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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 31699 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 30857 | . . 3 ⊢ 0ℎ ∈ ℋ | |
2 | norm0 30982 | . . . . 5 ⊢ (normℎ‘0ℎ) = 0 | |
3 | 0le1 11767 | . . . . 5 ⊢ 0 ≤ 1 | |
4 | 2, 3 | eqbrtri 5164 | . . . 4 ⊢ (normℎ‘0ℎ) ≤ 1 |
5 | eqid 2725 | . . . 4 ⊢ (abs‘(𝑇‘0ℎ)) = (abs‘(𝑇‘0ℎ)) | |
6 | 4, 5 | pm3.2i 469 | . . 3 ⊢ ((normℎ‘0ℎ) ≤ 1 ∧ (abs‘(𝑇‘0ℎ)) = (abs‘(𝑇‘0ℎ))) |
7 | fveq2 6892 | . . . . . 6 ⊢ (𝑦 = 0ℎ → (normℎ‘𝑦) = (normℎ‘0ℎ)) | |
8 | 7 | breq1d 5153 | . . . . 5 ⊢ (𝑦 = 0ℎ → ((normℎ‘𝑦) ≤ 1 ↔ (normℎ‘0ℎ) ≤ 1)) |
9 | 2fveq3 6897 | . . . . . 6 ⊢ (𝑦 = 0ℎ → (abs‘(𝑇‘𝑦)) = (abs‘(𝑇‘0ℎ))) | |
10 | 9 | eqeq2d 2736 | . . . . 5 ⊢ (𝑦 = 0ℎ → ((abs‘(𝑇‘0ℎ)) = (abs‘(𝑇‘𝑦)) ↔ (abs‘(𝑇‘0ℎ)) = (abs‘(𝑇‘0ℎ)))) |
11 | 8, 10 | anbi12d 630 | . . . 4 ⊢ (𝑦 = 0ℎ → (((normℎ‘𝑦) ≤ 1 ∧ (abs‘(𝑇‘0ℎ)) = (abs‘(𝑇‘𝑦))) ↔ ((normℎ‘0ℎ) ≤ 1 ∧ (abs‘(𝑇‘0ℎ)) = (abs‘(𝑇‘0ℎ))))) |
12 | 11 | rspcev 3601 | . . 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 6905 | . . 3 ⊢ (abs‘(𝑇‘0ℎ)) ∈ V | |
15 | eqeq1 2729 | . . . . 5 ⊢ (𝑥 = (abs‘(𝑇‘0ℎ)) → (𝑥 = (abs‘(𝑇‘𝑦)) ↔ (abs‘(𝑇‘0ℎ)) = (abs‘(𝑇‘𝑦)))) | |
16 | 15 | anbi2d 628 | . . . 4 ⊢ (𝑥 = (abs‘(𝑇‘0ℎ)) → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ (abs‘(𝑇‘0ℎ)) = (abs‘(𝑇‘𝑦))))) |
17 | 16 | rexbidv 3169 | . . 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 3659 | . 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 230 | 1 ⊢ (abs‘(𝑇‘0ℎ)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1533 ∈ wcel 2098 {cab 2702 ∃wrex 3060 class class class wbr 5143 ‘cfv 6543 0cc0 11138 1c1 11139 ≤ cle 11279 abscabs 15213 ℋchba 30773 normℎcno 30777 0ℎc0v 30778 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-hv0cl 30857 ax-hvmul0 30864 ax-hfi 30933 ax-his3 30938 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-seq 13999 df-exp 14059 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-hnorm 30822 |
This theorem is referenced by: nmfnrepnf 31734 |
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