Theorem nmooval 28546

Description: The operator norm function. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nmoofval.1	⊢ 𝑋 = (BaseSet‘𝑈)
nmoofval.2	⊢ 𝑌 = (BaseSet‘𝑊)
nmoofval.3	⊢ 𝐿 = (normCV‘𝑈)
nmoofval.4	⊢ 𝑀 = (normCV‘𝑊)
nmoofval.6	⊢ 𝑁 = (𝑈 normOpOLD 𝑊)

Assertion

Ref	Expression
nmooval	⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) = sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧)))}, ℝ*, < ))

Distinct variable groups:   𝑥,𝑧,𝑈   𝑥,𝑊,𝑧   𝑧,𝑋   𝑥,𝑌   𝑥,𝑇,𝑧

Allowed substitution hints:   𝐿(𝑥,𝑧)   𝑀(𝑥,𝑧)   𝑁(𝑥,𝑧)   𝑋(𝑥)   𝑌(𝑧)

Proof of Theorem nmooval

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmoofval.2	. . . . 5 ⊢ 𝑌 = (BaseSet‘𝑊)
2	1	fvexi 6659	. . . 4 ⊢ 𝑌 ∈ V
3		nmoofval.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
4	3	fvexi 6659	. . . 4 ⊢ 𝑋 ∈ V
5	2, 4	elmap 8418	. . 3 ⊢ (𝑇 ∈ (𝑌 ↑m 𝑋) ↔ 𝑇:𝑋⟶𝑌)
6		nmoofval.3	. . . . . 6 ⊢ 𝐿 = (normCV‘𝑈)
7		nmoofval.4	. . . . . 6 ⊢ 𝑀 = (normCV‘𝑊)
8		nmoofval.6	. . . . . 6 ⊢ 𝑁 = (𝑈 normOpOLD 𝑊)
9	3, 1, 6, 7, 8	nmoofval 28545	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ (𝑌 ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < )))
10	9	fveq1d 6647	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑁‘𝑇) = ((𝑡 ∈ (𝑌 ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < ))‘𝑇))
11		fveq1 6644	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑧) = (𝑇‘𝑧))
12	11	fveq2d 6649	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (𝑀‘(𝑡‘𝑧)) = (𝑀‘(𝑇‘𝑧)))
13	12	eqeq2d 2809	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (𝑥 = (𝑀‘(𝑡‘𝑧)) ↔ 𝑥 = (𝑀‘(𝑇‘𝑧))))
14	13	anbi2d 631	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧))) ↔ ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧)))))
15	14	rexbidv 3256	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧))) ↔ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧)))))
16	15	abbidv 2862	. . . . . 6 ⊢ (𝑡 = 𝑇 → {𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))} = {𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧)))})
17	16	supeq1d 8894	. . . . 5 ⊢ (𝑡 = 𝑇 → sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧)))}, ℝ*, < ))
18		eqid 2798	. . . . 5 ⊢ (𝑡 ∈ (𝑌 ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < )) = (𝑡 ∈ (𝑌 ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < ))
19		xrltso 12522	. . . . . 6 ⊢ < Or ℝ*
20	19	supex 8911	. . . . 5 ⊢ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧)))}, ℝ*, < ) ∈ V
21	17, 18, 20	fvmpt 6745	. . . 4 ⊢ (𝑇 ∈ (𝑌 ↑m 𝑋) → ((𝑡 ∈ (𝑌 ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < ))‘𝑇) = sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧)))}, ℝ*, < ))
22	10, 21	sylan9eq 2853	. . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑇 ∈ (𝑌 ↑m 𝑋)) → (𝑁‘𝑇) = sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧)))}, ℝ*, < ))
23	5, 22	sylan2br 597	. 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) = sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧)))}, ℝ*, < ))
24	23	3impa 1107	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) = sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧)))}, ℝ*, < ))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {cab 2776  ∃wrex 3107   class class class wbr 5030   ↦ cmpt 5110  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ↑_m cmap 8389  supcsup 8888  1c1 10527  ℝ^*cxr 10663   < clt 10664   ≤ cle 10665  NrmCVeccnv 28367  BaseSetcba 28369  norm_CVcnmcv 28373   normOp_OLD cnmoo 28524

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-pre-lttri 10600  ax-pre-lttrn 10601

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-po 5438  df-so 5439  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-sup 8890  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-nmoo 28528

This theorem is referenced by:  nmoxr  28549  nmooge0  28550  nmorepnf  28551  nmoolb  28554  nmoubi  28555  nmoo0  28574  nmlno0lem  28576