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 31238

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 31126	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → (𝐿‘𝑥) ∈ ℝ)
5	1, 4	mpan 703	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 → (𝐿‘𝑥) ∈ ℝ)
6		remulcl 11210	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ (𝐿‘𝑥) ∈ ℝ) → (𝐴 · (𝐿‘𝑥)) ∈ ℝ)
7	5, 6	sylan2 605	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → (𝐴 · (𝐿‘𝑥)) ∈ ℝ)
8	7	adantr 486	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → (𝐴 · (𝐿‘𝑥)) ∈ ℝ)
9		recn 11215	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
10	9	abscld 15526	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → (abs‘𝐴) ∈ ℝ)
11		remulcl 11210	. . . . . . . . . . . 12 ⊢ (((abs‘𝐴) ∈ ℝ ∧ (𝐿‘𝑥) ∈ ℝ) → ((abs‘𝐴) · (𝐿‘𝑥)) ∈ ℝ)
12	10, 5, 11	syl2an 608	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → ((abs‘𝐴) · (𝐿‘𝑥)) ∈ ℝ)
13	12	adantr 486	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → ((abs‘𝐴) · (𝐿‘𝑥)) ∈ ℝ)
14	10	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → (abs‘𝐴) ∈ ℝ)
15		simpl 488	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℝ)
16	10	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → (abs‘𝐴) ∈ ℝ)
17	2, 3	nvge0 31138	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → 0 ≤ (𝐿‘𝑥))
18	1, 17	mpan 703	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 → 0 ≤ (𝐿‘𝑥))
19	5, 18	jca 521	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 → ((𝐿‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐿‘𝑥)))
20	19	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → ((𝐿‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐿‘𝑥)))
21		leabs 15386	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ (abs‘𝐴))
22	21	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → 𝐴 ≤ (abs‘𝐴))
23		lemul1a 12094	. . . . . . . . . . . 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 1400	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → (𝐴 · (𝐿‘𝑥)) ≤ ((abs‘𝐴) · (𝐿‘𝑥)))
25	24	adantr 486	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → (𝐴 · (𝐿‘𝑥)) ≤ ((abs‘𝐴) · (𝐿‘𝑥)))
26	5	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → (𝐿‘𝑥) ∈ ℝ)
27		1red 11234	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → 1 ∈ ℝ)
28	9	absge0d 15534	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ → 0 ≤ (abs‘𝐴))
29	28	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘𝐴))
30	16, 29	jca 521	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)))
31	26, 27, 30	3jca 1146	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) → ((𝐿‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))))
32		lemul2a 12095	. . . . . . . . . . . 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 592	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → ((abs‘𝐴) · (𝐿‘𝑥)) ≤ ((abs‘𝐴) · 1))
34	10	recnd 11262	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ → (abs‘𝐴) ∈ ℂ)
35	34	mulridd 11251	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → ((abs‘𝐴) · 1) = (abs‘𝐴))
36	35	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → ((abs‘𝐴) · 1) = (abs‘𝐴))
37	33, 36	breqtrd 5135	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → ((abs‘𝐴) · (𝐿‘𝑥)) ≤ (abs‘𝐴))
38	8, 13, 14, 25, 37	letrd 11392	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → (𝐴 · (𝐿‘𝑥)) ≤ (abs‘𝐴))
39	38	adantlll 731	. . . . . . . 8 ⊢ ((((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) ∧ (𝐿‘𝑥) ≤ 1) → (𝐴 · (𝐿‘𝑥)) ≤ (abs‘𝐴))
40		nmoubi.w	. . . . . . . . . . . 12 ⊢ 𝑊 ∈ NrmCVec
41		ffvelcdm 7077	. . . . . . . . . . . 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 31126	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑇‘𝑥) ∈ 𝑌) → (𝑀‘(𝑇‘𝑥)) ∈ ℝ)
45	40, 41, 44	sylancr 599	. . . . . . . . . . 11 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑥 ∈ 𝑋) → (𝑀‘(𝑇‘𝑥)) ∈ ℝ)
46	45	adantlr 728	. . . . . . . . . 10 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) → (𝑀‘(𝑇‘𝑥)) ∈ ℝ)
47	7	adantll 727	. . . . . . . . . 10 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) → (𝐴 · (𝐿‘𝑥)) ∈ ℝ)
48	10	ad2antlr 740	. . . . . . . . . 10 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑋) → (abs‘𝐴) ∈ ℝ)
49		letr 11329	. . . . . . . . . 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 1398	. . . . . . . . 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 486	. . . . . . . 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 707	. . . . . . 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 418	. . . . . 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 3176	. . . 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 412	. . 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 11284	. . . . 5 ⊢ (𝐴 ∈ ℝ → (abs‘𝐴) ∈ ℝ^*)
58		nmoubi.3	. . . . . 6 ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊)
59	2, 42, 3, 43, 58, 1, 40	nmoubi 31237	. . . . 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 605	. . . 4 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) → ((𝑁‘𝑇) ≤ (abs‘𝐴) ↔ ∀𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ (abs‘𝐴))))
61	60	biimpar 483	. . 3 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ ∀𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ (abs‘𝐴))) → (𝑁‘𝑇) ≤ (abs‘𝐴))
62	56, 61	syldan 603	. 2 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ) ∧ ∀𝑥 ∈ 𝑋 (𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) → (𝑁‘𝑇) ≤ (abs‘𝐴))
63	62	3impa 1127	1 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ ∧ ∀𝑥 ∈ 𝑋 (𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) → (𝑁‘𝑇) ≤ (abs‘𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3078 class class class wbr 5107 ⟶wf 6533 ‘cfv 6537 (class class class)co 7416 ℝcr 11124 0cc0 11125 1c1 11126 · cmul 11130 ℝ^*cxr 11267 ≤ cle 11269 abscabs 15321 NrmCVeccnv 31049 BaseSetcba 31051 norm_CVcnmcv 31055 normOp_OLD cnmoo 31206

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-er 8699 df-map 8831 df-en 8956 df-dom 8957 df-sdom 8958 df-sup 9415 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-n0 12530 df-z 12617 df-uz 12889 df-rp 13043 df-seq 14066 df-exp 14126 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-grpo 30958 df-gid 30959 df-ginv 30960 df-ablo 31010 df-vc 31024 df-nv 31057 df-va 31060 df-ba 31061 df-sm 31062 df-0v 31063 df-nmcv 31065 df-nmoo 31210

This theorem is used by: nmoub2i 31239 isblo3i 31266

W3C validator