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Mirrors > Home > HSE Home > Th. List > nmfnlb | Structured version Visualization version GIF version |
Description: A lower bound for a functional norm. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmfnlb | ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (abs‘(𝑇‘𝐴)) ≤ (normfn‘𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmfnsetre 31906 | . . . . 5 ⊢ (𝑇: ℋ⟶ℂ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} ⊆ ℝ) | |
2 | ressxr 11303 | . . . . 5 ⊢ ℝ ⊆ ℝ* | |
3 | 1, 2 | sstrdi 4008 | . . . 4 ⊢ (𝑇: ℋ⟶ℂ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} ⊆ ℝ*) |
4 | 3 | 3ad2ant1 1132 | . . 3 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} ⊆ ℝ*) |
5 | fveq2 6907 | . . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (normℎ‘𝑦) = (normℎ‘𝐴)) | |
6 | 5 | breq1d 5158 | . . . . . . . 8 ⊢ (𝑦 = 𝐴 → ((normℎ‘𝑦) ≤ 1 ↔ (normℎ‘𝐴) ≤ 1)) |
7 | 2fveq3 6912 | . . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (abs‘(𝑇‘𝑦)) = (abs‘(𝑇‘𝐴))) | |
8 | 7 | eqeq2d 2746 | . . . . . . . 8 ⊢ (𝑦 = 𝐴 → ((abs‘(𝑇‘𝐴)) = (abs‘(𝑇‘𝑦)) ↔ (abs‘(𝑇‘𝐴)) = (abs‘(𝑇‘𝐴)))) |
9 | 6, 8 | anbi12d 632 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → (((normℎ‘𝑦) ≤ 1 ∧ (abs‘(𝑇‘𝐴)) = (abs‘(𝑇‘𝑦))) ↔ ((normℎ‘𝐴) ≤ 1 ∧ (abs‘(𝑇‘𝐴)) = (abs‘(𝑇‘𝐴))))) |
10 | eqid 2735 | . . . . . . . 8 ⊢ (abs‘(𝑇‘𝐴)) = (abs‘(𝑇‘𝐴)) | |
11 | 10 | biantru 529 | . . . . . . 7 ⊢ ((normℎ‘𝐴) ≤ 1 ↔ ((normℎ‘𝐴) ≤ 1 ∧ (abs‘(𝑇‘𝐴)) = (abs‘(𝑇‘𝐴)))) |
12 | 9, 11 | bitr4di 289 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (((normℎ‘𝑦) ≤ 1 ∧ (abs‘(𝑇‘𝐴)) = (abs‘(𝑇‘𝑦))) ↔ (normℎ‘𝐴) ≤ 1)) |
13 | 12 | rspcev 3622 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (abs‘(𝑇‘𝐴)) = (abs‘(𝑇‘𝑦)))) |
14 | fvex 6920 | . . . . . 6 ⊢ (abs‘(𝑇‘𝐴)) ∈ V | |
15 | eqeq1 2739 | . . . . . . . 8 ⊢ (𝑥 = (abs‘(𝑇‘𝐴)) → (𝑥 = (abs‘(𝑇‘𝑦)) ↔ (abs‘(𝑇‘𝐴)) = (abs‘(𝑇‘𝑦)))) | |
16 | 15 | anbi2d 630 | . . . . . . 7 ⊢ (𝑥 = (abs‘(𝑇‘𝐴)) → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ (abs‘(𝑇‘𝐴)) = (abs‘(𝑇‘𝑦))))) |
17 | 16 | rexbidv 3177 | . . . . . 6 ⊢ (𝑥 = (abs‘(𝑇‘𝐴)) → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (abs‘(𝑇‘𝐴)) = (abs‘(𝑇‘𝑦))))) |
18 | 14, 17 | elab 3681 | . . . . 5 ⊢ ((abs‘(𝑇‘𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (abs‘(𝑇‘𝐴)) = (abs‘(𝑇‘𝑦)))) |
19 | 13, 18 | sylibr 234 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (abs‘(𝑇‘𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}) |
20 | 19 | 3adant1 1129 | . . 3 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (abs‘(𝑇‘𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}) |
21 | supxrub 13363 | . . 3 ⊢ (({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} ⊆ ℝ* ∧ (abs‘(𝑇‘𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}) → (abs‘(𝑇‘𝐴)) ≤ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < )) | |
22 | 4, 20, 21 | syl2anc 584 | . 2 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (abs‘(𝑇‘𝐴)) ≤ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < )) |
23 | nmfnval 31905 | . . 3 ⊢ (𝑇: ℋ⟶ℂ → (normfn‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < )) | |
24 | 23 | 3ad2ant1 1132 | . 2 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (normfn‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < )) |
25 | 22, 24 | breqtrrd 5176 | 1 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (abs‘(𝑇‘𝐴)) ≤ (normfn‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 {cab 2712 ∃wrex 3068 ⊆ wss 3963 class class class wbr 5148 ⟶wf 6559 ‘cfv 6563 supcsup 9478 ℂcc 11151 ℝcr 11152 1c1 11154 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 abscabs 15270 ℋchba 30948 normℎcno 30952 normfncnmf 30980 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-hilex 31028 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-nmfn 31874 |
This theorem is referenced by: nmfnge0 31956 nmbdfnlbi 32078 nmcfnlbi 32081 |
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