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Theorem nmoub3i 29114

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 29002	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → (𝐿‘𝑥) ∈ ℝ)
5	1, 4	mpan 686	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 → (𝐿‘𝑥) ∈ ℝ)
6		remulcl 10940	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ (𝐿‘𝑥) ∈ ℝ) → (𝐴 · (𝐿‘𝑥)) ∈ ℝ)
7	5, 6	sylan2 592	. . . . . . . . . . 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 10945	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
10	9	abscld 15129	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → (abs‘𝐴) ∈ ℝ)
11		remulcl 10940	. . . . . . . . . . . 12 ⊢ (((abs‘𝐴) ∈ ℝ ∧ (𝐿‘𝑥) ∈ ℝ) → ((abs‘𝐴) · (𝐿‘𝑥)) ∈ ℝ)
12	10, 5, 11	syl2an 595	. . . . . . . . . . 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 722	. . . . . . . . . 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 29014	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → 0 ≤ (𝐿‘𝑥))
18	1, 17	mpan 686	. . . . . . . . . . . . . 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 14992	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ (abs‘𝐴))
22	21	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → 𝐴 ≤ (abs‘𝐴))
23		lemul1a 11812	. . . . . . . . . . . 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 1371	. . . . . . . . . . 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 10960	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → 1 ∈ ℝ)
28	9	absge0d 15137	. . . . . . . . . . . . . . 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 1126	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → ((𝐿‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))))
32		lemul2a 11813	. . . . . . . . . . . 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 579	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → ((abs‘𝐴) · (𝐿‘𝑥)) ≤ ((abs‘𝐴) · 1))
34	10	recnd 10987	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ → (abs‘𝐴) ∈ ℂ)
35	34	mulid1d 10976	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → ((abs‘𝐴) · 1) = (abs‘𝐴))
36	35	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → ((abs‘𝐴) · 1) = (abs‘𝐴))
37	33, 36	breqtrd 5104	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → ((abs‘𝐴) · (𝐿‘𝑥)) ≤ (abs‘𝐴))
38	8, 13, 14, 25, 37	letrd 11115	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → (𝐴 · (𝐿‘𝑥)) ≤ (abs‘𝐴))
39	38	adantlll 714	. . . . . . . 8 ⊢ ((((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → (𝐴 · (𝐿‘𝑥)) ≤ (abs‘𝐴))
40		nmoubi.w	. . . . . . . . . . . 12 ⊢ 𝑊 ∈ NrmCVec
41		ffvelrn 6953	. . . . . . . . . . . 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 29002	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑇‘𝑥) ∈ 𝑌) → (𝑀‘(𝑇‘𝑥)) ∈ ℝ)
45	40, 41, 44	sylancr 586	. . . . . . . . . . 11 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑥 ∈ 𝑋) → (𝑀‘(𝑇‘𝑥)) ∈ ℝ)
46	45	adantlr 711	. . . . . . . . . 10 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) → (𝑀‘(𝑇‘𝑥)) ∈ ℝ)
47	7	adantll 710	. . . . . . . . . 10 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) → (𝐴 · (𝐿‘𝑥)) ∈ ℝ)
48	10	ad2antlr 723	. . . . . . . . . 10 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) → (abs‘𝐴) ∈ ℝ)
49		letr 11052	. . . . . . . . . 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 1369	. . . . . . . . 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 690	. . . . . . 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 3104	. . . 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 11009	. . . . 5 ⊢ (𝐴 ∈ ℝ → (abs‘𝐴) ∈ ℝ^*)
58		nmoubi.3	. . . . . 6 ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊)
59	2, 42, 3, 43, 58, 1, 40	nmoubi 29113	. . . . 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 592	. . . 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 590	. 2 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ ∀𝑥 ∈ 𝑋 (𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) → (𝑁‘𝑇) ≤ (abs‘𝐴))
63	62	3impa 1108	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 395 ∧ w3a 1085 = wceq 1541 ∈ wcel 2109 ∀wral 3065 class class class wbr 5078 ⟶wf 6426 ‘cfv 6430 (class class class)co 7268 ℝcr 10854 0cc0 10855 1c1 10856 · cmul 10860 ℝ^*cxr 10992 ≤ cle 10994 abscabs 14926 NrmCVeccnv 28925 BaseSetcba 28927 norm_CVcnmcv 28931 normOp_OLD cnmoo 29082

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-er 8472 df-map 8591 df-en 8708 df-dom 8709 df-sdom 8710 df-sup 9162 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-n0 12217 df-z 12303 df-uz 12565 df-rp 12713 df-seq 13703 df-exp 13764 df-cj 14791 df-re 14792 df-im 14793 df-sqrt 14927 df-abs 14928 df-grpo 28834 df-gid 28835 df-ginv 28836 df-ablo 28886 df-vc 28900 df-nv 28933 df-va 28936 df-ba 28937 df-sm 28938 df-0v 28939 df-nmcv 28941 df-nmoo 29086

This theorem is referenced by: nmoub2i 29115 isblo3i 29142

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