Theorem nmoubi 27627

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 27618	. . . . . 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 1414	. . . . 5 ⊢ (𝑇:𝑋⟶𝑌 → (𝑁‘𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))}, ℝ*, < ))
10	9	breq1d 4663	. . . 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 27619	. . . . . 6 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))} ⊆ ℝ)
13	2, 12	mpan 706	. . . . 5 ⊢ (𝑇:𝑋⟶𝑌 → {𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))} ⊆ ℝ)
14		ressxr 10083	. . . . 5 ⊢ ℝ ⊆ ℝ*
15	13, 14	syl6ss 3615	. . . 4 ⊢ (𝑇:𝑋⟶𝑌 → {𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))} ⊆ ℝ*)
16		supxrleub 12156	. . . 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 488	. . 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 268	. 2 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ*) → ((𝑁‘𝑇) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))}𝑧 ≤ 𝐴))
19		eqeq1 2626	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑦 = (𝑀‘(𝑇‘𝑥)) ↔ 𝑧 = (𝑀‘(𝑇‘𝑥))))
20	19	anbi2d 740	. . . . 5 ⊢ (𝑦 = 𝑧 → (((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥))) ↔ ((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥)))))
21	20	rexbidv 3052	. . . 4 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥))) ↔ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥)))))
22	21	ralab 3367	. . 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 3224	. . . 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 468	. . . . . . . 8 ⊢ ((((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴) ↔ ((𝑧 = (𝑀‘(𝑇‘𝑥)) ∧ (𝐿‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴))
25		impexp 462	. . . . . . . 8 ⊢ (((𝑧 = (𝑀‘(𝑇‘𝑥)) ∧ (𝐿‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ (𝑧 = (𝑀‘(𝑇‘𝑥)) → ((𝐿‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)))
26	24, 25	bitri 264	. . . . . . 7 ⊢ ((((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴) ↔ (𝑧 = (𝑀‘(𝑇‘𝑥)) → ((𝐿‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)))
27	26	albii 1747	. . . . . 6 ⊢ (∀𝑧(((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴) ↔ ∀𝑧(𝑧 = (𝑀‘(𝑇‘𝑥)) → ((𝐿‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)))
28		fvex 6201	. . . . . . 7 ⊢ (𝑀‘(𝑇‘𝑥)) ∈ V
29		breq1 4656	. . . . . . . 8 ⊢ (𝑧 = (𝑀‘(𝑇‘𝑥)) → (𝑧 ≤ 𝐴 ↔ (𝑀‘(𝑇‘𝑥)) ≤ 𝐴))
30	29	imbi2d 330	. . . . . . 7 ⊢ (𝑧 = (𝑀‘(𝑇‘𝑥)) → (((𝐿‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴) ↔ ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 𝐴)))
31	28, 30	ceqsalv 3233	. . . . . 6 ⊢ (∀𝑧(𝑧 = (𝑀‘(𝑇‘𝑥)) → ((𝐿‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)) ↔ ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 𝐴))
32	27, 31	bitri 264	. . . . 5 ⊢ (∀𝑧(((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴) ↔ ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 𝐴))
33	32	ralbii 2980	. . . 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 3023	. . . . 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 1747	. . . 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 290	. . 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 267	. 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	syl6bb 276	1 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ*) → ((𝑁‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913   ⊆ wss 3574   class class class wbr 4653  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  supcsup 8346  ℝcr 9935  1c1 9937  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075  NrmCVeccnv 27439  BaseSetcba 27441  norm_CVcnmcv 27445   normOp_OLD cnmoo 27596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-nmcv 27455  df-nmoo 27600

This theorem is referenced by:  nmoub3i  27628  nmobndi  27630  ubthlem2  27727