Theorem eigposi 31860

Description: A sufficient condition (first conjunct pair, that holds when 𝑇 is a positive operator) for an eigenvalue 𝐵 (second conjunct pair) to be nonnegative. Remark (ii) in [Hughes] p. 137. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)

Hypotheses

Ref	Expression
eigpos.1	⊢ 𝐴 ∈ ℋ
eigpos.2	⊢ 𝐵 ∈ ℂ

Assertion

Ref	Expression
eigposi	⊢ ((((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵))

Proof of Theorem eigposi

Step	Hyp	Ref	Expression
1		oveq2 7364	. . . . . . . 8 ⊢ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) → (𝐴 ·ih (𝑇‘𝐴)) = (𝐴 ·ih (𝐵 ·ℎ 𝐴)))
2	1	eleq1d 2819	. . . . . . 7 ⊢ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) → ((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ↔ (𝐴 ·ih (𝐵 ·ℎ 𝐴)) ∈ ℝ))
3		eigpos.1	. . . . . . . . 9 ⊢ 𝐴 ∈ ℋ
4		eigpos.2	. . . . . . . . . 10 ⊢ 𝐵 ∈ ℂ
5	4, 3	hvmulcli 31038	. . . . . . . . 9 ⊢ (𝐵 ·ℎ 𝐴) ∈ ℋ
6		hire 31118	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ (𝐵 ·ℎ 𝐴) ∈ ℋ) → ((𝐴 ·ih (𝐵 ·ℎ 𝐴)) ∈ ℝ ↔ (𝐴 ·ih (𝐵 ·ℎ 𝐴)) = ((𝐵 ·ℎ 𝐴) ·ih 𝐴)))
7	3, 5, 6	mp2an 692	. . . . . . . 8 ⊢ ((𝐴 ·ih (𝐵 ·ℎ 𝐴)) ∈ ℝ ↔ (𝐴 ·ih (𝐵 ·ℎ 𝐴)) = ((𝐵 ·ℎ 𝐴) ·ih 𝐴))
8		oveq1 7363	. . . . . . . . 9 ⊢ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) → ((𝑇‘𝐴) ·ih 𝐴) = ((𝐵 ·ℎ 𝐴) ·ih 𝐴))
9	1, 8	eqeq12d 2750	. . . . . . . 8 ⊢ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) → ((𝐴 ·ih (𝑇‘𝐴)) = ((𝑇‘𝐴) ·ih 𝐴) ↔ (𝐴 ·ih (𝐵 ·ℎ 𝐴)) = ((𝐵 ·ℎ 𝐴) ·ih 𝐴)))
10	7, 9	bitr4id 290	. . . . . . 7 ⊢ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) → ((𝐴 ·ih (𝐵 ·ℎ 𝐴)) ∈ ℝ ↔ (𝐴 ·ih (𝑇‘𝐴)) = ((𝑇‘𝐴) ·ih 𝐴)))
11	2, 10	bitrd 279	. . . . . 6 ⊢ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) → ((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ↔ (𝐴 ·ih (𝑇‘𝐴)) = ((𝑇‘𝐴) ·ih 𝐴)))
12	11	adantr 480	. . . . 5 ⊢ (((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ) → ((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ↔ (𝐴 ·ih (𝑇‘𝐴)) = ((𝑇‘𝐴) ·ih 𝐴)))
13	3, 4	eigrei 31858	. . . . 5 ⊢ (((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ) → ((𝐴 ·ih (𝑇‘𝐴)) = ((𝑇‘𝐴) ·ih 𝐴) ↔ 𝐵 ∈ ℝ))
14	12, 13	bitrd 279	. . . 4 ⊢ (((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ) → ((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ↔ 𝐵 ∈ ℝ))
15	14	biimpac 478	. . 3 ⊢ (((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → 𝐵 ∈ ℝ)
16	15	adantlr 715	. 2 ⊢ ((((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → 𝐵 ∈ ℝ)
17		hiidrcl 31119	. . . . 5 ⊢ (𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ)
18	3, 17	mp1i 13	. . . 4 ⊢ ((((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → (𝐴 ·ih 𝐴) ∈ ℝ)
19		ax-his4 31109	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (𝐴 ·ih 𝐴))
20	3, 19	mpan 690	. . . . 5 ⊢ (𝐴 ≠ 0ℎ → 0 < (𝐴 ·ih 𝐴))
21	20	ad2antll 729	. . . 4 ⊢ ((((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → 0 < (𝐴 ·ih 𝐴))
22	18, 21	elrpd 12944	. . 3 ⊢ ((((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → (𝐴 ·ih 𝐴) ∈ ℝ+)
23		simplr 768	. . . 4 ⊢ ((((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → 0 ≤ (𝐴 ·ih (𝑇‘𝐴)))
24	1	ad2antrl 728	. . . . 5 ⊢ ((((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → (𝐴 ·ih (𝑇‘𝐴)) = (𝐴 ·ih (𝐵 ·ℎ 𝐴)))
25		his5 31110	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐴 ·ih (𝐵 ·ℎ 𝐴)) = ((∗‘𝐵) · (𝐴 ·ih 𝐴)))
26	4, 3, 3, 25	mp3an 1463	. . . . . 6 ⊢ (𝐴 ·ih (𝐵 ·ℎ 𝐴)) = ((∗‘𝐵) · (𝐴 ·ih 𝐴))
27	16	cjred 15147	. . . . . . 7 ⊢ ((((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → (∗‘𝐵) = 𝐵)
28	27	oveq1d 7371	. . . . . 6 ⊢ ((((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → ((∗‘𝐵) · (𝐴 ·ih 𝐴)) = (𝐵 · (𝐴 ·ih 𝐴)))
29	26, 28	eqtrid 2781	. . . . 5 ⊢ ((((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → (𝐴 ·ih (𝐵 ·ℎ 𝐴)) = (𝐵 · (𝐴 ·ih 𝐴)))
30	24, 29	eqtrd 2769	. . . 4 ⊢ ((((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → (𝐴 ·ih (𝑇‘𝐴)) = (𝐵 · (𝐴 ·ih 𝐴)))
31	23, 30	breqtrd 5122	. . 3 ⊢ ((((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → 0 ≤ (𝐵 · (𝐴 ·ih 𝐴)))
32	16, 22, 31	prodge0ld 13013	. 2 ⊢ ((((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → 0 ≤ 𝐵)
33	16, 32	jca 511	1 ⊢ ((((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2930   class class class wbr 5096  ‘cfv 6490  (class class class)co 7356  ℂcc 11022  ℝcr 11023  0cc0 11024   · cmul 11029   < clt 11164   ≤ cle 11165  ∗ccj 15017   ℋchba 30943   ·_ℎ csm 30945   ·_ih csp 30946  0_ℎc0v 30948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101  ax-hfvmul 31029  ax-hfi 31103  ax-his1 31106  ax-his3 31108  ax-his4 31109

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-div 11793  df-nn 12144  df-2 12206  df-rp 12904  df-cj 15020  df-re 15021  df-im 15022

This theorem is referenced by: (None)