Theorem eigposi 31344

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 7419	. . . . . . . 8 ⊢ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) → (𝐴 ·ih (𝑇‘𝐴)) = (𝐴 ·ih (𝐵 ·ℎ 𝐴)))
2	1	eleq1d 2818	. . . . . . 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 30522	. . . . . . . . 9 ⊢ (𝐵 ·ℎ 𝐴) ∈ ℋ
6		hire 30602	. . . . . . . . 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 690	. . . . . . . 8 ⊢ ((𝐴 ·ih (𝐵 ·ℎ 𝐴)) ∈ ℝ ↔ (𝐴 ·ih (𝐵 ·ℎ 𝐴)) = ((𝐵 ·ℎ 𝐴) ·ih 𝐴))
8		oveq1 7418	. . . . . . . . 9 ⊢ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) → ((𝑇‘𝐴) ·ih 𝐴) = ((𝐵 ·ℎ 𝐴) ·ih 𝐴))
9	1, 8	eqeq12d 2748	. . . . . . . 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 289	. . . . . . 7 ⊢ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) → ((𝐴 ·ih (𝐵 ·ℎ 𝐴)) ∈ ℝ ↔ (𝐴 ·ih (𝑇‘𝐴)) = ((𝑇‘𝐴) ·ih 𝐴)))
11	2, 10	bitrd 278	. . . . . 6 ⊢ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) → ((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ↔ (𝐴 ·ih (𝑇‘𝐴)) = ((𝑇‘𝐴) ·ih 𝐴)))
12	11	adantr 481	. . . . 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 31342	. . . . 5 ⊢ (((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ) → ((𝐴 ·ih (𝑇‘𝐴)) = ((𝑇‘𝐴) ·ih 𝐴) ↔ 𝐵 ∈ ℝ))
14	12, 13	bitrd 278	. . . 4 ⊢ (((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ) → ((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ↔ 𝐵 ∈ ℝ))
15	14	biimpac 479	. . 3 ⊢ (((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → 𝐵 ∈ ℝ)
16	15	adantlr 713	. 2 ⊢ ((((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → 𝐵 ∈ ℝ)
17		hiidrcl 30603	. . . . 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 30593	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (𝐴 ·ih 𝐴))
20	3, 19	mpan 688	. . . . 5 ⊢ (𝐴 ≠ 0ℎ → 0 < (𝐴 ·ih 𝐴))
21	20	ad2antll 727	. . . 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 13017	. . 3 ⊢ ((((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → (𝐴 ·ih 𝐴) ∈ ℝ+)
23		simplr 767	. . . 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 726	. . . . 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 30594	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐴 ·ih (𝐵 ·ℎ 𝐴)) = ((∗‘𝐵) · (𝐴 ·ih 𝐴)))
26	4, 3, 3, 25	mp3an 1461	. . . . . 6 ⊢ (𝐴 ·ih (𝐵 ·ℎ 𝐴)) = ((∗‘𝐵) · (𝐴 ·ih 𝐴))
27	16	cjred 15177	. . . . . . 7 ⊢ ((((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → (∗‘𝐵) = 𝐵)
28	27	oveq1d 7426	. . . . . 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 2784	. . . . 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 2772	. . . 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 5174	. . 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 13086	. 2 ⊢ ((((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → 0 ≤ 𝐵)
33	16, 32	jca 512	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 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2940   class class class wbr 5148  ‘cfv 6543  (class class class)co 7411  ℂcc 11110  ℝcr 11111  0cc0 11112   · cmul 11117   < clt 11252   ≤ cle 11253  ∗ccj 15047   ℋchba 30427   ·_ℎ csm 30429   ·_ih csp 30430  0_ℎc0v 30432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-hfvmul 30513  ax-hfi 30587  ax-his1 30590  ax-his3 30592  ax-his4 30593

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  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 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-2 12279  df-rp 12979  df-cj 15050  df-re 15051  df-im 15052

This theorem is referenced by: (None)