Theorem eigposi 31855

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 7439	. . . . . . . 8 ⊢ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) → (𝐴 ·ih (𝑇‘𝐴)) = (𝐴 ·ih (𝐵 ·ℎ 𝐴)))
2	1	eleq1d 2826	. . . . . . 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 31033	. . . . . . . . 9 ⊢ (𝐵 ·ℎ 𝐴) ∈ ℋ
6		hire 31113	. . . . . . . . 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 7438	. . . . . . . . 9 ⊢ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) → ((𝑇‘𝐴) ·ih 𝐴) = ((𝐵 ·ℎ 𝐴) ·ih 𝐴))
9	1, 8	eqeq12d 2753	. . . . . . . 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 31853	. . . . 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 31114	. . . . 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 31104	. . . . . 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 13074	. . 3 ⊢ ((((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → (𝐴 ·ih 𝐴) ∈ ℝ+)
23		simplr 769	. . . 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 31105	. . . . . . 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 15265	. . . . . . 7 ⊢ ((((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → (∗‘𝐵) = 𝐵)
28	27	oveq1d 7446	. . . . . 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 2789	. . . . 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 2777	. . . 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 5169	. . 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 13143	. 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 2940   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431  ℂcc 11153  ℝcr 11154  0cc0 11155   · cmul 11160   < clt 11295   ≤ cle 11296  ∗ccj 15135   ℋchba 30938   ·_ℎ csm 30940   ·_ih csp 30941  0_ℎc0v 30943

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 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-hfvmul 31024  ax-hfi 31098  ax-his1 31101  ax-his3 31103  ax-his4 31104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-2 12329  df-rp 13035  df-cj 15138  df-re 15139  df-im 15140

This theorem is referenced by: (None)