Theorem nmopub 29943

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 29891	. . . 4 ⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥)))}, ℝ*, < ))
2	1	adantr 484	. . 3 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → (normop‘𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥)))}, ℝ*, < ))
3	2	breq1d 5049	. 2 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → ((normop‘𝑇) ≤ 𝐴 ↔ sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥)))}, ℝ*, < ) ≤ 𝐴))
4		nmopsetretALT 29898	. . . . 5 ⊢ (𝑇: ℋ⟶ ℋ → {𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥)))} ⊆ ℝ)
5		ressxr 10842	. . . . 5 ⊢ ℝ ⊆ ℝ*
6	4, 5	sstrdi 3899	. . . 4 ⊢ (𝑇: ℋ⟶ ℋ → {𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥)))} ⊆ ℝ*)
7		supxrleub 12881	. . . 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 583	. . 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 464	. . . . . . 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 633	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝑦 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) ↔ (𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1)))
12	9, 11	syl5bb 286	. . . . . 6 ⊢ (𝑦 = 𝑧 → (((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥))) ↔ (𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1)))
13	12	rexbidv 3206	. . . . 5 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥))) ↔ ∃𝑥 ∈ ℋ (𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1)))
14	13	ralab 3595	. . . 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 3147	. . . . 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 454	. . . . . . . 8 ⊢ (((𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ (𝑧 = (normℎ‘(𝑇‘𝑥)) → ((normℎ‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)))
17	16	albii 1827	. . . . . . 7 ⊢ (∀𝑧((𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ ∀𝑧(𝑧 = (normℎ‘(𝑇‘𝑥)) → ((normℎ‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)))
18		fvex 6708	. . . . . . . 8 ⊢ (normℎ‘(𝑇‘𝑥)) ∈ V
19		breq1 5042	. . . . . . . . 9 ⊢ (𝑧 = (normℎ‘(𝑇‘𝑥)) → (𝑧 ≤ 𝐴 ↔ (normℎ‘(𝑇‘𝑥)) ≤ 𝐴))
20	19	imbi2d 344	. . . . . . . 8 ⊢ (𝑧 = (normℎ‘(𝑇‘𝑥)) → (((normℎ‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴) ↔ ((normℎ‘𝑥) ≤ 1 → (normℎ‘(𝑇‘𝑥)) ≤ 𝐴)))
21	18, 20	ceqsalv 3433	. . . . . . 7 ⊢ (∀𝑧(𝑧 = (normℎ‘(𝑇‘𝑥)) → ((normℎ‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)) ↔ ((normℎ‘𝑥) ≤ 1 → (normℎ‘(𝑇‘𝑥)) ≤ 𝐴))
22	17, 21	bitri 278	. . . . . 6 ⊢ (∀𝑧((𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ ((normℎ‘𝑥) ≤ 1 → (normℎ‘(𝑇‘𝑥)) ≤ 𝐴))
23	22	ralbii 3078	. . . . 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 3188	. . . . . 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 1827	. . . . 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 304	. . . 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 281	. . 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 290	. 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 282	1 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → ((normop‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (normℎ‘(𝑇‘𝑥)) ≤ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1541   = wceq 1543   ∈ wcel 2112  {cab 2714  ∀wral 3051  ∃wrex 3052   ⊆ wss 3853   class class class wbr 5039  ⟶wf 6354  ‘cfv 6358  supcsup 9034  ℝcr 10693  1c1 10695  ℝ^*cxr 10831   < clt 10832   ≤ cle 10833   ℋchba 28954  norm_ℎcno 28958  norm_opcnop 28980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501  ax-cnex 10750  ax-resscn 10751  ax-1cn 10752  ax-icn 10753  ax-addcl 10754  ax-addrcl 10755  ax-mulcl 10756  ax-mulrcl 10757  ax-mulcom 10758  ax-addass 10759  ax-mulass 10760  ax-distr 10761  ax-i2m1 10762  ax-1ne0 10763  ax-1rid 10764  ax-rnegex 10765  ax-rrecex 10766  ax-cnre 10767  ax-pre-lttri 10768  ax-pre-lttrn 10769  ax-pre-ltadd 10770  ax-pre-mulgt0 10771  ax-pre-sup 10772  ax-hilex 29034  ax-hv0cl 29038  ax-hvmul0 29045  ax-hfi 29114  ax-his1 29117  ax-his3 29119  ax-his4 29120

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7148  df-ov 7194  df-oprab 7195  df-mpo 7196  df-om 7623  df-2nd 7740  df-wrecs 8025  df-recs 8086  df-rdg 8124  df-er 8369  df-map 8488  df-en 8605  df-dom 8606  df-sdom 8607  df-sup 9036  df-pnf 10834  df-mnf 10835  df-xr 10836  df-ltxr 10837  df-le 10838  df-sub 11029  df-neg 11030  df-div 11455  df-nn 11796  df-2 11858  df-3 11859  df-n0 12056  df-z 12142  df-uz 12404  df-rp 12552  df-seq 13540  df-exp 13601  df-cj 14627  df-re 14628  df-im 14629  df-sqrt 14763  df-hnorm 29003  df-nmop 29874

This theorem is referenced by:  nmopub2tALT  29944  nmophmi  30066  nmopadjlem  30124  nmoptrii  30129  nmopcoi  30130  nmopcoadji  30136