Theorem eigre 29615

Description: A necessary and sufficient condition (that holds when 𝑇 is a Hermitian operator) for an eigenvalue 𝐵 to be real. Generalization of Equation 1.30 of [Hughes] p. 49. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
eigre	⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → ((𝐴 ·ih (𝑇‘𝐴)) = ((𝑇‘𝐴) ·ih 𝐴) ↔ 𝐵 ∈ ℝ))

Proof of Theorem eigre

Step	Hyp	Ref	Expression
1		fveq2 6673	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (𝑇‘𝐴) = (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)))
2		oveq2 7167	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (𝐵 ·ℎ 𝐴) = (𝐵 ·ℎ if(𝐴 ∈ ℋ, 𝐴, 0ℎ)))
3	1, 2	eqeq12d 2840	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ↔ (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) = (𝐵 ·ℎ if(𝐴 ∈ ℋ, 𝐴, 0ℎ))))
4		neeq1 3081	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (𝐴 ≠ 0ℎ ↔ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ≠ 0ℎ))
5	3, 4	anbi12d 632	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ) ↔ ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) = (𝐵 ·ℎ if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ∧ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ≠ 0ℎ)))
6		id 22	. . . . . . 7 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → 𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ))
7	6, 1	oveq12d 7177	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (𝐴 ·ih (𝑇‘𝐴)) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ))))
8	1, 6	oveq12d 7177	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → ((𝑇‘𝐴) ·ih 𝐴) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ·ih if(𝐴 ∈ ℋ, 𝐴, 0ℎ)))
9	7, 8	eqeq12d 2840	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → ((𝐴 ·ih (𝑇‘𝐴)) = ((𝑇‘𝐴) ·ih 𝐴) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ·ih if(𝐴 ∈ ℋ, 𝐴, 0ℎ))))
10	9	bibi1d 346	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (((𝐴 ·ih (𝑇‘𝐴)) = ((𝑇‘𝐴) ·ih 𝐴) ↔ 𝐵 ∈ ℝ) ↔ ((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ·ih if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ↔ 𝐵 ∈ ℝ)))
11	5, 10	imbi12d 347	. . 3 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → ((((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ) → ((𝐴 ·ih (𝑇‘𝐴)) = ((𝑇‘𝐴) ·ih 𝐴) ↔ 𝐵 ∈ ℝ)) ↔ (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) = (𝐵 ·ℎ if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ∧ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ≠ 0ℎ) → ((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ·ih if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ↔ 𝐵 ∈ ℝ))))
12		oveq1 7166	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ ℂ, 𝐵, 0) → (𝐵 ·ℎ if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) = (if(𝐵 ∈ ℂ, 𝐵, 0) ·ℎ if(𝐴 ∈ ℋ, 𝐴, 0ℎ)))
13	12	eqeq2d 2835	. . . . 5 ⊢ (𝐵 = if(𝐵 ∈ ℂ, 𝐵, 0) → ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) = (𝐵 ·ℎ if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ↔ (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) = (if(𝐵 ∈ ℂ, 𝐵, 0) ·ℎ if(𝐴 ∈ ℋ, 𝐴, 0ℎ))))
14	13	anbi1d 631	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℂ, 𝐵, 0) → (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) = (𝐵 ·ℎ if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ∧ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ≠ 0ℎ) ↔ ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) = (if(𝐵 ∈ ℂ, 𝐵, 0) ·ℎ if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ∧ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ≠ 0ℎ)))
15		eleq1 2903	. . . . 5 ⊢ (𝐵 = if(𝐵 ∈ ℂ, 𝐵, 0) → (𝐵 ∈ ℝ ↔ if(𝐵 ∈ ℂ, 𝐵, 0) ∈ ℝ))
16	15	bibi2d 345	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℂ, 𝐵, 0) → (((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ·ih if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ↔ 𝐵 ∈ ℝ) ↔ ((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ·ih if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ↔ if(𝐵 ∈ ℂ, 𝐵, 0) ∈ ℝ)))
17	14, 16	imbi12d 347	. . 3 ⊢ (𝐵 = if(𝐵 ∈ ℂ, 𝐵, 0) → ((((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) = (𝐵 ·ℎ if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ∧ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ≠ 0ℎ) → ((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ·ih if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ↔ 𝐵 ∈ ℝ)) ↔ (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) = (if(𝐵 ∈ ℂ, 𝐵, 0) ·ℎ if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ∧ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ≠ 0ℎ) → ((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ·ih if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ↔ if(𝐵 ∈ ℂ, 𝐵, 0) ∈ ℝ))))
18		ifhvhv0 28802	. . . 4 ⊢ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ ℋ
19		0cn 10636	. . . . 5 ⊢ 0 ∈ ℂ
20	19	elimel 4537	. . . 4 ⊢ if(𝐵 ∈ ℂ, 𝐵, 0) ∈ ℂ
21	18, 20	eigrei 29614	. . 3 ⊢ (((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) = (if(𝐵 ∈ ℂ, 𝐵, 0) ·ℎ if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ∧ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ≠ 0ℎ) → ((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ))) = ((𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ·ih if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ↔ if(𝐵 ∈ ℂ, 𝐵, 0) ∈ ℝ))
22	11, 17, 21	dedth2h 4527	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) → (((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ) → ((𝐴 ·ih (𝑇‘𝐴)) = ((𝑇‘𝐴) ·ih 𝐴) ↔ 𝐵 ∈ ℝ)))
23	22	imp 409	1 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → ((𝐴 ·ih (𝑇‘𝐴)) = ((𝑇‘𝐴) ·ih 𝐴) ↔ 𝐵 ∈ ℝ))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113   ≠ wne 3019  ifcif 4470  ‘cfv 6358  (class class class)co 7159  ℂcc 10538  ℝcr 10539  0cc0 10540   ℋchba 28699   ·_ℎ csm 28701   ·_ih csp 28702  0_ℎc0v 28704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617  ax-hv0cl 28783  ax-hfvmul 28785  ax-hfi 28859  ax-his1 28862  ax-his3 28864  ax-his4 28865

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-po 5477  df-so 5478  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-er 8292  df-en 8513  df-dom 8514  df-sdom 8515  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-div 11301  df-2 11703  df-cj 14461  df-re 14462  df-im 14463

This theorem is referenced by:  eighmre  29743