Hilbert Space Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  HSE Home  >  Th. List  >  eigorthi Structured version   Visualization version   GIF version

Theorem eigorthi 29608
 Description: A necessary and sufficient condition (that holds when 𝑇 is a Hermitian operator) for two eigenvectors 𝐴 and 𝐵 to be orthogonal. Generalization of Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Jan-2005.) (New usage is discouraged.)
Hypotheses
Ref Expression
eigorthi.1 𝐴 ∈ ℋ
eigorthi.2 𝐵 ∈ ℋ
eigorthi.3 𝐶 ∈ ℂ
eigorthi.4 𝐷 ∈ ℂ
Assertion
Ref Expression
eigorthi ((((𝑇𝐴) = (𝐶 · 𝐴) ∧ (𝑇𝐵) = (𝐷 · 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) → ((𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵) ↔ (𝐴 ·ih 𝐵) = 0))

Proof of Theorem eigorthi
StepHypRef Expression
1 oveq2 7158 . . . 4 ((𝑇𝐵) = (𝐷 · 𝐵) → (𝐴 ·ih (𝑇𝐵)) = (𝐴 ·ih (𝐷 · 𝐵)))
2 eigorthi.4 . . . . 5 𝐷 ∈ ℂ
3 eigorthi.1 . . . . 5 𝐴 ∈ ℋ
4 eigorthi.2 . . . . 5 𝐵 ∈ ℋ
5 his5 28857 . . . . 5 ((𝐷 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝐷 · 𝐵)) = ((∗‘𝐷) · (𝐴 ·ih 𝐵)))
62, 3, 4, 5mp3an 1457 . . . 4 (𝐴 ·ih (𝐷 · 𝐵)) = ((∗‘𝐷) · (𝐴 ·ih 𝐵))
71, 6syl6eq 2872 . . 3 ((𝑇𝐵) = (𝐷 · 𝐵) → (𝐴 ·ih (𝑇𝐵)) = ((∗‘𝐷) · (𝐴 ·ih 𝐵)))
8 oveq1 7157 . . . 4 ((𝑇𝐴) = (𝐶 · 𝐴) → ((𝑇𝐴) ·ih 𝐵) = ((𝐶 · 𝐴) ·ih 𝐵))
9 eigorthi.3 . . . . 5 𝐶 ∈ ℂ
10 ax-his3 28855 . . . . 5 ((𝐶 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐶 · 𝐴) ·ih 𝐵) = (𝐶 · (𝐴 ·ih 𝐵)))
119, 3, 4, 10mp3an 1457 . . . 4 ((𝐶 · 𝐴) ·ih 𝐵) = (𝐶 · (𝐴 ·ih 𝐵))
128, 11syl6eq 2872 . . 3 ((𝑇𝐴) = (𝐶 · 𝐴) → ((𝑇𝐴) ·ih 𝐵) = (𝐶 · (𝐴 ·ih 𝐵)))
137, 12eqeqan12rd 2840 . 2 (((𝑇𝐴) = (𝐶 · 𝐴) ∧ (𝑇𝐵) = (𝐷 · 𝐵)) → ((𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵) ↔ ((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵))))
143, 4hicli 28852 . . . . . . . 8 (𝐴 ·ih 𝐵) ∈ ℂ
152cjcli 14522 . . . . . . . . 9 (∗‘𝐷) ∈ ℂ
16 mulcan2 11272 . . . . . . . . 9 (((∗‘𝐷) ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ ((𝐴 ·ih 𝐵) ∈ ℂ ∧ (𝐴 ·ih 𝐵) ≠ 0)) → (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) ↔ (∗‘𝐷) = 𝐶))
1715, 9, 16mp3an12 1447 . . . . . . . 8 (((𝐴 ·ih 𝐵) ∈ ℂ ∧ (𝐴 ·ih 𝐵) ≠ 0) → (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) ↔ (∗‘𝐷) = 𝐶))
1814, 17mpan 688 . . . . . . 7 ((𝐴 ·ih 𝐵) ≠ 0 → (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) ↔ (∗‘𝐷) = 𝐶))
19 eqcom 2828 . . . . . . 7 ((∗‘𝐷) = 𝐶𝐶 = (∗‘𝐷))
2018, 19syl6bb 289 . . . . . 6 ((𝐴 ·ih 𝐵) ≠ 0 → (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) ↔ 𝐶 = (∗‘𝐷)))
2120biimpcd 251 . . . . 5 (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) → ((𝐴 ·ih 𝐵) ≠ 0 → 𝐶 = (∗‘𝐷)))
2221necon1d 3038 . . . 4 (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) → (𝐶 ≠ (∗‘𝐷) → (𝐴 ·ih 𝐵) = 0))
2322com12 32 . . 3 (𝐶 ≠ (∗‘𝐷) → (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) → (𝐴 ·ih 𝐵) = 0))
24 oveq2 7158 . . . 4 ((𝐴 ·ih 𝐵) = 0 → ((∗‘𝐷) · (𝐴 ·ih 𝐵)) = ((∗‘𝐷) · 0))
25 oveq2 7158 . . . . 5 ((𝐴 ·ih 𝐵) = 0 → (𝐶 · (𝐴 ·ih 𝐵)) = (𝐶 · 0))
269mul01i 10824 . . . . . 6 (𝐶 · 0) = 0
2715mul01i 10824 . . . . . 6 ((∗‘𝐷) · 0) = 0
2826, 27eqtr4i 2847 . . . . 5 (𝐶 · 0) = ((∗‘𝐷) · 0)
2925, 28syl6eq 2872 . . . 4 ((𝐴 ·ih 𝐵) = 0 → (𝐶 · (𝐴 ·ih 𝐵)) = ((∗‘𝐷) · 0))
3024, 29eqtr4d 2859 . . 3 ((𝐴 ·ih 𝐵) = 0 → ((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)))
3123, 30impbid1 227 . 2 (𝐶 ≠ (∗‘𝐷) → (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) ↔ (𝐴 ·ih 𝐵) = 0))
3213, 31sylan9bb 512 1 ((((𝑇𝐴) = (𝐶 · 𝐴) ∧ (𝑇𝐵) = (𝐷 · 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) → ((𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵) ↔ (𝐴 ·ih 𝐵) = 0))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ‘cfv 6350  (class class class)co 7150  ℂcc 10529  0cc0 10531   · cmul 10536  ∗ccj 14449   ℋchba 28690   ·ℎ csm 28692   ·ih csp 28693 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-hfvmul 28776  ax-hfi 28850  ax-his1 28853  ax-his3 28855 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  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 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-po 5469  df-so 5470  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-2 11694  df-cj 14452  df-re 14453  df-im 14454 This theorem is referenced by:  eigorth  29609
 Copyright terms: Public domain W3C validator