Theorem eigposi 31817

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 7413	. . . . . . . 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 30995	. . . . . . . . 9 ⊢ (𝐵 ·ℎ 𝐴) ∈ ℋ
6		hire 31075	. . . . . . . . 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 7412	. . . . . . . . 9 ⊢ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) → ((𝑇‘𝐴) ·ih 𝐴) = ((𝐵 ·ℎ 𝐴) ·ih 𝐴))
9	1, 8	eqeq12d 2751	. . . . . . . 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 31815	. . . . 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 31076	. . . . 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 31066	. . . . . 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 13048	. . 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 31067	. . . . . . 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 15245	. . . . . . 7 ⊢ ((((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → (∗‘𝐵) = 𝐵)
28	27	oveq1d 7420	. . . . . 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 2782	. . . . 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 2770	. . . 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 5145	. . 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 13117	. 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 1540   ∈ wcel 2108   ≠ wne 2932   class class class wbr 5119  ‘cfv 6531  (class class class)co 7405  ℂcc 11127  ℝcr 11128  0cc0 11129   · cmul 11134   < clt 11269   ≤ cle 11270  ∗ccj 15115   ℋchba 30900   ·_ℎ csm 30902   ·_ih csp 30903  0_ℎc0v 30905

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206  ax-hfvmul 30986  ax-hfi 31060  ax-his1 31063  ax-his3 31065  ax-his4 31066

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-div 11895  df-nn 12241  df-2 12303  df-rp 13009  df-cj 15118  df-re 15119  df-im 15120

This theorem is referenced by: (None)