![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > nmopsetretALT | Structured version Visualization version GIF version |
Description: The set in the supremum of the operator norm definition df-nmop 28999 is a set of reals. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
nmopsetretALT | ⊢ (𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffvelrn 6512 | . . . . . . . 8 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ) | |
2 | normcl 28283 | . . . . . . . 8 ⊢ ((𝑇‘𝑦) ∈ ℋ → (normℎ‘(𝑇‘𝑦)) ∈ ℝ) | |
3 | 1, 2 | syl 17 | . . . . . . 7 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (normℎ‘(𝑇‘𝑦)) ∈ ℝ) |
4 | eleq1 2819 | . . . . . . 7 ⊢ (𝑥 = (normℎ‘(𝑇‘𝑦)) → (𝑥 ∈ ℝ ↔ (normℎ‘(𝑇‘𝑦)) ∈ ℝ)) | |
5 | 3, 4 | syl5ibr 236 | . . . . . 6 ⊢ (𝑥 = (normℎ‘(𝑇‘𝑦)) → ((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → 𝑥 ∈ ℝ)) |
6 | 5 | impcom 445 | . . . . 5 ⊢ (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑥 = (normℎ‘(𝑇‘𝑦))) → 𝑥 ∈ ℝ) |
7 | 6 | adantrl 754 | . . . 4 ⊢ (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))) → 𝑥 ∈ ℝ) |
8 | 7 | exp31 631 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → (𝑦 ∈ ℋ → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦))) → 𝑥 ∈ ℝ))) |
9 | 8 | rexlimdv 3160 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦))) → 𝑥 ∈ ℝ)) |
10 | 9 | abssdv 3809 | 1 ⊢ (𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1624 ∈ wcel 2131 {cab 2738 ∃wrex 3043 ⊆ wss 3707 class class class wbr 4796 ⟶wf 6037 ‘cfv 6041 ℝcr 10119 1c1 10121 ≤ cle 10259 ℋchil 28077 normℎcno 28081 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-cnex 10176 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196 ax-pre-mulgt0 10197 ax-pre-sup 10198 ax-hv0cl 28161 ax-hvmul0 28168 ax-hfi 28237 ax-his1 28240 ax-his3 28242 ax-his4 28243 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rmo 3050 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4581 df-iun 4666 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-pred 5833 df-ord 5879 df-on 5880 df-lim 5881 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-om 7223 df-2nd 7326 df-wrecs 7568 df-recs 7629 df-rdg 7667 df-er 7903 df-en 8114 df-dom 8115 df-sdom 8116 df-sup 8505 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-sub 10452 df-neg 10453 df-div 10869 df-nn 11205 df-2 11263 df-3 11264 df-n0 11477 df-z 11562 df-uz 11872 df-rp 12018 df-seq 12988 df-exp 13047 df-cj 14030 df-re 14031 df-im 14032 df-sqrt 14166 df-hnorm 28126 |
This theorem is referenced by: nmopub 29068 |
Copyright terms: Public domain | W3C validator |