Theorem eigre 31639

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 6892	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (𝑇‘𝐴) = (𝑇‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)))
2		oveq2 7423	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (𝐵 ·ℎ 𝐴) = (𝐵 ·ℎ if(𝐴 ∈ ℋ, 𝐴, 0ℎ)))
3	1, 2	eqeq12d 2744	. . . . 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 2999	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (𝐴 ≠ 0ℎ ↔ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ≠ 0ℎ))
5	3, 4	anbi12d 631	. . . 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 7433	. . . . . 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 7433	. . . . . 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 2744	. . . . 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 343	. . . 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 344	. . 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 7422	. . . . . 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 2739	. . . . 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 630	. . . 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 2817	. . . . 5 ⊢ (𝐵 = if(𝐵 ∈ ℂ, 𝐵, 0) → (𝐵 ∈ ℝ ↔ if(𝐵 ∈ ℂ, 𝐵, 0) ∈ ℝ))
16	15	bibi2d 342	. . . 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 344	. . 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 30826	. . . 4 ⊢ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ ℋ
19		0cn 11231	. . . . 5 ⊢ 0 ∈ ℂ
20	19	elimel 4594	. . . 4 ⊢ if(𝐵 ∈ ℂ, 𝐵, 0) ∈ ℂ
21	18, 20	eigrei 31638	. . 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 4584	. 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 406	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 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099   ≠ wne 2936  ifcif 4525  ‘cfv 6543  (class class class)co 7415  ℂcc 11131  ℝcr 11132  0cc0 11133   ℋchba 30723   ·_ℎ csm 30725   ·_ih csp 30726  0_ℎc0v 30728

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 2699  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210  ax-hv0cl 30807  ax-hfvmul 30809  ax-hfi 30883  ax-his1 30886  ax-his3 30888  ax-his4 30889

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-po 5585  df-so 5586  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-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-er 8719  df-en 8959  df-dom 8960  df-sdom 8961  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-div 11897  df-2 12300  df-cj 15073  df-re 15074  df-im 15075

This theorem is referenced by:  eighmre  31767