Theorem nmoubi 30751

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 30742	. . . . . 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 1453	. . . . 5 ⊢ (𝑇:𝑋⟶𝑌 → (𝑁‘𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))}, ℝ*, < ))
10	9	breq1d 5112	. . . 4 ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) ≤ 𝐴 ↔ sup({𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))}, ℝ*, < ) ≤ 𝐴))
11	10	adantr 480	. . 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 30743	. . . . . 6 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))} ⊆ ℝ)
13	2, 12	mpan 690	. . . . 5 ⊢ (𝑇:𝑋⟶𝑌 → {𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))} ⊆ ℝ)
14		ressxr 11194	. . . . 5 ⊢ ℝ ⊆ ℝ*
15	13, 14	sstrdi 3956	. . . 4 ⊢ (𝑇:𝑋⟶𝑌 → {𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))} ⊆ ℝ*)
16		supxrleub 13262	. . . 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 580	. . 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 279	. 2 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ*) → ((𝑁‘𝑇) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))}𝑧 ≤ 𝐴))
19		eqeq1 2733	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑦 = (𝑀‘(𝑇‘𝑥)) ↔ 𝑧 = (𝑀‘(𝑇‘𝑥))))
20	19	anbi2d 630	. . . . 5 ⊢ (𝑦 = 𝑧 → (((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥))) ↔ ((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥)))))
21	20	rexbidv 3157	. . . 4 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥))) ↔ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥)))))
22	21	ralab 3661	. . 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 3261	. . . 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 464	. . . . . . . 8 ⊢ ((((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴) ↔ ((𝑧 = (𝑀‘(𝑇‘𝑥)) ∧ (𝐿‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴))
25		impexp 450	. . . . . . . 8 ⊢ (((𝑧 = (𝑀‘(𝑇‘𝑥)) ∧ (𝐿‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ (𝑧 = (𝑀‘(𝑇‘𝑥)) → ((𝐿‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)))
26	24, 25	bitri 275	. . . . . . 7 ⊢ ((((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴) ↔ (𝑧 = (𝑀‘(𝑇‘𝑥)) → ((𝐿‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)))
27	26	albii 1819	. . . . . 6 ⊢ (∀𝑧(((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴) ↔ ∀𝑧(𝑧 = (𝑀‘(𝑇‘𝑥)) → ((𝐿‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)))
28		fvex 6853	. . . . . . 7 ⊢ (𝑀‘(𝑇‘𝑥)) ∈ V
29		breq1 5105	. . . . . . . 8 ⊢ (𝑧 = (𝑀‘(𝑇‘𝑥)) → (𝑧 ≤ 𝐴 ↔ (𝑀‘(𝑇‘𝑥)) ≤ 𝐴))
30	29	imbi2d 340	. . . . . . 7 ⊢ (𝑧 = (𝑀‘(𝑇‘𝑥)) → (((𝐿‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴) ↔ ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 𝐴)))
31	28, 30	ceqsalv 3484	. . . . . 6 ⊢ (∀𝑧(𝑧 = (𝑀‘(𝑇‘𝑥)) → ((𝐿‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)) ↔ ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 𝐴))
32	27, 31	bitri 275	. . . . 5 ⊢ (∀𝑧(((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴) ↔ ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 𝐴))
33	32	ralbii 3075	. . . 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 3160	. . . . 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 1819	. . . 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 301	. . 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 278	. 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 287	1 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ*) → ((𝑁‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053   ⊆ wss 3911   class class class wbr 5102  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  supcsup 9367  ℝcr 11043  1c1 11045  ℝ^*cxr 11183   < clt 11184   ≤ cle 11185  NrmCVeccnv 30563  BaseSetcba 30565  norm_CVcnmcv 30569   normOp_OLD cnmoo 30720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-pre-sup 11122

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-po 5539  df-so 5540  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-er 8648  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-sup 9369  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-vc 30538  df-nv 30571  df-va 30574  df-ba 30575  df-sm 30576  df-0v 30577  df-nmcv 30579  df-nmoo 30724

This theorem is referenced by:  nmoub3i  30752  nmobndi  30754  ubthlem2  30850