antlr 726	. . . . . . . . . 10 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) → (abs‘𝐴) ∈ ℝ)
49		letr 11308	. . . . . . . . . 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 482	. . . . . . . 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 693	. . . . . . 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 414	. . . . . 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 3168	. . . 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 408	. . 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 11264	. . . . 5 ⊢ (𝐴 ∈ ℝ → (abs‘𝐴) ∈ ℝ^*)
58		nmoubi.3	. . . . . 6 ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊)
59	2, 42, 3, 43, 58, 1, 40	nmoubi 30025	. . . . 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 594	. . . 4 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) → ((𝑁‘𝑇) ≤ (abs‘𝐴) ↔ ∀𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ (abs‘𝐴))))
61	60	biimpar 479	. . 3 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ ∀𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ (abs‘𝐴))) → (𝑁‘𝑇) ≤ (abs‘𝐴))
62	56, 61	syldan 592	. 2 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ ∀𝑥 ∈ 𝑋 (𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) → (𝑁‘𝑇) ≤ (abs‘𝐴))
63	62	3impa 1111	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 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 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 abscabs 15181 NrmCVeccnv 29837 BaseSetcba 29839 norm_CVcnmcv 29843 normOp_OLD cnmoo 29994

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-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-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-grpo 29746 df-gid 29747 df-ginv 29748 df-ablo 29798 df-vc 29812 df-nv 29845 df-va 29848 df-ba 29849 df-sm 29850 df-0v 29851 df-nmcv 29853 df-nmoo 29998

This theorem is referenced by: nmoub2i 30027 isblo3i 30054

W3C validator