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Mirrors > Home > HSE Home > Th. List > nmfn0 | Structured version Visualization version GIF version |
Description: The norm of the identically zero functional is zero. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmfn0 | ⊢ (normfn‘( ℋ × {0})) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0lnfn 29764 | . . 3 ⊢ ( ℋ × {0}) ∈ LinFn | |
2 | lnfnf 29663 | . . 3 ⊢ (( ℋ × {0}) ∈ LinFn → ( ℋ × {0}): ℋ⟶ℂ) | |
3 | nmfnval 29655 | . . 3 ⊢ (( ℋ × {0}): ℋ⟶ℂ → (normfn‘( ℋ × {0})) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦)))}, ℝ*, < )) | |
4 | 1, 2, 3 | mp2b 10 | . 2 ⊢ (normfn‘( ℋ × {0})) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦)))}, ℝ*, < ) |
5 | c0ex 10637 | . . . . . . . . . . . 12 ⊢ 0 ∈ V | |
6 | 5 | fvconst2 6968 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ ℋ → (( ℋ × {0})‘𝑦) = 0) |
7 | 6 | fveq2d 6676 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℋ → (abs‘(( ℋ × {0})‘𝑦)) = (abs‘0)) |
8 | abs0 14647 | . . . . . . . . . 10 ⊢ (abs‘0) = 0 | |
9 | 7, 8 | syl6eq 2874 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (abs‘(( ℋ × {0})‘𝑦)) = 0) |
10 | 9 | eqeq2d 2834 | . . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝑥 = (abs‘(( ℋ × {0})‘𝑦)) ↔ 𝑥 = 0)) |
11 | 10 | anbi2d 630 | . . . . . . 7 ⊢ (𝑦 ∈ ℋ → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = 0))) |
12 | 11 | rexbiia 3248 | . . . . . 6 ⊢ (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = 0)) |
13 | ax-hv0cl 28782 | . . . . . . . 8 ⊢ 0ℎ ∈ ℋ | |
14 | 0le1 11165 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
15 | fveq2 6672 | . . . . . . . . . . 11 ⊢ (𝑦 = 0ℎ → (normℎ‘𝑦) = (normℎ‘0ℎ)) | |
16 | norm0 28907 | . . . . . . . . . . 11 ⊢ (normℎ‘0ℎ) = 0 | |
17 | 15, 16 | syl6eq 2874 | . . . . . . . . . 10 ⊢ (𝑦 = 0ℎ → (normℎ‘𝑦) = 0) |
18 | 17 | breq1d 5078 | . . . . . . . . 9 ⊢ (𝑦 = 0ℎ → ((normℎ‘𝑦) ≤ 1 ↔ 0 ≤ 1)) |
19 | 18 | rspcev 3625 | . . . . . . . 8 ⊢ ((0ℎ ∈ ℋ ∧ 0 ≤ 1) → ∃𝑦 ∈ ℋ (normℎ‘𝑦) ≤ 1) |
20 | 13, 14, 19 | mp2an 690 | . . . . . . 7 ⊢ ∃𝑦 ∈ ℋ (normℎ‘𝑦) ≤ 1 |
21 | r19.41v 3349 | . . . . . . 7 ⊢ (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = 0) ↔ (∃𝑦 ∈ ℋ (normℎ‘𝑦) ≤ 1 ∧ 𝑥 = 0)) | |
22 | 20, 21 | mpbiran 707 | . . . . . 6 ⊢ (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = 0) ↔ 𝑥 = 0) |
23 | 12, 22 | bitri 277 | . . . . 5 ⊢ (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦))) ↔ 𝑥 = 0) |
24 | 23 | abbii 2888 | . . . 4 ⊢ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦)))} = {𝑥 ∣ 𝑥 = 0} |
25 | df-sn 4570 | . . . 4 ⊢ {0} = {𝑥 ∣ 𝑥 = 0} | |
26 | 24, 25 | eqtr4i 2849 | . . 3 ⊢ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦)))} = {0} |
27 | 26 | supeq1i 8913 | . 2 ⊢ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦)))}, ℝ*, < ) = sup({0}, ℝ*, < ) |
28 | xrltso 12537 | . . 3 ⊢ < Or ℝ* | |
29 | 0xr 10690 | . . 3 ⊢ 0 ∈ ℝ* | |
30 | supsn 8938 | . . 3 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0) | |
31 | 28, 29, 30 | mp2an 690 | . 2 ⊢ sup({0}, ℝ*, < ) = 0 |
32 | 4, 27, 31 | 3eqtri 2850 | 1 ⊢ (normfn‘( ℋ × {0})) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 {cab 2801 ∃wrex 3141 {csn 4569 class class class wbr 5068 Or wor 5475 × cxp 5555 ⟶wf 6353 ‘cfv 6357 supcsup 8906 ℂcc 10537 0cc0 10539 1c1 10540 ℝ*cxr 10676 < clt 10677 ≤ cle 10678 abscabs 14595 ℋchba 28698 normℎcno 28702 0ℎc0v 28703 normfncnmf 28730 LinFnclf 28733 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-hilex 28778 ax-hfvadd 28779 ax-hv0cl 28782 ax-hfvmul 28784 ax-hvmul0 28789 ax-hfi 28858 ax-his3 28863 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-hnorm 28747 df-nmfn 29624 df-lnfn 29627 |
This theorem is referenced by: nmbdfnlb 29829 branmfn 29884 |
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