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Mirrors > Home > HSE Home > Th. List > kbpj | Structured version Visualization version GIF version |
Description: If a vector 𝐴 has norm 1, the outer product ∣ 𝐴〉 〈𝐴 ∣ is the projector onto the subspace spanned by 𝐴. http://en.wikipedia.org/wiki/Bra-ket#Linear%5Foperators. (Contributed by NM, 30-May-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
kbpj | ⊢ ((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) = 1) → (𝐴 ketbra 𝐴) = (projℎ‘(span‘{𝐴}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 6798 | . . . . . . . . 9 ⊢ ((normℎ‘𝐴) = 1 → ((normℎ‘𝐴)↑2) = (1↑2)) | |
2 | sq1 13158 | . . . . . . . . 9 ⊢ (1↑2) = 1 | |
3 | 1, 2 | syl6eq 2821 | . . . . . . . 8 ⊢ ((normℎ‘𝐴) = 1 → ((normℎ‘𝐴)↑2) = 1) |
4 | 3 | oveq2d 6807 | . . . . . . 7 ⊢ ((normℎ‘𝐴) = 1 → ((𝑥 ·ih 𝐴) / ((normℎ‘𝐴)↑2)) = ((𝑥 ·ih 𝐴) / 1)) |
5 | hicl 28270 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝑥 ·ih 𝐴) ∈ ℂ) | |
6 | 5 | ancoms 446 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ·ih 𝐴) ∈ ℂ) |
7 | 6 | div1d 10993 | . . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑥 ·ih 𝐴) / 1) = (𝑥 ·ih 𝐴)) |
8 | 4, 7 | sylan9eqr 2827 | . . . . . 6 ⊢ (((𝐴 ∈ ℋ ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝐴) = 1) → ((𝑥 ·ih 𝐴) / ((normℎ‘𝐴)↑2)) = (𝑥 ·ih 𝐴)) |
9 | 8 | an32s 631 | . . . . 5 ⊢ (((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) = 1) ∧ 𝑥 ∈ ℋ) → ((𝑥 ·ih 𝐴) / ((normℎ‘𝐴)↑2)) = (𝑥 ·ih 𝐴)) |
10 | 9 | oveq1d 6806 | . . . 4 ⊢ (((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) = 1) ∧ 𝑥 ∈ ℋ) → (((𝑥 ·ih 𝐴) / ((normℎ‘𝐴)↑2)) ·ℎ 𝐴) = ((𝑥 ·ih 𝐴) ·ℎ 𝐴)) |
11 | simpll 750 | . . . . 5 ⊢ (((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) = 1) ∧ 𝑥 ∈ ℋ) → 𝐴 ∈ ℋ) | |
12 | simpr 471 | . . . . 5 ⊢ (((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) = 1) ∧ 𝑥 ∈ ℋ) → 𝑥 ∈ ℋ) | |
13 | ax-1ne0 10205 | . . . . . . . . 9 ⊢ 1 ≠ 0 | |
14 | neeq1 3005 | . . . . . . . . 9 ⊢ ((normℎ‘𝐴) = 1 → ((normℎ‘𝐴) ≠ 0 ↔ 1 ≠ 0)) | |
15 | 13, 14 | mpbiri 248 | . . . . . . . 8 ⊢ ((normℎ‘𝐴) = 1 → (normℎ‘𝐴) ≠ 0) |
16 | normne0 28320 | . . . . . . . 8 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0ℎ)) | |
17 | 15, 16 | syl5ib 234 | . . . . . . 7 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) = 1 → 𝐴 ≠ 0ℎ)) |
18 | 17 | imp 393 | . . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) = 1) → 𝐴 ≠ 0ℎ) |
19 | 18 | adantr 466 | . . . . 5 ⊢ (((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) = 1) ∧ 𝑥 ∈ ℋ) → 𝐴 ≠ 0ℎ) |
20 | pjspansn 28769 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝑥 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((projℎ‘(span‘{𝐴}))‘𝑥) = (((𝑥 ·ih 𝐴) / ((normℎ‘𝐴)↑2)) ·ℎ 𝐴)) | |
