Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
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 28844 | . . 3 ⊢ ( ℋ × {0}) ∈ LinFn | |
2 | lnfnf 28743 | . . 3 ⊢ (( ℋ × {0}) ∈ LinFn → ( ℋ × {0}): ℋ⟶ℂ) | |
3 | nmfnval 28735 | . . 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 10034 | . . . . . . . . . . . 12 ⊢ 0 ∈ V | |
6 | 5 | fvconst2 6469 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ ℋ → (( ℋ × {0})‘𝑦) = 0) |
7 | 6 | fveq2d 6195 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℋ → (abs‘(( ℋ × {0})‘𝑦)) = (abs‘0)) |
8 | abs0 14025 | . . . . . . . . . 10 ⊢ (abs‘0) = 0 | |
9 | 7, 8 | syl6eq 2672 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (abs‘(( ℋ × {0})‘𝑦)) = 0) |
10 | 9 | eqeq2d 2632 | . . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝑥 = (abs‘(( ℋ × {0})‘𝑦)) ↔ 𝑥 = 0)) |
11 | 10 | anbi2d 740 | . . . . . . 7 ⊢ (𝑦 ∈ ℋ → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = 0))) |
12 | 11 | rexbiia 3040 | . . . . . 6 ⊢ (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = 0)) |
13 | ax-hv0cl 27860 | . . . . . . . 8 ⊢ 0ℎ ∈ ℋ | |
14 | 0le1 10551 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
15 | fveq2 6191 | . . . . . . . . . . 11 ⊢ (𝑦 = 0ℎ → (normℎ‘𝑦) = (normℎ‘0ℎ)) | |
16 | norm0 27985 | . . . . . . . . . . 11 ⊢ (normℎ‘0ℎ) = 0 | |
17 | 15, 16 | syl6eq 2672 | . . . . . . . . . 10 ⊢ (𝑦 = 0ℎ → (normℎ‘𝑦) = 0) |
18 | 17 | breq1d 4663 | . . . . . . . . 9 ⊢ (𝑦 = 0ℎ → ((normℎ‘𝑦) ≤ 1 ↔ 0 ≤ 1)) |
19 | 18 | rspcev 3309 | . . . . . . . 8 ⊢ ((0ℎ ∈ ℋ ∧ 0 ≤ 1) → ∃𝑦 ∈ ℋ (normℎ‘𝑦) ≤ 1) |
20 | 13, 14, 19 | mp2an 708 | . . . . . . 7 ⊢ ∃𝑦 ∈ ℋ (normℎ‘𝑦) ≤ 1 |
21 | r19.41v 3089 | . . . . . . 7 ⊢ (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = 0) ↔ (∃𝑦 ∈ ℋ (normℎ‘𝑦) ≤ 1 ∧ 𝑥 = 0)) | |
22 | 20, 21 | mpbiran 953 | . . . . . 6 ⊢ (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = 0) ↔ 𝑥 = 0) |
23 | 12, 22 | bitri 264 | . . . . 5 ⊢ (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦))) ↔ 𝑥 = 0) |
24 | 23 | abbii 2739 | . . . 4 ⊢ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦)))} = {𝑥 ∣ 𝑥 = 0} |
25 | df-sn 4178 | . . . 4 ⊢ {0} = {𝑥 ∣ 𝑥 = 0} | |
26 | 24, 25 | eqtr4i 2647 | . . 3 ⊢ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦)))} = {0} |
27 | 26 | supeq1i 8353 | . 2 ⊢ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦)))}, ℝ*, < ) = sup({0}, ℝ*, < ) |
28 | xrltso 11974 | . . 3 ⊢ < Or ℝ* | |
29 | 0xr 10086 | . . 3 ⊢ 0 ∈ ℝ* | |
30 | supsn 8378 | . . 3 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0) | |
31 | 28, 29, 30 | mp2an 708 | . 2 ⊢ sup({0}, ℝ*, < ) = 0 |
32 | 4, 27, 31 | 3eqtri 2648 | 1 ⊢ (normfn‘( ℋ × {0})) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∈ wcel 1990 {cab 2608 ∃wrex 2913 {csn 4177 class class class wbr 4653 Or wor 5034 × cxp 5112 ⟶wf 5884 ‘cfv 5888 supcsup 8346 ℂcc 9934 0cc0 9936 1c1 9937 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 abscabs 13974 ℋchil 27776 normℎcno 27780 0ℎc0v 27781 normfncnmf 27808 LinFnclf 27811 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-hilex 27856 ax-hfvadd 27857 ax-hv0cl 27860 ax-hfvmul 27862 ax-hvmul0 27867 ax-hfi 27936 ax-his3 27941 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-hnorm 27825 df-nmfn 28704 df-lnfn 28707 |
This theorem is referenced by: nmbdfnlb 28909 branmfn 28964 |
Copyright terms: Public domain | W3C validator |