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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 29762 | . . 3 ⊢ ( ℋ × {0}) ∈ LinFn | |
2 | lnfnf 29661 | . . 3 ⊢ (( ℋ × {0}) ∈ LinFn → ( ℋ × {0}): ℋ⟶ℂ) | |
3 | nmfnval 29653 | . . 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 10635 | . . . . . . . . . . . 12 ⊢ 0 ∈ V | |
6 | 5 | fvconst2 6966 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ ℋ → (( ℋ × {0})‘𝑦) = 0) |
7 | 6 | fveq2d 6674 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℋ → (abs‘(( ℋ × {0})‘𝑦)) = (abs‘0)) |
8 | abs0 14645 | . . . . . . . . . 10 ⊢ (abs‘0) = 0 | |
9 | 7, 8 | syl6eq 2872 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (abs‘(( ℋ × {0})‘𝑦)) = 0) |
10 | 9 | eqeq2d 2832 | . . . . . . . 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 3246 | . . . . . 6 ⊢ (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = 0)) |
13 | ax-hv0cl 28780 | . . . . . . . 8 ⊢ 0ℎ ∈ ℋ | |
14 | 0le1 11163 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
15 | fveq2 6670 | . . . . . . . . . . 11 ⊢ (𝑦 = 0ℎ → (normℎ‘𝑦) = (normℎ‘0ℎ)) | |
16 | norm0 28905 | . . . . . . . . . . 11 ⊢ (normℎ‘0ℎ) = 0 | |
17 | 15, 16 | syl6eq 2872 | . . . . . . . . . 10 ⊢ (𝑦 = 0ℎ → (normℎ‘𝑦) = 0) |
18 | 17 | breq1d 5076 | . . . . . . . . 9 ⊢ (𝑦 = 0ℎ → ((normℎ‘𝑦) ≤ 1 ↔ 0 ≤ 1)) |
19 | 18 | rspcev 3623 | . . . . . . . 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 3347 | . . . . . . 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 2886 | . . . 4 ⊢ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦)))} = {𝑥 ∣ 𝑥 = 0} |
25 | df-sn 4568 | . . . 4 ⊢ {0} = {𝑥 ∣ 𝑥 = 0} | |
26 | 24, 25 | eqtr4i 2847 | . . 3 ⊢ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦)))} = {0} |
27 | 26 | supeq1i 8911 | . 2 ⊢ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(( ℋ × {0})‘𝑦)))}, ℝ*, < ) = sup({0}, ℝ*, < ) |
28 | xrltso 12535 | . . 3 ⊢ < Or ℝ* | |
29 | 0xr 10688 | . . 3 ⊢ 0 ∈ ℝ* | |
30 | supsn 8936 | . . 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 2848 | 1 ⊢ (normfn‘( ℋ × {0})) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 {cab 2799 ∃wrex 3139 {csn 4567 class class class wbr 5066 Or wor 5473 × cxp 5553 ⟶wf 6351 ‘cfv 6355 supcsup 8904 ℂcc 10535 0cc0 10537 1c1 10538 ℝ*cxr 10674 < clt 10675 ≤ cle 10676 abscabs 14593 ℋchba 28696 normℎcno 28700 0ℎc0v 28701 normfncnmf 28728 LinFnclf 28731 |
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-hilex 28776 ax-hfvadd 28777 ax-hv0cl 28780 ax-hfvmul 28782 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-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 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-abs 14595 df-hnorm 28745 df-nmfn 29622 df-lnfn 29625 |
This theorem is referenced by: nmbdfnlb 29827 branmfn 29882 |
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