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Mirrors > Home > HSE Home > Th. List > eigvalcl | Structured version Visualization version GIF version |
Description: An eigenvalue is a complex number. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
eigvalcl | ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((eigval‘𝑇)‘𝐴) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eigvalval 29533 | . 2 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((eigval‘𝑇)‘𝐴) = (((𝑇‘𝐴) ·ih 𝐴) / ((normℎ‘𝐴)↑2))) | |
2 | eleigveccl 29532 | . . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → 𝐴 ∈ ℋ) | |
3 | ffvelrn 6672 | . . . . 5 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → (𝑇‘𝐴) ∈ ℋ) | |
4 | hicl 28651 | . . . . 5 ⊢ (((𝑇‘𝐴) ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑇‘𝐴) ·ih 𝐴) ∈ ℂ) | |
5 | 3, 4 | sylancom 580 | . . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑇‘𝐴) ·ih 𝐴) ∈ ℂ) |
6 | 2, 5 | syldan 583 | . . 3 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((𝑇‘𝐴) ·ih 𝐴) ∈ ℂ) |
7 | normcl 28696 | . . . . . 6 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ) | |
8 | 7 | recnd 10466 | . . . . 5 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℂ) |
9 | 2, 8 | syl 17 | . . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → (normℎ‘𝐴) ∈ ℂ) |
10 | 9 | sqcld 13321 | . . 3 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((normℎ‘𝐴)↑2) ∈ ℂ) |
11 | eleigvec 29530 | . . . . 5 ⊢ (𝑇: ℋ⟶ ℋ → (𝐴 ∈ (eigvec‘𝑇) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴)))) | |
12 | 11 | biimpa 469 | . . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴))) |
13 | sqne0 13302 | . . . . . . . 8 ⊢ ((normℎ‘𝐴) ∈ ℂ → (((normℎ‘𝐴)↑2) ≠ 0 ↔ (normℎ‘𝐴) ≠ 0)) | |
14 | 8, 13 | syl 17 | . . . . . . 7 ⊢ (𝐴 ∈ ℋ → (((normℎ‘𝐴)↑2) ≠ 0 ↔ (normℎ‘𝐴) ≠ 0)) |
15 | normne0 28701 | . . . . . . 7 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0ℎ)) | |
16 | 14, 15 | bitr2d 272 | . . . . . 6 ⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ ↔ ((normℎ‘𝐴)↑2) ≠ 0)) |
17 | 16 | biimpa 469 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴)↑2) ≠ 0) |
18 | 17 | 3adant3 1113 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴)) → ((normℎ‘𝐴)↑2) ≠ 0) |
19 | 12, 18 | syl 17 | . . 3 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((normℎ‘𝐴)↑2) ≠ 0) |
20 | 6, 10, 19 | divcld 11215 | . 2 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → (((𝑇‘𝐴) ·ih 𝐴) / ((normℎ‘𝐴)↑2)) ∈ ℂ) |
21 | 1, 20 | eqeltrd 2859 | 1 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((eigval‘𝑇)‘𝐴) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ≠ wne 2960 ∃wrex 3082 ⟶wf 6181 ‘cfv 6185 (class class class)co 6974 ℂcc 10331 0cc0 10333 / cdiv 11096 2c2 11493 ↑cexp 13242 ℋchba 28490 ·ℎ csm 28492 ·ih csp 28493 normℎcno 28494 0ℎc0v 28495 eigveccei 28530 eigvalcel 28531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-inf2 8896 ax-cc 9653 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 ax-addf 10412 ax-mulf 10413 ax-hilex 28570 ax-hfvadd 28571 ax-hvcom 28572 ax-hvass 28573 ax-hv0cl 28574 ax-hvaddid 28575 ax-hfvmul 28576 ax-hvmulid 28577 ax-hvmulass 28578 ax-hvdistr1 28579 ax-hvdistr2 28580 ax-hvmul0 28581 ax-hfi 28650 ax-his1 28653 ax-his2 28654 ax-his3 28655 ax-his4 28656 ax-hcompl 28773 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-iin 4791 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-of 7225 df-om 7395 df-1st 7499 df-2nd 7500 df-supp 7632 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-2o 7904 df-oadd 7907 df-omul 7908 df-er 8087 df-map 8206 df-pm 8207 df-ixp 8258 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-fsupp 8627 df-fi 8668 df-sup 8699 df-inf 8700 df-oi 8767 df-card 9160 df-acn 9163 df-cda 9386 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-z 11792 df-dec 11910 df-uz 12057 df-q 12161 df-rp 12203 df-xneg 12322 df-xadd 12323 df-xmul 12324 df-ioo 12556 df-ico 12558 df-icc 12559 df-fz 12707 df-fzo 12848 df-fl 12975 df-seq 13183 df-exp 13243 df-hash 13504 df-cj 14317 df-re 14318 df-im 14319 df-sqrt 14453 df-abs 14454 df-clim 14704 df-rlim 14705 df-sum 14902 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-starv 16434 df-sca 16435 df-vsca 16436 df-ip 16437 df-tset 16438 df-ple 16439 df-ds 16441 df-unif 16442 df-hom 16443 df-cco 16444 df-rest 16550 df-topn 16551 df-0g 16569 df-gsum 16570 df-topgen 16571 df-pt 16572 df-prds 16575 df-xrs 16629 df-qtop 16634 df-imas 16635 df-xps 16637 df-mre 16727 df-mrc 16728 df-acs 16730 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-submnd 17816 df-mulg 18024 df-cntz 18230 df-cmn 18680 df-psmet 20254 df-xmet 20255 df-met 20256 df-bl 20257 df-mopn 20258 df-fbas 20259 df-fg 20260 df-cnfld 20263 df-top 21221 df-topon 21238 df-topsp 21260 df-bases 21273 df-cld 21346 df-ntr 21347 df-cls 21348 df-nei 21425 df-cn 21554 df-cnp 21555 df-lm 21556 df-haus 21642 df-tx 21889 df-hmeo 22082 df-fil 22173 df-fm 22265 df-flim 22266 df-flf 22267 df-xms 22648 df-ms 22649 df-tms 22650 df-cfil 23576 df-cau 23577 df-cmet 23578 df-grpo 28062 df-gid 28063 df-ginv 28064 df-gdiv 28065 df-ablo 28114 df-vc 28128 df-nv 28161 df-va 28164 df-ba 28165 df-sm 28166 df-0v 28167 df-vs 28168 df-nmcv 28169 df-ims 28170 df-dip 28270 df-ssp 28291 df-ph 28382 df-cbn 28433 df-hnorm 28539 df-hba 28540 df-hvsub 28542 df-hlim 28543 df-hcau 28544 df-sh 28778 df-ch 28792 df-oc 28823 df-ch0 28824 df-span 28882 df-eigvec 29426 df-eigval 29427 |
This theorem is referenced by: eighmre 29536 |
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