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Mirrors > Home > HSE Home > Th. List > nmfnrepnf | Structured version Visualization version GIF version |
Description: The norm of a Hilbert space functional is either real or plus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmfnrepnf | ⊢ (𝑇: ℋ⟶ℂ → ((normfn‘𝑇) ∈ ℝ ↔ (normfn‘𝑇) ≠ +∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmfnsetre 31905 | . . 3 ⊢ (𝑇: ℋ⟶ℂ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} ⊆ ℝ) | |
2 | nmfnsetn0 31906 | . . . 4 ⊢ (abs‘(𝑇‘0ℎ)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} | |
3 | 2 | ne0ii 4349 | . . 3 ⊢ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} ≠ ∅ |
4 | supxrre2 13369 | . . 3 ⊢ (({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} ⊆ ℝ ∧ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} ≠ ∅) → (sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < ) ∈ ℝ ↔ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < ) ≠ +∞)) | |
5 | 1, 3, 4 | sylancl 586 | . 2 ⊢ (𝑇: ℋ⟶ℂ → (sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < ) ∈ ℝ ↔ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < ) ≠ +∞)) |
6 | nmfnval 31904 | . . 3 ⊢ (𝑇: ℋ⟶ℂ → (normfn‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < )) | |
7 | 6 | eleq1d 2823 | . 2 ⊢ (𝑇: ℋ⟶ℂ → ((normfn‘𝑇) ∈ ℝ ↔ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < ) ∈ ℝ)) |
8 | 6 | neeq1d 2997 | . 2 ⊢ (𝑇: ℋ⟶ℂ → ((normfn‘𝑇) ≠ +∞ ↔ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < ) ≠ +∞)) |
9 | 5, 7, 8 | 3bitr4d 311 | 1 ⊢ (𝑇: ℋ⟶ℂ → ((normfn‘𝑇) ∈ ℝ ↔ (normfn‘𝑇) ≠ +∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 {cab 2711 ≠ wne 2937 ∃wrex 3067 ⊆ wss 3962 ∅c0 4338 class class class wbr 5147 ⟶wf 6558 ‘cfv 6562 supcsup 9477 ℂcc 11150 ℝcr 11151 1c1 11153 +∞cpnf 11289 ℝ*cxr 11291 < clt 11292 ≤ cle 11293 abscabs 15269 ℋchba 30947 normℎcno 30951 0ℎc0v 30952 normfncnmf 30979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-hilex 31027 ax-hv0cl 31031 ax-hvmul0 31038 ax-hfi 31107 ax-his3 31112 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-hnorm 30996 df-nmfn 31873 |
This theorem is referenced by: (None) |
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