Theorem nmoubi 31197

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 31188	. . . . . 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 1480	. . . . 5 ⊢ (𝑇:𝑋⟶𝑌 → (𝑁‘𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))}, ℝ*, < ))
10	9	breq1d 5121	. . . 4 ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) ≤ 𝐴 ↔ sup({𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))}, ℝ*, < ) ≤ 𝐴))
11	10	adantr 486	. . 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 31189	. . . . . 6 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))} ⊆ ℝ)
13	2, 12	mpan 703	. . . . 5 ⊢ (𝑇:𝑋⟶𝑌 → {𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))} ⊆ ℝ)
14		ressxr 11270	. . . . 5 ⊢ ℝ ⊆ ℝ*
15	13, 14	sstrdi 3950	. . . 4 ⊢ (𝑇:𝑋⟶𝑌 → {𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))} ⊆ ℝ*)
16		supxrleub 13370	. . . 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 592	. . 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 282	. 2 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ*) → ((𝑁‘𝑇) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥)))}𝑧 ≤ 𝐴))
19		eqeq1 2769	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑦 = (𝑀‘(𝑇‘𝑥)) ↔ 𝑧 = (𝑀‘(𝑇‘𝑥))))
20	19	anbi2d 642	. . . . 5 ⊢ (𝑦 = 𝑧 → (((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥))) ↔ ((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥)))))
21	20	rexbidv 3191	. . . 4 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑥))) ↔ ∃𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥)))))
22	21	ralab 3658	. . 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 3293	. . . 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 470	. . . . . . . 8 ⊢ ((((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴) ↔ ((𝑧 = (𝑀‘(𝑇‘𝑥)) ∧ (𝐿‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴))
25		impexp 456	. . . . . . . 8 ⊢ (((𝑧 = (𝑀‘(𝑇‘𝑥)) ∧ (𝐿‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ (𝑧 = (𝑀‘(𝑇‘𝑥)) → ((𝐿‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)))
26	24, 25	bitri 278	. . . . . . 7 ⊢ ((((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴) ↔ (𝑧 = (𝑀‘(𝑇‘𝑥)) → ((𝐿‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)))
27	26	albii 1852	. . . . . 6 ⊢ (∀𝑧(((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴) ↔ ∀𝑧(𝑧 = (𝑀‘(𝑇‘𝑥)) → ((𝐿‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)))
28		fvex 6898	. . . . . . 7 ⊢ (𝑀‘(𝑇‘𝑥)) ∈ V
29		breq1 5114	. . . . . . . 8 ⊢ (𝑧 = (𝑀‘(𝑇‘𝑥)) → (𝑧 ≤ 𝐴 ↔ (𝑀‘(𝑇‘𝑥)) ≤ 𝐴))
30	29	imbi2d 343	. . . . . . 7 ⊢ (𝑧 = (𝑀‘(𝑇‘𝑥)) → (((𝐿‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴) ↔ ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 𝐴)))
31	28, 30	ceqsalv 3496	. . . . . 6 ⊢ (∀𝑧(𝑧 = (𝑀‘(𝑇‘𝑥)) → ((𝐿‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)) ↔ ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 𝐴))
32	27, 31	bitri 278	. . . . 5 ⊢ (∀𝑧(((𝐿‘𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇‘𝑥))) → 𝑧 ≤ 𝐴) ↔ ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 𝐴))
33	32	ralbii 3113	. . . 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 3194	. . . . 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 1852	. . . 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 304	. . 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 281	. 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 290	1 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ*) → ((𝑁‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 𝐴)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2146  {cab 2743  ∀wral 3081  ∃wrex 3091   ⊆ wss 3906   class class class wbr 5111  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419  supcsup 9407  ℝcr 11116  1c1 11118  ℝ^*cxr 11259   < clt 11260   ≤ cle 11261  NrmCVeccnv 31009  BaseSetcba 31011  norm_CVcnmcv 31015   normOp_OLD cnmoo 31166

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11173  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193  ax-pre-mulgt0 11194  ax-pre-sup 11195

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-po 5571  df-so 5572  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-er 8700  df-map 8832  df-en 8950  df-dom 8951  df-sdom 8952  df-sup 9409  df-pnf 11262  df-mnf 11263  df-xr 11264  df-ltxr 11265  df-le 11266  df-sub 11460  df-neg 11461  df-vc 30984  df-nv 31017  df-va 31020  df-ba 31021  df-sm 31022  df-0v 31023  df-nmcv 31025  df-nmoo 31170

This theorem is used by:  nmoub3i  31198  nmobndi  31200  ubthlem2  31296