nmoub3i - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > nmoub3i		Structured version Visualization version GIF version

Theorem nmoub3i 31068

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 30956	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → (𝐿‘𝑥) ∈ ℝ)
5	1, 4	mpan 702	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 → (𝐿‘𝑥) ∈ ℝ)
6		remulcl 11187	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ (𝐿‘𝑥) ∈ ℝ) → (𝐴 · (𝐿‘𝑥)) ∈ ℝ)
7	5, 6	sylan2 604	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → (𝐴 · (𝐿‘𝑥)) ∈ ℝ)
8	7	adantr 485	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → (𝐴 · (𝐿‘𝑥)) ∈ ℝ)
9		recn 11192	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
10	9	abscld 15492	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → (abs‘𝐴) ∈ ℝ)
11		remulcl 11187	. . . . . . . . . . . 12 ⊢ (((abs‘𝐴) ∈ ℝ ∧ (𝐿‘𝑥) ∈ ℝ) → ((abs‘𝐴) · (𝐿‘𝑥)) ∈ ℝ)
12	10, 5, 11	syl2an 607	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → ((abs‘𝐴) · (𝐿‘𝑥)) ∈ ℝ)
13	12	adantr 485	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → ((abs‘𝐴) · (𝐿‘𝑥)) ∈ ℝ)
14	10	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → (abs‘𝐴) ∈ ℝ)
15		simpl 487	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℝ)
16	10	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → (abs‘𝐴) ∈ ℝ)
17	2, 3	nvge0 30968	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → 0 ≤ (𝐿‘𝑥))
18	1, 17	mpan 702	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 → 0 ≤ (𝐿‘𝑥))
19	5, 18	jca 520	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 → ((𝐿‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐿‘𝑥)))
20	19	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → ((𝐿‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐿‘𝑥)))
21		leabs 15352	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ (abs‘𝐴))
22	21	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → 𝐴 ≤ (abs‘𝐴))
23		lemul1a 12071	. . . . . . . . . . . 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 1398	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → (𝐴 · (𝐿‘𝑥)) ≤ ((abs‘𝐴) · (𝐿‘𝑥)))
25	24	adantr 485	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → (𝐴 · (𝐿‘𝑥)) ≤ ((abs‘𝐴) · (𝐿‘𝑥)))
26	5	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → (𝐿‘𝑥) ∈ ℝ)
27		1red 11211	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → 1 ∈ ℝ)
28	9	absge0d 15500	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ → 0 ≤ (abs‘𝐴))
29	28	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘𝐴))
30	16, 29	jca 520	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)))
31	26, 27, 30	3jca 1144	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → ((𝐿‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))))
32		lemul2a 12072	. . . . . . . . . . . 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 591	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → ((abs‘𝐴) · (𝐿‘𝑥)) ≤ ((abs‘𝐴) · 1))
34	10	recnd 11239	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ → (abs‘𝐴) ∈ ℂ)
35	34	mulridd 11228	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → ((abs‘𝐴) · 1) = (abs‘𝐴))
36	35	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → ((abs‘𝐴) · 1) = (abs‘𝐴))
37	33, 36	breqtrd 5141	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → ((abs‘𝐴) · (𝐿‘𝑥)) ≤ (abs‘𝐴))
38	8, 13, 14, 25, 37	letrd 11369	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → (𝐴 · (𝐿‘𝑥)) ≤ (abs‘𝐴))
39	38	adantlll 730	. . . . . . . 8 ⊢ ((((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → (𝐴 · (𝐿‘𝑥)) ≤ (abs‘𝐴))
40		nmoubi.w	. . . . . . . . . . . 12 ⊢ 𝑊 ∈ NrmCVec
41		ffvelcdm 7079	. . . . . . . . . . . 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 30956	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑇‘𝑥) ∈ 𝑌) → (𝑀‘(𝑇‘𝑥)) ∈ ℝ)
45	40, 41, 44	sylancr 598	. . . . . . . . . . 11 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑥 ∈ 𝑋) → (𝑀‘(𝑇‘𝑥)) ∈ ℝ)
46	45	adantlr 727	. . . . . . . . . 10 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) → (𝑀‘(𝑇‘𝑥)) ∈ ℝ)
47	7	adantll 726	. . . . . . . . . 10 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) → (𝐴 · (𝐿‘𝑥)) ∈ ℝ)
48	10	ad2antlr 739	. . . . . . . . . 10 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) → (abs‘𝐴) ∈ ℝ)
49		letr 11306	. . . . . . . . . 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 1396	. . . . . . . . 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 485	. . . . . . . 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 706	. . . . . . 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 417	. . . . . 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 87	. . . . 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 3183	. . . 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 411	. . 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 11261	. . . . 5 ⊢ (𝐴 ∈ ℝ → (abs‘𝐴) ∈ ℝ^*)
58		nmoubi.3	. . . . . 6 ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊)
59	2, 42, 3, 43, 58, 1, 40	nmoubi 31067	. . . . 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 604	. . . 4 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) → ((𝑁‘𝑇) ≤ (abs‘𝐴) ↔ ∀𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ (abs‘𝐴))))
61	60	biimpar 482	. . 3 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ ∀𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ (abs‘𝐴))) → (𝑁‘𝑇) ≤ (abs‘𝐴))
62	56, 61	syldan 602	. 2 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ ∀𝑥 ∈ 𝑋 (𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) → (𝑁‘𝑇) ≤ (abs‘𝐴))
63	62	3impa 1125	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 209 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ∀wral 3085 class class class wbr 5113 ⟶wf 6535 ‘cfv 6539 (class class class)co 7413 ℝcr 11101 0cc0 11102 1c1 11103 · cmul 11107 ℝ^*cxr 11244 ≤ cle 11246 abscabs 15287 NrmCVeccnv 30879 BaseSetcba 30881 norm_CVcnmcv 30885 normOp_OLD cnmoo 31036

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5273 ax-pow 5339 ax-pr 5407 ax-un 7735 ax-cnex 11158 ax-resscn 11159 ax-1cn 11160 ax-icn 11161 ax-addcl 11162 ax-addrcl 11163 ax-mulcl 11164 ax-mulrcl 11165 ax-mulcom 11166 ax-addass 11167 ax-mulass 11168 ax-distr 11169 ax-i2m1 11170 ax-1ne0 11171 ax-1rid 11172 ax-rnegex 11173 ax-rrecex 11174 ax-cnre 11175 ax-pre-lttri 11176 ax-pre-lttrn 11177 ax-pre-ltadd 11178 ax-pre-mulgt0 11179 ax-pre-sup 11180

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5559 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-pred 6305 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6495 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7865 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-er 8696 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9404 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11445 df-neg 11446 df-div 11874 df-nn 12236 df-2 12305 df-3 12306 df-n0 12507 df-z 12594 df-uz 12865 df-rp 13019 df-seq 14040 df-exp 14100 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-grpo 30788 df-gid 30789 df-ginv 30790 df-ablo 30840 df-vc 30854 df-nv 30887 df-va 30890 df-ba 30891 df-sm 30892 df-0v 30893 df-nmcv 30895 df-nmoo 31040

This theorem is referenced by: nmoub2i 31069 isblo3i 31096

W3C validator