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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 29622 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 28780 | . . 3 ⊢ 0ℎ ∈ ℋ | |
2 | norm0 28905 | . . . . 5 ⊢ (normℎ‘0ℎ) = 0 | |
3 | 0le1 11163 | . . . . 5 ⊢ 0 ≤ 1 | |
4 | 2, 3 | eqbrtri 5087 | . . . 4 ⊢ (normℎ‘0ℎ) ≤ 1 |
5 | eqid 2821 | . . . 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 6670 | . . . . . 6 ⊢ (𝑦 = 0ℎ → (normℎ‘𝑦) = (normℎ‘0ℎ)) | |
8 | 7 | breq1d 5076 | . . . . 5 ⊢ (𝑦 = 0ℎ → ((normℎ‘𝑦) ≤ 1 ↔ (normℎ‘0ℎ) ≤ 1)) |
9 | 2fveq3 6675 | . . . . . 6 ⊢ (𝑦 = 0ℎ → (abs‘(𝑇‘𝑦)) = (abs‘(𝑇‘0ℎ))) | |
10 | 9 | eqeq2d 2832 | . . . . 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 3623 | . . 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 6683 | . . 3 ⊢ (abs‘(𝑇‘0ℎ)) ∈ V | |
15 | eqeq1 2825 | . . . . 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 3297 | . . 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 3667 | . 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 1537 ∈ wcel 2114 {cab 2799 ∃wrex 3139 class class class wbr 5066 ‘cfv 6355 0cc0 10537 1c1 10538 ≤ cle 10676 abscabs 14593 ℋchba 28696 normℎcno 28700 0ℎc0v 28701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-hv0cl 28780 ax-hvmul0 28787 ax-hfi 28856 ax-his3 28861 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-hnorm 28745 |
This theorem is referenced by: nmfnrepnf 29657 |
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