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Mirrors > Home > MPE Home > Th. List > nmosetn0 | Structured version Visualization version GIF version |
Description: The set in the supremum of the operator norm definition df-nmoo 30467 is nonempty. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmosetn0.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nmosetn0.5 | ⊢ 𝑍 = (0vec‘𝑈) |
nmosetn0.4 | ⊢ 𝑀 = (normCV‘𝑈) |
Ref | Expression |
---|---|
nmosetn0 | ⊢ (𝑈 ∈ NrmCVec → (𝑁‘(𝑇‘𝑍)) ∈ {𝑥 ∣ ∃𝑦 ∈ 𝑋 ((𝑀‘𝑦) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇‘𝑦)))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmosetn0.1 | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
2 | nmosetn0.5 | . . . 4 ⊢ 𝑍 = (0vec‘𝑈) | |
3 | 1, 2 | nvzcl 30356 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋) |
4 | nmosetn0.4 | . . . . . 6 ⊢ 𝑀 = (normCV‘𝑈) | |
5 | 2, 4 | nvz0 30390 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → (𝑀‘𝑍) = 0) |
6 | 0le1 11734 | . . . . 5 ⊢ 0 ≤ 1 | |
7 | 5, 6 | eqbrtrdi 5177 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → (𝑀‘𝑍) ≤ 1) |
8 | eqid 2724 | . . . 4 ⊢ (𝑁‘(𝑇‘𝑍)) = (𝑁‘(𝑇‘𝑍)) | |
9 | 7, 8 | jctir 520 | . . 3 ⊢ (𝑈 ∈ NrmCVec → ((𝑀‘𝑍) ≤ 1 ∧ (𝑁‘(𝑇‘𝑍)) = (𝑁‘(𝑇‘𝑍)))) |
10 | fveq2 6881 | . . . . . 6 ⊢ (𝑦 = 𝑍 → (𝑀‘𝑦) = (𝑀‘𝑍)) | |
11 | 10 | breq1d 5148 | . . . . 5 ⊢ (𝑦 = 𝑍 → ((𝑀‘𝑦) ≤ 1 ↔ (𝑀‘𝑍) ≤ 1)) |
12 | 2fveq3 6886 | . . . . . 6 ⊢ (𝑦 = 𝑍 → (𝑁‘(𝑇‘𝑦)) = (𝑁‘(𝑇‘𝑍))) | |
13 | 12 | eqeq2d 2735 | . . . . 5 ⊢ (𝑦 = 𝑍 → ((𝑁‘(𝑇‘𝑍)) = (𝑁‘(𝑇‘𝑦)) ↔ (𝑁‘(𝑇‘𝑍)) = (𝑁‘(𝑇‘𝑍)))) |
14 | 11, 13 | anbi12d 630 | . . . 4 ⊢ (𝑦 = 𝑍 → (((𝑀‘𝑦) ≤ 1 ∧ (𝑁‘(𝑇‘𝑍)) = (𝑁‘(𝑇‘𝑦))) ↔ ((𝑀‘𝑍) ≤ 1 ∧ (𝑁‘(𝑇‘𝑍)) = (𝑁‘(𝑇‘𝑍))))) |
15 | 14 | rspcev 3604 | . . 3 ⊢ ((𝑍 ∈ 𝑋 ∧ ((𝑀‘𝑍) ≤ 1 ∧ (𝑁‘(𝑇‘𝑍)) = (𝑁‘(𝑇‘𝑍)))) → ∃𝑦 ∈ 𝑋 ((𝑀‘𝑦) ≤ 1 ∧ (𝑁‘(𝑇‘𝑍)) = (𝑁‘(𝑇‘𝑦)))) |
16 | 3, 9, 15 | syl2anc 583 | . 2 ⊢ (𝑈 ∈ NrmCVec → ∃𝑦 ∈ 𝑋 ((𝑀‘𝑦) ≤ 1 ∧ (𝑁‘(𝑇‘𝑍)) = (𝑁‘(𝑇‘𝑦)))) |
17 | fvex 6894 | . . 3 ⊢ (𝑁‘(𝑇‘𝑍)) ∈ V | |
18 | eqeq1 2728 | . . . . 5 ⊢ (𝑥 = (𝑁‘(𝑇‘𝑍)) → (𝑥 = (𝑁‘(𝑇‘𝑦)) ↔ (𝑁‘(𝑇‘𝑍)) = (𝑁‘(𝑇‘𝑦)))) | |
19 | 18 | anbi2d 628 | . . . 4 ⊢ (𝑥 = (𝑁‘(𝑇‘𝑍)) → (((𝑀‘𝑦) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇‘𝑦))) ↔ ((𝑀‘𝑦) ≤ 1 ∧ (𝑁‘(𝑇‘𝑍)) = (𝑁‘(𝑇‘𝑦))))) |
20 | 19 | rexbidv 3170 | . . 3 ⊢ (𝑥 = (𝑁‘(𝑇‘𝑍)) → (∃𝑦 ∈ 𝑋 ((𝑀‘𝑦) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇‘𝑦))) ↔ ∃𝑦 ∈ 𝑋 ((𝑀‘𝑦) ≤ 1 ∧ (𝑁‘(𝑇‘𝑍)) = (𝑁‘(𝑇‘𝑦))))) |
21 | 17, 20 | elab 3660 | . 2 ⊢ ((𝑁‘(𝑇‘𝑍)) ∈ {𝑥 ∣ ∃𝑦 ∈ 𝑋 ((𝑀‘𝑦) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇‘𝑦)))} ↔ ∃𝑦 ∈ 𝑋 ((𝑀‘𝑦) ≤ 1 ∧ (𝑁‘(𝑇‘𝑍)) = (𝑁‘(𝑇‘𝑦)))) |
22 | 16, 21 | sylibr 233 | 1 ⊢ (𝑈 ∈ NrmCVec → (𝑁‘(𝑇‘𝑍)) ∈ {𝑥 ∣ ∃𝑦 ∈ 𝑋 ((𝑀‘𝑦) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇‘𝑦)))}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {cab 2701 ∃wrex 3062 class class class wbr 5138 ‘cfv 6533 0cc0 11106 1c1 11107 ≤ cle 11246 NrmCVeccnv 30306 BaseSetcba 30308 0veccn0v 30310 normCVcnmcv 30312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-rp 12972 df-seq 13964 df-exp 14025 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-grpo 30215 df-gid 30216 df-ginv 30217 df-ablo 30267 df-vc 30281 df-nv 30314 df-va 30317 df-ba 30318 df-sm 30319 df-0v 30320 df-nmcv 30322 |
This theorem is referenced by: nmooge0 30489 nmorepnf 30490 |
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