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Mirrors > Home > HSE Home > Th. List > nmop0 | Structured version Visualization version GIF version |
Description: The norm of the zero operator is zero. (Contributed by NM, 8-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmop0 | ⊢ (normop‘ 0hop ) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ho0f 29528 | . . 3 ⊢ 0hop : ℋ⟶ ℋ | |
2 | nmopval 29633 | . . 3 ⊢ ( 0hop : ℋ⟶ ℋ → (normop‘ 0hop ) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘( 0hop ‘𝑦)))}, ℝ*, < )) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (normop‘ 0hop ) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘( 0hop ‘𝑦)))}, ℝ*, < ) |
4 | ho0val 29527 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ ℋ → ( 0hop ‘𝑦) = 0ℎ) | |
5 | 4 | fveq2d 6674 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℋ → (normℎ‘( 0hop ‘𝑦)) = (normℎ‘0ℎ)) |
6 | norm0 28905 | . . . . . . . . . 10 ⊢ (normℎ‘0ℎ) = 0 | |
7 | 5, 6 | syl6eq 2872 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (normℎ‘( 0hop ‘𝑦)) = 0) |
8 | 7 | eqeq2d 2832 | . . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝑥 = (normℎ‘( 0hop ‘𝑦)) ↔ 𝑥 = 0)) |
9 | 8 | anbi2d 630 | . . . . . . 7 ⊢ (𝑦 ∈ ℋ → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘( 0hop ‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = 0))) |
10 | 9 | rexbiia 3246 | . . . . . 6 ⊢ (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘( 0hop ‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = 0)) |
11 | ax-hv0cl 28780 | . . . . . . . 8 ⊢ 0ℎ ∈ ℋ | |
12 | 0le1 11163 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
13 | fveq2 6670 | . . . . . . . . . . 11 ⊢ (𝑦 = 0ℎ → (normℎ‘𝑦) = (normℎ‘0ℎ)) | |
14 | 13, 6 | syl6eq 2872 | . . . . . . . . . 10 ⊢ (𝑦 = 0ℎ → (normℎ‘𝑦) = 0) |
15 | 14 | breq1d 5076 | . . . . . . . . 9 ⊢ (𝑦 = 0ℎ → ((normℎ‘𝑦) ≤ 1 ↔ 0 ≤ 1)) |
16 | 15 | rspcev 3623 | . . . . . . . 8 ⊢ ((0ℎ ∈ ℋ ∧ 0 ≤ 1) → ∃𝑦 ∈ ℋ (normℎ‘𝑦) ≤ 1) |
17 | 11, 12, 16 | mp2an 690 | . . . . . . 7 ⊢ ∃𝑦 ∈ ℋ (normℎ‘𝑦) ≤ 1 |
18 | r19.41v 3347 | . . . . . . 7 ⊢ (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = 0) ↔ (∃𝑦 ∈ ℋ (normℎ‘𝑦) ≤ 1 ∧ 𝑥 = 0)) | |
19 | 17, 18 | mpbiran 707 | . . . . . 6 ⊢ (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = 0) ↔ 𝑥 = 0) |
20 | 10, 19 | bitri 277 | . . . . 5 ⊢ (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘( 0hop ‘𝑦))) ↔ 𝑥 = 0) |
21 | 20 | abbii 2886 | . . . 4 ⊢ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘( 0hop ‘𝑦)))} = {𝑥 ∣ 𝑥 = 0} |
22 | df-sn 4568 | . . . 4 ⊢ {0} = {𝑥 ∣ 𝑥 = 0} | |
23 | 21, 22 | eqtr4i 2847 | . . 3 ⊢ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘( 0hop ‘𝑦)))} = {0} |
24 | 23 | supeq1i 8911 | . 2 ⊢ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘( 0hop ‘𝑦)))}, ℝ*, < ) = sup({0}, ℝ*, < ) |
25 | xrltso 12535 | . . 3 ⊢ < Or ℝ* | |
26 | 0xr 10688 | . . 3 ⊢ 0 ∈ ℝ* | |
27 | supsn 8936 | . . 3 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0) | |
28 | 25, 26, 27 | mp2an 690 | . 2 ⊢ sup({0}, ℝ*, < ) = 0 |
29 | 3, 24, 28 | 3eqtri 2848 | 1 ⊢ (normop‘ 0hop ) = 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 ⟶wf 6351 ‘cfv 6355 supcsup 8904 0cc0 10537 1c1 10538 ℝ*cxr 10674 < clt 10675 ≤ cle 10676 ℋchba 28696 normℎcno 28700 0ℎc0v 28701 0hop ch0o 28720 normopcnop 28722 |
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-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cc 9857 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-pre-sup 10615 ax-addf 10616 ax-mulf 10617 ax-hilex 28776 ax-hfvadd 28777 ax-hvcom 28778 ax-hvass 28779 ax-hv0cl 28780 ax-hvaddid 28781 ax-hfvmul 28782 ax-hvmulid 28783 ax-hvmulass 28784 ax-hvdistr1 28785 ax-hvdistr2 28786 ax-hvmul0 28787 ax-hfi 28856 ax-his1 28859 ax-his2 28860 ax-his3 28861 ax-his4 28862 ax-hcompl 28979 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 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-int 4877 df-iun 4921 df-iin 4922 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-se 5515 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-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-omul 8107 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-acn 9371 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-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-rlim 14846 df-sum 15043 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-cn 21835 df-cnp 21836 df-lm 21837 df-haus 21923 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cfil 23858 df-cau 23859 df-cmet 23860 df-grpo 28270 df-gid 28271 df-ginv 28272 df-gdiv 28273 df-ablo 28322 df-vc 28336 df-nv 28369 df-va 28372 df-ba 28373 df-sm 28374 df-0v 28375 df-vs 28376 df-nmcv 28377 df-ims 28378 df-dip 28478 df-ssp 28499 df-ph 28590 df-cbn 28640 df-hnorm 28745 df-hba 28746 df-hvsub 28748 df-hlim 28749 df-hcau 28750 df-sh 28984 df-ch 28998 df-oc 29029 df-ch0 29030 df-shs 29085 df-pjh 29172 df-h0op 29525 df-nmop 29616 |
This theorem is referenced by: nmop0h 29768 0bdop 29770 nmlnop0iALT 29772 pjbdlni 29926 |
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