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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 31887 | . . 3 ⊢ ( ℋ × {0}) ∈ LinFn | |
2 | lnfnf 31786 | . . 3 ⊢ (( ℋ × {0}) ∈ LinFn → ( ℋ × {0}): ℋ⟶ℂ) | |
3 | nmfnval 31778 | . . 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 11245 | . . . . . . . . . . . 12 ⊢ 0 ∈ V | |
6 | 5 | fvconst2 7216 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ ℋ → (( ℋ × {0})‘𝑦) = 0) |
7 | 6 | fveq2d 6900 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℋ → (abs‘(( ℋ × {0})‘𝑦)) = (abs‘0)) |
8 | abs0 15276 | . . . . . . . . . 10 ⊢ (abs‘0) = 0 | |
9 | 7, 8 | eqtrdi 2781 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (abs‘(( ℋ × {0})‘𝑦)) = 0) |
10 | 9 | eqeq2d 2736 | . . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝑥 = (abs‘(( ℋ × {0})‘𝑦)) ↔ 𝑥 = 0)) |
11 | 10 | anbi2d 628 | . . . . . . 7 ⊢ (𝑦 ∈ ℋ → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = 0))) |
12 | 11 | rexbiia 3081 | . . . . . 6 ⊢ (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = 0)) |
13 | ax-hv0cl 30905 | . . . . . . . 8 ⊢ 0ℎ ∈ ℋ | |
14 | 0le1 11774 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
15 | fveq2 6896 | . . . . . . . . . . 11 ⊢ (𝑦 = 0ℎ → (normℎ‘𝑦) = (normℎ‘0ℎ)) | |
16 | norm0 31030 | . . . . . . . . . . 11 ⊢ (normℎ‘0ℎ) = 0 | |
17 | 15, 16 | eqtrdi 2781 | . . . . . . . . . 10 ⊢ (𝑦 = 0ℎ → (normℎ‘𝑦) = 0) |
18 | 17 | breq1d 5159 | . . . . . . . . 9 ⊢ (𝑦 = 0ℎ → ((normℎ‘𝑦) ≤ 1 ↔ 0 ≤ 1)) |
19 | 18 | rspcev 3606 | . . . . . . . 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 3178 | . . . . . . 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 274 | . . . . 5 ⊢ (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦))) ↔ 𝑥 = 0) |
24 | 23 | abbii 2795 | . . . 4 ⊢ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦)))} = {𝑥 ∣ 𝑥 = 0} |
25 | df-sn 4631 | . . . 4 ⊢ {0} = {𝑥 ∣ 𝑥 = 0} | |
26 | 24, 25 | eqtr4i 2756 | . . 3 ⊢ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦)))} = {0} |
27 | 26 | supeq1i 9477 | . 2 ⊢ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦)))}, ℝ*, < ) = sup({0}, ℝ*, < ) |
28 | xrltso 13160 | . . 3 ⊢ < Or ℝ* | |
29 | 0xr 11298 | . . 3 ⊢ 0 ∈ ℝ* | |
30 | supsn 9502 | . . 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 2757 | 1 ⊢ (normfn‘( ℋ × {0})) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1533 ∈ wcel 2098 {cab 2702 ∃wrex 3059 {csn 4630 class class class wbr 5149 Or wor 5589 × cxp 5676 ⟶wf 6545 ‘cfv 6549 supcsup 9470 ℂcc 11143 0cc0 11145 1c1 11146 ℝ*cxr 11284 < clt 11285 ≤ cle 11286 abscabs 15225 ℋchba 30821 normℎcno 30825 0ℎc0v 30826 normfncnmf 30853 LinFnclf 30856 |
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 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-hilex 30901 ax-hfvadd 30902 ax-hv0cl 30905 ax-hfvmul 30907 ax-hvmul0 30912 ax-hfi 30981 ax-his3 30986 |
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 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9472 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-2 12313 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-seq 14008 df-exp 14068 df-cj 15090 df-re 15091 df-im 15092 df-sqrt 15226 df-abs 15227 df-hnorm 30870 df-nmfn 31747 df-lnfn 31750 |
This theorem is referenced by: nmbdfnlb 31952 branmfn 32007 |
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