Theorem nmoubi 30843

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 30834	. . . . . 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 1454	. . . . 5 ⊢ (𝑇:𝑋⟶𝑌 → (𝑁‘𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))}, ℝ*, < ))
10	9	breq1d 5095	. . . 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 30835	. . . . . 6 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))} ⊆ ℝ)
13	2, 12	mpan 691	. . . . 5 ⊢ (𝑇:𝑋⟶𝑌 → {𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))} ⊆ ℝ)
14		ressxr 11189	. . . . 5 ⊢ ℝ ⊆ ℝ*
15	13, 14	sstrdi 3934	. . . 4 ⊢ (𝑇:𝑋⟶𝑌 → {𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))} ⊆ ℝ*)
16		supxrleub 13278	. . . 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 581	. . 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 2740	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑦 = (𝑀‘(𝑇‘𝑥)) ↔ 𝑧 = (𝑀‘(𝑇‘𝑥))))
20	19	anbi2d 631	. . . . 5 ⊢ (𝑦 = 𝑧 → (((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥))) ↔ ((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥)))))
21	20	rexbidv 3161	. . . 4 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥))) ↔ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥)))))
22	21	ralab 3639	. . 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 3263	. . . 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 1821	. . . . . 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 5088	. . . . . . . 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 3469	. . . . . 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 3083	. . . 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 3164	. . . . 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 1821	. . . 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 1540   = wceq 1542   ∈ wcel 2114  {cab 2714  ∀wral 3051  ∃wrex 3061   ⊆ wss 3889   class class class wbr 5085  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  supcsup 9353  ℝcr 11037  1c1 11039  ℝ^*cxr 11178   < clt 11179   ≤ cle 11180  NrmCVeccnv 30655  BaseSetcba 30657  norm_CVcnmcv 30661   normOp_OLD cnmoo 30812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  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 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-sup 9355  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-vc 30630  df-nv 30663  df-va 30666  df-ba 30667  df-sm 30668  df-0v 30669  df-nmcv 30671  df-nmoo 30816

This theorem is referenced by:  nmoub3i  30844  nmobndi  30846  ubthlem2  30942