21 | 11, 12, 19, 20 | syl3anc 1476 | . . . 4 ⊢ (((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) = 1) ∧ 𝑥 ∈ ℋ) → ((projℎ‘(span‘{𝐴}))‘𝑥) = (((𝑥 ·ih 𝐴) / ((normℎ‘𝐴)↑2)) ·ℎ 𝐴)) |
22 | kbval 29146 | . . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ketbra 𝐴)‘𝑥) = ((𝑥 ·ih 𝐴) ·ℎ 𝐴)) | |
23 | 22 | 3anidm12 1529 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ketbra 𝐴)‘𝑥) = ((𝑥 ·ih 𝐴) ·ℎ 𝐴)) |
24 | 23 | adantlr 694 | . . . 4 ⊢ (((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) = 1) ∧ 𝑥 ∈ ℋ) → ((𝐴 ketbra 𝐴)‘𝑥) = ((𝑥 ·ih 𝐴) ·ℎ 𝐴)) |
25 | 10, 21, 24 | 3eqtr4rd 2816 | . . 3 ⊢ (((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) = 1) ∧ 𝑥 ∈ ℋ) → ((𝐴 ketbra 𝐴)‘𝑥) = ((projℎ‘(span‘{𝐴}))‘𝑥)) |
26 | 25 | ralrimiva 3115 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) = 1) → ∀𝑥 ∈ ℋ ((𝐴 ketbra 𝐴)‘𝑥) = ((projℎ‘(span‘{𝐴}))‘𝑥)) |
27 | kbop 29145 | . . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐴 ketbra 𝐴): ℋ⟶ ℋ) | |
28 | 27 | anidms 556 | . . . . 5 ⊢ (𝐴 ∈ ℋ → (𝐴 ketbra 𝐴): ℋ⟶ ℋ) |
29 | ffn 6183 | . . . . 5 ⊢ ((𝐴 ketbra 𝐴): ℋ⟶ ℋ → (𝐴 ketbra 𝐴) Fn ℋ) | |
30 | 28, 29 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℋ → (𝐴 ketbra 𝐴) Fn ℋ) |
31 | spansnch 28752 | . . . . 5 ⊢ (𝐴 ∈ ℋ → (span‘{𝐴}) ∈ Cℋ ) | |
32 | pjfn 28901 | . . . . 5 ⊢ ((span‘{𝐴}) ∈ Cℋ → (projℎ‘(span‘{𝐴})) Fn ℋ) | |
33 | 31, 32 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℋ → (projℎ‘(span‘{𝐴})) Fn ℋ) |
34 | eqfnfv 6452 | . . . 4 ⊢ (((𝐴 ketbra 𝐴) Fn ℋ ∧ (projℎ‘(span‘{𝐴})) Fn ℋ) → ((𝐴 ketbra 𝐴) = (projℎ‘(span‘{𝐴})) ↔ ∀𝑥 ∈ ℋ ((𝐴 ketbra 𝐴)‘𝑥) = ((projℎ‘(span‘{𝐴}))‘𝑥))) | |
35 | 30, 33, 34 | syl2anc 573 | . . 3 ⊢ (𝐴 ∈ ℋ → ((𝐴 ketbra 𝐴) = (projℎ‘(span‘{𝐴})) ↔ ∀𝑥 ∈ ℋ ((𝐴 ketbra 𝐴)‘𝑥) = ((projℎ‘(span‘{𝐴}))‘𝑥))) |
36 | 35 | adantr 466 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) = 1) → ((𝐴 ketbra 𝐴) = (projℎ‘(span‘{𝐴})) ↔ ∀𝑥 ∈ ℋ ((𝐴 ketbra 𝐴)‘𝑥) = ((projℎ‘(span‘{𝐴}))‘𝑥))) |
37 | 26, 36 | mpbird 247 | 1 ⊢ ((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) = 1) → (𝐴 ketbra 𝐴) = (projℎ‘(span‘{𝐴}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∀wral 3061 {csn 4316 Fn wfn 6024 ⟶wf 6025 ‘cfv 6029 (class class class)co 6791 ℂcc 10134 0cc0 10136 1c1 10137 / cdiv 10884 2c2 11270 ↑cexp 13060 ℋchil 28109 ·ℎ csm 28111 ·ih csp 28112 normℎcno 28113 0ℎc0v 28114 Cℋ cch 28119 spancspn 28122 projℎcpjh 28127 ketbra ck 28147 |
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 7094 ax-inf2 8700 ax-cc 9457 ax-cnex 10192 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213 ax-pre-sup 10214 ax-addf 10215 ax-mulf 10216 ax-hilex 28189 ax-hfvadd 28190 ax-hvcom 28191 ax-hvass 28192 ax-hv0cl 28193 ax-hvaddid 28194 ax-hfvmul 28195 ax-hvmulid 28196 ax-hvmulass 28197 ax-hvdistr1 28198 ax-hvdistr2 28199 ax-hvmul0 28200 ax-hfi 28269 ax-his1 28272 ax-his2 28273 ax-his3 28274 ax-his4 28275 ax-hcompl 28392 |
