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Mirrors > Home > HSE Home > Th. List > ho02i | Structured version Visualization version GIF version |
Description: A condition implying that a Hilbert space operator is identically zero. Lemma 3.2(S10) of [Beran] p. 95. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ho0.1 | ⊢ 𝑇: ℋ⟶ ℋ |
Ref | Expression |
---|---|
ho02i | ⊢ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = 0 ↔ 𝑇 = 0hop ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralcom 3246 | . 2 ⊢ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = 0 ↔ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = 0) | |
2 | ho0.1 | . . . . 5 ⊢ 𝑇: ℋ⟶ ℋ | |
3 | 2 | ffvelrni 6500 | . . . 4 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℋ) |
4 | hial02 28296 | . . . . 5 ⊢ ((𝑇‘𝑦) ∈ ℋ → (∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = 0 ↔ (𝑇‘𝑦) = 0ℎ)) | |
5 | hial0 28295 | . . . . 5 ⊢ ((𝑇‘𝑦) ∈ ℋ → (∀𝑥 ∈ ℋ ((𝑇‘𝑦) ·ih 𝑥) = 0 ↔ (𝑇‘𝑦) = 0ℎ)) | |
6 | 4, 5 | bitr4d 271 | . . . 4 ⊢ ((𝑇‘𝑦) ∈ ℋ → (∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = 0 ↔ ∀𝑥 ∈ ℋ ((𝑇‘𝑦) ·ih 𝑥) = 0)) |
7 | 3, 6 | syl 17 | . . 3 ⊢ (𝑦 ∈ ℋ → (∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = 0 ↔ ∀𝑥 ∈ ℋ ((𝑇‘𝑦) ·ih 𝑥) = 0)) |
8 | 7 | ralbiia 3128 | . 2 ⊢ (∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = 0 ↔ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ ((𝑇‘𝑦) ·ih 𝑥) = 0) |
9 | 2 | ho01i 29023 | . 2 ⊢ (∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ ((𝑇‘𝑦) ·ih 𝑥) = 0 ↔ 𝑇 = 0hop ) |
10 | 1, 8, 9 | 3bitri 286 | 1 ⊢ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = 0 ↔ 𝑇 = 0hop ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1631 ∈ wcel 2145 ∀wral 3061 ⟶wf 6025 ‘cfv 6029 (class class class)co 6792 0cc0 10138 ℋchil 28112 ·ih csp 28115 0ℎc0v 28117 0hop ch0o 28136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 ax-cc 9459 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 ax-addf 10217 ax-mulf 10218 ax-hilex 28192 ax-hfvadd 28193 ax-hvcom 28194 ax-hvass 28195 ax-hv0cl 28196 ax-hvaddid 28197 ax-hfvmul 28198 ax-hvmulid 28199 ax-hvmulass 28200 ax-hvdistr1 28201 ax-hvdistr2 28202 ax-hvmul0 28203 ax-hfi 28272 ax-his1 28275 ax-his2 28276 ax-his3 28277 ax-his4 28278 ax-hcompl 28395 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-isom 6038 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-of 7044 df-om 7213 df-1st 7315 df-2nd 7316 df-supp 7447 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-2o 7714 df-oadd 7717 df-omul 7718 df-er 7896 df-map 8011 df-pm 8012 df-ixp 8063 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-fsupp 8432 df-fi 8473 df-sup 8504 df-inf 8505 df-oi 8571 df-card 8965 df-acn 8968 df-cda 9192 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-n0 11496 df-z 11581 df-dec 11697 df-uz 11890 df-q 11993 df-rp 12032 df-xneg 12147 df-xadd 12148 df-xmul 12149 df-ioo 12380 df-ico 12382 df-icc 12383 df-fz 12530 df-fzo 12670 df-fl 12797 df-seq 13005 df-exp 13064 df-hash 13318 df-cj 14043 df-re 14044 df-im 14045 df-sqrt 14179 df-abs 14180 df-clim 14423 df-rlim 14424 df-sum 14621 df-struct 16062 df-ndx 16063 df-slot 16064 df-base 16066 df-sets 16067 df-ress 16068 df-plusg 16158 df-mulr 16159 df-starv 16160 df-sca 16161 df-vsca 16162 df-ip 16163 df-tset 16164 df-ple 16165 df-ds 16168 df-unif 16169 df-hom 16170 df-cco 16171 df-rest 16287 df-topn 16288 df-0g 16306 df-gsum 16307 df-topgen 16308 df-pt 16309 df-prds 16312 df-xrs 16366 df-qtop 16371 df-imas 16372 df-xps 16374 df-mre 16450 df-mrc 16451 df-acs 16453 df-mgm 17446 df-sgrp 17488 df-mnd 17499 df-submnd 17540 df-mulg 17745 df-cntz 17953 df-cmn 18398 df-psmet 19949 df-xmet 19950 df-met 19951 df-bl 19952 df-mopn 19953 df-fbas 19954 df-fg 19955 df-cnfld 19958 df-top 20915 df-topon 20932 df-topsp 20954 df-bases 20967 df-cld 21040 df-ntr 21041 df-cls 21042 df-nei 21119 df-cn 21248 df-cnp 21249 df-lm 21250 df-haus 21336 df-tx 21582 df-hmeo 21775 df-fil 21866 df-fm 21958 df-flim 21959 df-flf 21960 df-xms 22341 df-ms 22342 df-tms 22343 df-cfil 23268 df-cau 23269 df-cmet 23270 df-grpo 27683 df-gid 27684 df-ginv 27685 df-gdiv 27686 df-ablo 27735 df-vc 27750 df-nv 27783 df-va 27786 df-ba 27787 df-sm 27788 df-0v 27789 df-vs 27790 df-nmcv 27791 df-ims 27792 df-dip 27892 df-ssp 27913 df-ph 28004 df-cbn 28055 df-hnorm 28161 df-hba 28162 df-hvsub 28164 df-hlim 28165 df-hcau 28166 df-sh 28400 df-ch 28414 df-oc 28445 df-ch0 28446 df-shs 28503 df-pjh 28590 df-h0op 28943 |
This theorem is referenced by: (None) |
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