Theorem nmopub 30558

Description: An upper bound for an operator norm. (Contributed by NM, 7-Mar-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
nmopub	⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → ((normop‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (normℎ‘(𝑇‘𝑥)) ≤ 𝐴)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑇

Proof of Theorem nmopub

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmopval 30506	. . . 4 ⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥)))}, ℝ*, < ))
2	1	adantr 481	. . 3 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → (normop‘𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥)))}, ℝ*, < ))
3	2	breq1d 5102	. 2 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → ((normop‘𝑇) ≤ 𝐴 ↔ sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥)))}, ℝ*, < ) ≤ 𝐴))
4		nmopsetretALT 30513	. . . . 5 ⊢ (𝑇: ℋ⟶ ℋ → {𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥)))} ⊆ ℝ)
5		ressxr 11120	. . . . 5 ⊢ ℝ ⊆ ℝ*
6	4, 5	sstrdi 3944	. . . 4 ⊢ (𝑇: ℋ⟶ ℋ → {𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥)))} ⊆ ℝ*)
7		supxrleub 13161	. . . 4 ⊢ (({𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥)))} ⊆ ℝ* ∧ 𝐴 ∈ ℝ*) → (sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥)))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥)))}𝑧 ≤ 𝐴))
8	6, 7	sylan 580	. . 3 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → (sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥)))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥)))}𝑧 ≤ 𝐴))
9		ancom 461	. . . . . . 7 ⊢ (((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥))) ↔ (𝑦 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1))
10		eqeq1 2740	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑦 = (normℎ‘(𝑇‘𝑥)) ↔ 𝑧 = (normℎ‘(𝑇‘𝑥))))
11	10	anbi1d 630	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝑦 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) ↔ (𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1)))
12	9, 11	bitrid 282	. . . . . 6 ⊢ (𝑦 = 𝑧 → (((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥))) ↔ (𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1)))
13	12	rexbidv 3171	. . . . 5 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥))) ↔ ∃𝑥 ∈ ℋ (𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1)))
14	13	ralab 3638	. . . 4 ⊢ (∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥)))}𝑧 ≤ 𝐴 ↔ ∀𝑧(∃𝑥 ∈ ℋ (𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴))
15		ralcom4 3265	. . . . 5 ⊢ (∀𝑥 ∈ ℋ ∀𝑧((𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ ∀𝑧∀𝑥 ∈ ℋ ((𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴))
16		impexp 451	. . . . . . . 8 ⊢ (((𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ (𝑧 = (normℎ‘(𝑇‘𝑥)) → ((normℎ‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)))
17	16	albii 1820	. . . . . . 7 ⊢ (∀𝑧((𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ ∀𝑧(𝑧 = (normℎ‘(𝑇‘𝑥)) → ((normℎ‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)))
18		fvex 6838	. . . . . . . 8 ⊢ (normℎ‘(𝑇‘𝑥)) ∈ V
19		breq1 5095	. . . . . . . . 9 ⊢ (𝑧 = (normℎ‘(𝑇‘𝑥)) → (𝑧 ≤ 𝐴 ↔ (normℎ‘(𝑇‘𝑥)) ≤ 𝐴))
20	19	imbi2d 340	. . . . . . . 8 ⊢ (𝑧 = (normℎ‘(𝑇‘𝑥)) → (((normℎ‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴) ↔ ((normℎ‘𝑥) ≤ 1 → (normℎ‘(𝑇‘𝑥)) ≤ 𝐴)))
21	18, 20	ceqsalv 3476	. . . . . . 7 ⊢ (∀𝑧(𝑧 = (normℎ‘(𝑇‘𝑥)) → ((normℎ‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)) ↔ ((normℎ‘𝑥) ≤ 1 → (normℎ‘(𝑇‘𝑥)) ≤ 𝐴))
22	17, 21	bitri 274	. . . . . 6 ⊢ (∀𝑧((𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ ((normℎ‘𝑥) ≤ 1 → (normℎ‘(𝑇‘𝑥)) ≤ 𝐴))
23	22	ralbii 3092	. . . . 5 ⊢ (∀𝑥 ∈ ℋ ∀𝑧((𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (normℎ‘(𝑇‘𝑥)) ≤ 𝐴))
24		r19.23v 3175	. . . . . 6 ⊢ (∀𝑥 ∈ ℋ ((𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ (∃𝑥 ∈ ℋ (𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴))
25	24	albii 1820	. . . . 5 ⊢ (∀𝑧∀𝑥 ∈ ℋ ((𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ ∀𝑧(∃𝑥 ∈ ℋ (𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴))
26	15, 23, 25	3bitr3i 300	. . . 4 ⊢ (∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (normℎ‘(𝑇‘𝑥)) ≤ 𝐴) ↔ ∀𝑧(∃𝑥 ∈ ℋ (𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴))
27	14, 26	bitr4i 277	. . 3 ⊢ (∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥)))}𝑧 ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (normℎ‘(𝑇‘𝑥)) ≤ 𝐴))
28	8, 27	bitrdi 286	. 2 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → (sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥)))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (normℎ‘(𝑇‘𝑥)) ≤ 𝐴)))
29	3, 28	bitrd 278	1 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → ((normop‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (normℎ‘(𝑇‘𝑥)) ≤ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1538   = wceq 1540   ∈ wcel 2105  {cab 2713  ∀wral 3061  ∃wrex 3070   ⊆ wss 3898   class class class wbr 5092  ⟶wf 6475  ‘cfv 6479  supcsup 9297  ℝcr 10971  1c1 10973  ℝ^*cxr 11109   < clt 11110   ≤ cle 11111   ℋchba 29569  norm_ℎcno 29573  norm_opcnop 29595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049  ax-pre-sup 11050  ax-hilex 29649  ax-hv0cl 29653  ax-hvmul0 29660  ax-hfi 29729  ax-his1 29732  ax-his3 29734  ax-his4 29735

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-om 7781  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-er 8569  df-map 8688  df-en 8805  df-dom 8806  df-sdom 8807  df-sup 9299  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-div 11734  df-nn 12075  df-2 12137  df-3 12138  df-n0 12335  df-z 12421  df-uz 12684  df-rp 12832  df-seq 13823  df-exp 13884  df-cj 14909  df-re 14910  df-im 14911  df-sqrt 15045  df-hnorm 29618  df-nmop 30489

This theorem is referenced by:  nmopub2tALT  30559  nmophmi  30681  nmopadjlem  30739  nmoptrii  30744  nmopcoi  30745  nmopcoadji  30751