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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 29537 | . . 3 ⊢ ( ℋ × {0}) ∈ LinFn | |
2 | lnfnf 29436 | . . 3 ⊢ (( ℋ × {0}) ∈ LinFn → ( ℋ × {0}): ℋ⟶ℂ) | |
3 | nmfnval 29428 | . . 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 10427 | . . . . . . . . . . . 12 ⊢ 0 ∈ V | |
6 | 5 | fvconst2 6787 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ ℋ → (( ℋ × {0})‘𝑦) = 0) |
7 | 6 | fveq2d 6497 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℋ → (abs‘(( ℋ × {0})‘𝑦)) = (abs‘0)) |
8 | abs0 14500 | . . . . . . . . . 10 ⊢ (abs‘0) = 0 | |
9 | 7, 8 | syl6eq 2824 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (abs‘(( ℋ × {0})‘𝑦)) = 0) |
10 | 9 | eqeq2d 2782 | . . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝑥 = (abs‘(( ℋ × {0})‘𝑦)) ↔ 𝑥 = 0)) |
11 | 10 | anbi2d 619 | . . . . . . 7 ⊢ (𝑦 ∈ ℋ → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = 0))) |
12 | 11 | rexbiia 3187 | . . . . . 6 ⊢ (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = 0)) |
13 | ax-hv0cl 28553 | . . . . . . . 8 ⊢ 0ℎ ∈ ℋ | |
14 | 0le1 10958 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
15 | fveq2 6493 | . . . . . . . . . . 11 ⊢ (𝑦 = 0ℎ → (normℎ‘𝑦) = (normℎ‘0ℎ)) | |
16 | norm0 28678 | . . . . . . . . . . 11 ⊢ (normℎ‘0ℎ) = 0 | |
17 | 15, 16 | syl6eq 2824 | . . . . . . . . . 10 ⊢ (𝑦 = 0ℎ → (normℎ‘𝑦) = 0) |
18 | 17 | breq1d 4933 | . . . . . . . . 9 ⊢ (𝑦 = 0ℎ → ((normℎ‘𝑦) ≤ 1 ↔ 0 ≤ 1)) |
19 | 18 | rspcev 3529 | . . . . . . . 8 ⊢ ((0ℎ ∈ ℋ ∧ 0 ≤ 1) → ∃𝑦 ∈ ℋ (normℎ‘𝑦) ≤ 1) |
20 | 13, 14, 19 | mp2an 679 | . . . . . . 7 ⊢ ∃𝑦 ∈ ℋ (normℎ‘𝑦) ≤ 1 |
21 | r19.41v 3282 | . . . . . . 7 ⊢ (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = 0) ↔ (∃𝑦 ∈ ℋ (normℎ‘𝑦) ≤ 1 ∧ 𝑥 = 0)) | |
22 | 20, 21 | mpbiran 696 | . . . . . 6 ⊢ (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = 0) ↔ 𝑥 = 0) |
23 | 12, 22 | bitri 267 | . . . . 5 ⊢ (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦))) ↔ 𝑥 = 0) |
24 | 23 | abbii 2838 | . . . 4 ⊢ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦)))} = {𝑥 ∣ 𝑥 = 0} |
25 | df-sn 4436 | . . . 4 ⊢ {0} = {𝑥 ∣ 𝑥 = 0} | |
26 | 24, 25 | eqtr4i 2799 | . . 3 ⊢ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦)))} = {0} |
27 | 26 | supeq1i 8700 | . 2 ⊢ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦)))}, ℝ*, < ) = sup({0}, ℝ*, < ) |
28 | xrltso 12345 | . . 3 ⊢ < Or ℝ* | |
29 | 0xr 10481 | . . 3 ⊢ 0 ∈ ℝ* | |
30 | supsn 8725 | . . 3 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0) | |
31 | 28, 29, 30 | mp2an 679 | . 2 ⊢ sup({0}, ℝ*, < ) = 0 |
32 | 4, 27, 31 | 3eqtri 2800 | 1 ⊢ (normfn‘( ℋ × {0})) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 387 = wceq 1507 ∈ wcel 2050 {cab 2752 ∃wrex 3083 {csn 4435 class class class wbr 4923 Or wor 5319 × cxp 5399 ⟶wf 6178 ‘cfv 6182 supcsup 8693 ℂcc 10327 0cc0 10329 1c1 10330 ℝ*cxr 10467 < clt 10468 ≤ cle 10469 abscabs 14448 ℋchba 28469 normℎcno 28473 0ℎc0v 28474 normfncnmf 28501 LinFnclf 28504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 ax-hilex 28549 ax-hfvadd 28550 ax-hv0cl 28553 ax-hfvmul 28555 ax-hvmul0 28560 ax-hfi 28629 ax-his3 28634 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-2nd 7496 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-er 8083 df-map 8202 df-en 8301 df-dom 8302 df-sdom 8303 df-sup 8695 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-div 11093 df-nn 11434 df-2 11497 df-n0 11702 df-z 11788 df-uz 12053 df-rp 12199 df-seq 13179 df-exp 13239 df-cj 14313 df-re 14314 df-im 14315 df-sqrt 14449 df-abs 14450 df-hnorm 28518 df-nmfn 29397 df-lnfn 29400 |
This theorem is referenced by: nmbdfnlb 29602 branmfn 29657 |
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