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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 31888 | . 2 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((eigval‘𝑇)‘𝐴) = (((𝑇‘𝐴) ·ih 𝐴) / ((normℎ‘𝐴)↑2))) | |
2 | eleigveccl 31887 | . . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → 𝐴 ∈ ℋ) | |
3 | ffvelcdm 7085 | . . . . 5 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → (𝑇‘𝐴) ∈ ℋ) | |
4 | hicl 31008 | . . . . 5 ⊢ (((𝑇‘𝐴) ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑇‘𝐴) ·ih 𝐴) ∈ ℂ) | |
5 | 3, 4 | sylancom 586 | . . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑇‘𝐴) ·ih 𝐴) ∈ ℂ) |
6 | 2, 5 | syldan 589 | . . 3 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((𝑇‘𝐴) ·ih 𝐴) ∈ ℂ) |
7 | normcl 31053 | . . . . . 6 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ) | |
8 | 7 | recnd 11281 | . . . . 5 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℂ) |
9 | 2, 8 | syl 17 | . . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → (normℎ‘𝐴) ∈ ℂ) |
10 | 9 | sqcld 14155 | . . 3 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((normℎ‘𝐴)↑2) ∈ ℂ) |
11 | eleigvec 31885 | . . . . 5 ⊢ (𝑇: ℋ⟶ ℋ → (𝐴 ∈ (eigvec‘𝑇) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴)))) | |
12 | 11 | biimpa 475 | . . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴))) |
13 | sqne0 14134 | . . . . . . . 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 31058 | . . . . . . 7 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0ℎ)) | |
16 | 14, 15 | bitr2d 279 | . . . . . 6 ⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ ↔ ((normℎ‘𝐴)↑2) ≠ 0)) |
17 | 16 | biimpa 475 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘𝐴)↑2) ≠ 0) |
18 | 17 | 3adant3 1129 | . . . 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 12033 | . 2 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → (((𝑇‘𝐴) ·ih 𝐴) / ((normℎ‘𝐴)↑2)) ∈ ℂ) |
21 | 1, 20 | eqeltrd 2826 | 1 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((eigval‘𝑇)‘𝐴) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∃wrex 3060 ⟶wf 6540 ‘cfv 6544 (class class class)co 7414 ℂcc 11145 0cc0 11147 / cdiv 11910 2c2 12311 ↑cexp 14073 ℋchba 30847 ·ℎ csm 30849 ·ih csp 30850 normℎcno 30851 0ℎc0v 30852 eigveccei 30887 eigvalcel 30888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-inf2 9675 ax-cc 10467 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 ax-pre-sup 11225 ax-addf 11226 ax-mulf 11227 ax-hilex 30927 ax-hfvadd 30928 ax-hvcom 30929 ax-hvass 30930 ax-hv0cl 30931 ax-hvaddid 30932 ax-hfvmul 30933 ax-hvmulid 30934 ax-hvmulass 30935 ax-hvdistr1 30936 ax-hvdistr2 30937 ax-hvmul0 30938 ax-hfi 31007 ax-his1 31010 ax-his2 31011 ax-his3 31012 ax-his4 31013 ax-hcompl 31130 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4907 df-int 4948 df-iun 4996 df-iin 4997 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7680 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-oadd 8490 df-omul 8491 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9397 df-fi 9445 df-sup 9476 df-inf 9477 df-oi 9544 df-card 9973 df-acn 9976 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-div 11911 df-nn 12257 df-2 12319 df-3 12320 df-4 12321 df-5 12322 df-6 12323 df-7 12324 df-8 12325 df-9 12326 df-n0 12517 df-z 12603 df-dec 12722 df-uz 12867 df-q 12977 df-rp 13021 df-xneg 13138 df-xadd 13139 df-xmul 13140 df-ioo 13374 df-ico 13376 df-icc 13377 df-fz 13531 df-fzo 13674 df-fl 13804 df-seq 14014 df-exp 14074 df-hash 14341 df-cj 15097 df-re 15098 df-im 15099 df-sqrt 15233 df-abs 15234 df-clim 15483 df-rlim 15484 df-sum 15684 df-struct 17142 df-sets 17159 df-slot 17177 df-ndx 17189 df-base 17207 df-ress 17236 df-plusg 17272 df-mulr 17273 df-starv 17274 df-sca 17275 df-vsca 17276 df-ip 17277 df-tset 17278 df-ple 17279 df-ds 17281 df-unif 17282 df-hom 17283 df-cco 17284 df-rest 17430 df-topn 17431 df-0g 17449 df-gsum 17450 df-topgen 17451 df-pt 17452 df-prds 17455 df-xrs 17510 df-qtop 17515 df-imas 17516 df-xps 17518 df-mre 17592 df-mrc 17593 df-acs 17595 df-mgm 18626 df-sgrp 18705 df-mnd 18721 df-submnd 18767 df-mulg 19056 df-cntz 19305 df-cmn 19774 df-psmet 21329 df-xmet 21330 df-met 21331 df-bl 21332 df-mopn 21333 df-fbas 21334 df-fg 21335 df-cnfld 21338 df-top 22882 df-topon 22899 df-topsp 22921 df-bases 22935 df-cld 23009 df-ntr 23010 df-cls 23011 df-nei 23088 df-cn 23217 df-cnp 23218 df-lm 23219 df-haus 23305 df-tx 23552 df-hmeo 23745 df-fil 23836 df-fm 23928 df-flim 23929 df-flf 23930 df-xms 24312 df-ms 24313 df-tms 24314 df-cfil 25269 df-cau 25270 df-cmet 25271 df-grpo 30421 df-gid 30422 df-ginv 30423 df-gdiv 30424 df-ablo 30473 df-vc 30487 df-nv 30520 df-va 30523 df-ba 30524 df-sm 30525 df-0v 30526 df-vs 30527 df-nmcv 30528 df-ims 30529 df-dip 30629 df-ssp 30650 df-ph 30741 df-cbn 30791 df-hnorm 30896 df-hba 30897 df-hvsub 30899 df-hlim 30900 df-hcau 30901 df-sh 31135 df-ch 31149 df-oc 31180 df-ch0 31181 df-span 31237 df-eigvec 31781 df-eigval 31782 |
This theorem is referenced by: eighmre 31891 |
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