Theorem nmopub 32537

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 32485	. . . 4 ⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥)))}, ℝ*, < ))
2	1	adantr 486	. . 3 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → (normop‘𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥)))}, ℝ*, < ))
3	2	breq1d 5116	. 2 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → ((normop‘𝑇) ≤ 𝐴 ↔ sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥)))}, ℝ*, < ) ≤ 𝐴))
4		nmopsetretALT 32492	. . . . 5 ⊢ (𝑇: ℋ⟶ ℋ → {𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥)))} ⊆ ℝ)
5		ressxr 11377	. . . . 5 ⊢ ℝ ⊆ ℝ*
6	4, 5	sstrdi 3925	. . . 4 ⊢ (𝑇: ℋ⟶ ℋ → {𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥)))} ⊆ ℝ*)
7		supxrleub 13480	. . . 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 592	. . 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 466	. . . . . . 7 ⊢ (((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥))) ↔ (𝑦 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1))
10		eqeq1 2767	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑦 = (normℎ‘(𝑇‘𝑥)) ↔ 𝑧 = (normℎ‘(𝑇‘𝑥))))
11	10	anbi1d 643	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝑦 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) ↔ (𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1)))
12	9, 11	bitrid 286	. . . . . 6 ⊢ (𝑦 = 𝑧 → (((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥))) ↔ (𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1)))
13	12	rexbidv 3190	. . . . 5 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (normℎ‘(𝑇‘𝑥))) ↔ ∃𝑥 ∈ ℋ (𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1)))
14	13	ralab 3654	. . . 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 3292	. . . . 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 456	. . . . . . . 8 ⊢ (((𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ (𝑧 = (normℎ‘(𝑇‘𝑥)) → ((normℎ‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)))
17	16	albii 1852	. . . . . . 7 ⊢ (∀𝑧((𝑧 = (normℎ‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ ∀𝑧(𝑧 = (normℎ‘(𝑇‘𝑥)) → ((normℎ‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)))
18		fvex 6918	. . . . . . . 8 ⊢ (normℎ‘(𝑇‘𝑥)) ∈ V
19		breq1 5109	. . . . . . . . 9 ⊢ (𝑧 = (normℎ‘(𝑇‘𝑥)) → (𝑧 ≤ 𝐴 ↔ (normℎ‘(𝑇‘𝑥)) ≤ 𝐴))
20	19	imbi2d 343	. . . . . . . 8 ⊢ (𝑧 = (normℎ‘(𝑇‘𝑥)) → (((normℎ‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴) ↔ ((normℎ‘𝑥) ≤ 1 → (normℎ‘(𝑇‘𝑥)) ≤ 𝐴)))
21	18, 20	ceqsalv 3493	. . . . . . 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 3112	. . . . 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 3193	. . . . . 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 1852	. . . . 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   class class class wbr 5106  ⟶wf 6551  ‘cfv 6555  supcsup 9456  ℝcr 11223  1c1 11225  ℝ^*cxr 11366   < clt 11367   ≤ cle 11368   ℋchba 31548  norm_ℎcno 31552  norm_opcnop 31574

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301  ax-pre-sup 11302  ax-hilex 31628  ax-hv0cl 31632  ax-hvmul0 31639  ax-hfi 31708  ax-his1 31711  ax-his3 31713  ax-his4 31714

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-er 8742  df-map 8874  df-en 8999  df-dom 9000  df-sdom 9001  df-sup 9458  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-div 11998  df-nn 12360  df-2 12429  df-3 12430  df-n0 12631  df-z 12718  df-uz 12990  df-rp 13145  df-seq 14169  df-exp 14229  df-cj 15292  df-re 15293  df-im 15294  df-sqrt 15428  df-hnorm 31597  df-nmop 32468

This theorem is used by:  nmopub2tALT  32538  nmophmi  32660  nmopadjlem  32718  nmoptrii  32723  nmopcoi  32724  nmopcoadji  32730