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 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-of 7042 df-om 7211 df-1st 7313 df-2nd 7314 df-supp 7445 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-1o 7711 df-2o 7712 df-oadd 7715 df-omul 7716 df-er 7894 df-map 8009 df-pm 8010 df-ixp 8061 df-en 8108 df-dom 8109 df-sdom 8110 df-fin 8111 df-fsupp 8430 df-fi 8471 df-sup 8502 df-inf 8503 df-oi 8569 df-card 8963 df-acn 8966 df-cda 9190 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-div 10885 df-nn 11221 df-2 11279 df-3 11280 df-4 11281 df-5 11282 df-6 11283 df-7 11284 df-8 11285 df-9 11286 df-n0 11493 df-z 11578 df-dec 11694 df-uz 11887 df-q 11990 df-rp 12029 df-xneg 12144 df-xadd 12145 df-xmul 12146 df-ioo 12377 df-ico 12379 df-icc 12380 df-fz 12527 df-fzo 12667 df-fl 12794 df-seq 13002 df-exp 13061 df-hash 13315 df-cj 14040 df-re 14041 df-im 14042 df-sqrt 14176 df-abs 14177 df-clim 14420 df-rlim 14421 df-sum 14618 df-struct 16059 df-ndx 16060 df-slot 16061 df-base 16063 df-sets 16064 df-ress 16065 df-plusg 16155 df-mulr 16156 df-starv 16157 df-sca 16158 df-vsca 16159 df-ip 16160 df-tset 16161 df-ple 16162 df-ds 16165 df-unif 16166 df-hom 16167 df-cco 16168 df-rest 16284 df-topn 16285 df-0g 16303 df-gsum 16304 df-topgen 16305 df-pt 16306 df-prds 16309 df-xrs 16363 df-qtop 16368 df-imas 16369 df-xps 16371 df-mre 16447 df-mrc 16448 df-acs 16450 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-submnd 17537 df-mulg 17742 df-cntz 17950 df-cmn 18395 df-psmet 19946 df-xmet 19947 df-met 19948 df-bl 19949 df-mopn 19950 df-fbas 19951 df-fg 19952 df-cnfld 19955 df-top 20912 df-topon 20929 df-topsp 20951 df-bases 20964 df-cld 21037 df-ntr 21038 df-cls 21039 df-nei 21116 df-cn 21245 df-cnp 21246 df-lm 21247 df-haus 21333 df-tx 21579 df-hmeo 21772 df-fil 21863 df-fm 21955 df-flim 21956 df-flf 21957 df-xms 22338 df-ms 22339 df-tms 22340 df-cfil 23265 df-cau 23266 df-cmet 23267 df-grpo 27680 df-gid 27681 df-ginv 27682 df-gdiv 27683 df-ablo 27732 df-vc 27747 df-nv 27780 df-va 27783 df-ba 27784 df-sm 27785 df-0v 27786 df-vs 27787 df-nmcv 27788 df-ims 27789 df-dip 27889 df-ssp 27910 df-ph 28001 df-cbn 28052 df-hnorm 28158 df-hba 28159 df-hvsub 28161 df-hlim 28162 df-hcau 28163 df-sh 28397 df-ch 28411 df-oc 28442 df-ch0 28443 df-shs 28500 df-span 28501 df-pjh 28587 df-kb 29043 |
This theorem is referenced by: (None) |
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