Theorem nmoubi 30861

Description: An upper bound for an operator norm. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
nmoubi	⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ*) → ((𝑁‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 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 nmoubi

Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmoubi.u	. . . . . 6 ⊢ 𝑈 ∈ NrmCVec
2		nmoubi.w	. . . . . 6 ⊢ 𝑊 ∈ NrmCVec
3		nmoubi.1	. . . . . . 7 ⊢ 𝑋 = (BaseSet‘𝑈)
4		nmoubi.y	. . . . . . 7 ⊢ 𝑌 = (BaseSet‘𝑊)
5		nmoubi.l	. . . . . . 7 ⊢ 𝐿 = (normCV‘𝑈)
6		nmoubi.m	. . . . . . 7 ⊢ 𝑀 = (normCV‘𝑊)
7		nmoubi.3	. . . . . . 7 ⊢ 𝑁 = (𝑈 normOpOLD 𝑊)
8	3, 4, 5, 6, 7	nmooval 30852	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))}, ℝ*, < ))
9	1, 2, 8	mp3an12 1459	. . . . 5 ⊢ (𝑇:𝑋⟶𝑌 → (𝑁‘𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))}, ℝ*, < ))
10	9	breq1d 5082	. . . 4 ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) ≤ 𝐴 ↔ sup({𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))}, ℝ*, < ) ≤ 𝐴))
11	10	adantr 481	. . 3 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ*) → ((𝑁‘𝑇) ≤ 𝐴 ↔ sup({𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))}, ℝ*, < ) ≤ 𝐴))
12	4, 6	nmosetre 30853	. . . . . 6 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))} ⊆ ℝ)
13	2, 12	mpan 696	. . . . 5 ⊢ (𝑇:𝑋⟶𝑌 → {𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))} ⊆ ℝ)
14		ressxr 11180	. . . . 5 ⊢ ℝ ⊆ ℝ*
15	13, 14	sstrdi 3927	. . . 4 ⊢ (𝑇:𝑋⟶𝑌 → {𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))} ⊆ ℝ*)
16		supxrleub 13269	. . . 4 ⊢ (({𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))} ⊆ ℝ* ∧ 𝐴 ∈ ℝ*) → (sup({𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))}𝑧 ≤ 𝐴))
17	15, 16	sylan 586	. . 3 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ*) → (sup({𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))}𝑧 ≤ 𝐴))
18	11, 17	bitrd 280	. 2 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ*) → ((𝑁‘𝑇) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))}𝑧 ≤ 𝐴))
19		eqeq1 2743	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑦 = (𝑀‘(𝑇‘𝑥)) ↔ 𝑧 = (𝑀‘(𝑇‘𝑥))))
20	19	anbi2d 636	. . . . 5 ⊢ (𝑦 = 𝑧 → (((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥))) ↔ ((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥)))))
21	20	rexbidv 3163	. . . 4 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥))) ↔ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥)))))
22	21	ralab 3634	. . 3 ⊢ (∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))}𝑧 ≤ 𝐴 ↔ ∀𝑧(∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴))
23		ralcom4 3265	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑧(((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴) ↔ ∀𝑧∀𝑥 ∈ 𝑋 (((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴))
24		ancomst 465	. . . . . . . 8 ⊢ ((((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴) ↔ ((𝑧 = (𝑀‘(𝑇‘𝑥)) ∧ (𝐿‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴))
25		impexp 451	. . . . . . . 8 ⊢ (((𝑧 = (𝑀‘(𝑇‘𝑥)) ∧ (𝐿‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ (𝑧 = (𝑀‘(𝑇‘𝑥)) → ((𝐿‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)))
26	24, 25	bitri 276	. . . . . . 7 ⊢ ((((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴) ↔ (𝑧 = (𝑀‘(𝑇‘𝑥)) → ((𝐿‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)))
27	26	albii 1826	. . . . . 6 ⊢ (∀𝑧(((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴) ↔ ∀𝑧(𝑧 = (𝑀‘(𝑇‘𝑥)) → ((𝐿‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)))
28		fvex 6840	. . . . . . 7 ⊢ (𝑀‘(𝑇‘𝑥)) ∈ V
29		breq1 5075	. . . . . . . 8 ⊢ (𝑧 = (𝑀‘(𝑇‘𝑥)) → (𝑧 ≤ 𝐴 ↔ (𝑀‘(𝑇‘𝑥)) ≤ 𝐴))
30	29	imbi2d 341	. . . . . . 7 ⊢ (𝑧 = (𝑀‘(𝑇‘𝑥)) → (((𝐿‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴) ↔ ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 𝐴)))
31	28, 30	ceqsalv 3470	. . . . . 6 ⊢ (∀𝑧(𝑧 = (𝑀‘(𝑇‘𝑥)) → ((𝐿‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)) ↔ ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 𝐴))
32	27, 31	bitri 276	. . . . 5 ⊢ (∀𝑧(((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴) ↔ ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 𝐴))
33	32	ralbii 3085	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑧(((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴) ↔ ∀𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 𝐴))
34		r19.23v 3166	. . . . 5 ⊢ (∀𝑥 ∈ 𝑋 (((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴) ↔ (∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴))
35	34	albii 1826	. . . 4 ⊢ (∀𝑧∀𝑥 ∈ 𝑋 (((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴) ↔ ∀𝑧(∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴))
36	23, 33, 35	3bitr3i 302	. . 3 ⊢ (∀𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 𝐴) ↔ ∀𝑧(∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴))
37	22, 36	bitr4i 279	. 2 ⊢ (∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))}𝑧 ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 𝐴))
38	18, 37	bitrdi 288	1 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ*) → ((𝑁‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   = wceq 1547   ∈ wcel 2119  {cab 2717  ∀wral 3053  ∃wrex 3063   ⊆ wss 3883   class class class wbr 5072  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  supcsup 9343  ℝcr 11028  1c1 11030  ℝ^*cxr 11169   < clt 11170   ≤ cle 11171  NrmCVeccnv 30673  BaseSetcba 30675  norm_CVcnmcv 30679   normOp_OLD cnmoo 30830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-po 5526  df-so 5527  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-er 8633  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9345  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-vc 30648  df-nv 30681  df-va 30684  df-ba 30685  df-sm 30686  df-0v 30687  df-nmcv 30689  df-nmoo 30834

This theorem is referenced by:  nmoub3i  30862  nmobndi  30864  ubthlem2  30960