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Mirrors > Home > HSE Home > Th. List > nmfnsetn0 | Structured version Visualization version GIF version |
Description: The set in the supremum of the functional norm definition df-nmfn 29625 is nonempty. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmfnsetn0 | ⊢ (abs‘(𝑇‘0ℎ)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hv0cl 28783 | . . 3 ⊢ 0ℎ ∈ ℋ | |
2 | norm0 28908 | . . . . 5 ⊢ (normℎ‘0ℎ) = 0 | |
3 | 0le1 11166 | . . . . 5 ⊢ 0 ≤ 1 | |
4 | 2, 3 | eqbrtri 5090 | . . . 4 ⊢ (normℎ‘0ℎ) ≤ 1 |
5 | eqid 2824 | . . . 4 ⊢ (abs‘(𝑇‘0ℎ)) = (abs‘(𝑇‘0ℎ)) | |
6 | 4, 5 | pm3.2i 473 | . . 3 ⊢ ((normℎ‘0ℎ) ≤ 1 ∧ (abs‘(𝑇‘0ℎ)) = (abs‘(𝑇‘0ℎ))) |
7 | fveq2 6673 | . . . . . 6 ⊢ (𝑦 = 0ℎ → (normℎ‘𝑦) = (normℎ‘0ℎ)) | |
8 | 7 | breq1d 5079 | . . . . 5 ⊢ (𝑦 = 0ℎ → ((normℎ‘𝑦) ≤ 1 ↔ (normℎ‘0ℎ) ≤ 1)) |
9 | 2fveq3 6678 | . . . . . 6 ⊢ (𝑦 = 0ℎ → (abs‘(𝑇‘𝑦)) = (abs‘(𝑇‘0ℎ))) | |
10 | 9 | eqeq2d 2835 | . . . . 5 ⊢ (𝑦 = 0ℎ → ((abs‘(𝑇‘0ℎ)) = (abs‘(𝑇‘𝑦)) ↔ (abs‘(𝑇‘0ℎ)) = (abs‘(𝑇‘0ℎ)))) |
11 | 8, 10 | anbi12d 632 | . . . 4 ⊢ (𝑦 = 0ℎ → (((normℎ‘𝑦) ≤ 1 ∧ (abs‘(𝑇‘0ℎ)) = (abs‘(𝑇‘𝑦))) ↔ ((normℎ‘0ℎ) ≤ 1 ∧ (abs‘(𝑇‘0ℎ)) = (abs‘(𝑇‘0ℎ))))) |
12 | 11 | rspcev 3626 | . . 3 ⊢ ((0ℎ ∈ ℋ ∧ ((normℎ‘0ℎ) ≤ 1 ∧ (abs‘(𝑇‘0ℎ)) = (abs‘(𝑇‘0ℎ)))) → ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (abs‘(𝑇‘0ℎ)) = (abs‘(𝑇‘𝑦)))) |
13 | 1, 6, 12 | mp2an 690 | . 2 ⊢ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (abs‘(𝑇‘0ℎ)) = (abs‘(𝑇‘𝑦))) |
14 | fvex 6686 | . . 3 ⊢ (abs‘(𝑇‘0ℎ)) ∈ V | |
15 | eqeq1 2828 | . . . . 5 ⊢ (𝑥 = (abs‘(𝑇‘0ℎ)) → (𝑥 = (abs‘(𝑇‘𝑦)) ↔ (abs‘(𝑇‘0ℎ)) = (abs‘(𝑇‘𝑦)))) | |
16 | 15 | anbi2d 630 | . . . 4 ⊢ (𝑥 = (abs‘(𝑇‘0ℎ)) → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ (abs‘(𝑇‘0ℎ)) = (abs‘(𝑇‘𝑦))))) |
17 | 16 | rexbidv 3300 | . . 3 ⊢ (𝑥 = (abs‘(𝑇‘0ℎ)) → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (abs‘(𝑇‘0ℎ)) = (abs‘(𝑇‘𝑦))))) |
18 | 14, 17 | elab 3670 | . 2 ⊢ ((abs‘(𝑇‘0ℎ)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (abs‘(𝑇‘0ℎ)) = (abs‘(𝑇‘𝑦)))) |
19 | 13, 18 | mpbir 233 | 1 ⊢ (abs‘(𝑇‘0ℎ)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1536 ∈ wcel 2113 {cab 2802 ∃wrex 3142 class class class wbr 5069 ‘cfv 6358 0cc0 10540 1c1 10541 ≤ cle 10679 abscabs 14596 ℋchba 28699 normℎcno 28703 0ℎc0v 28704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-hv0cl 28783 ax-hvmul0 28790 ax-hfi 28859 ax-his3 28864 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-hnorm 28748 |
This theorem is referenced by: nmfnrepnf 29660 |
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