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Theorem eigposi 31702

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 ≠ 0_ℎ)) → (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵))

**Proof of Theorem eigposi**
Step	Hyp	Ref	Expression
1		oveq2 7425	. . . . . . . 8 ⊢ ((𝑇‘𝐴) = (𝐵 ·_ℎ 𝐴) → (𝐴 ·_ih (𝑇‘𝐴)) = (𝐴 ·_ih (𝐵 ·_ℎ 𝐴)))
2	1	eleq1d 2810	. . . . . . 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 30880	. . . . . . . . 9 ⊢ (𝐵 ·_ℎ 𝐴) ∈ ℋ
6		hire 30960	. . . . . . . . 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 7424	. . . . . . . . 9 ⊢ ((𝑇‘𝐴) = (𝐵 ·_ℎ 𝐴) → ((𝑇‘𝐴) ·_ih 𝐴) = ((𝐵 ·_ℎ 𝐴) ·_ih 𝐴))
9	1, 8	eqeq12d 2741	. . . . . . . 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 479	. . . . 5 ⊢ (((𝑇‘𝐴) = (𝐵 ·_ℎ 𝐴) ∧ 𝐴 ≠ 0_ℎ) → ((𝐴 ·_ih (𝑇‘𝐴)) ∈ ℝ ↔ (𝐴 ·_ih (𝑇‘𝐴)) = ((𝑇‘𝐴) ·_ih 𝐴)))
13	3, 4	eigrei 31700	. . . . 5 ⊢ (((𝑇‘𝐴) = (𝐵 ·_ℎ 𝐴) ∧ 𝐴 ≠ 0_ℎ) → ((𝐴 ·_ih (𝑇‘𝐴)) = ((𝑇‘𝐴) ·_ih 𝐴) ↔ 𝐵 ∈ ℝ))
14	12, 13	bitrd 278	. . . 4 ⊢ (((𝑇‘𝐴) = (𝐵 ·_ℎ 𝐴) ∧ 𝐴 ≠ 0_ℎ) → ((𝐴 ·_ih (𝑇‘𝐴)) ∈ ℝ ↔ 𝐵 ∈ ℝ))
15	14	biimpac 477	. . 3 ⊢ (((𝐴 ·_ih (𝑇‘𝐴)) ∈ ℝ ∧ ((𝑇‘𝐴) = (𝐵 ·_ℎ 𝐴) ∧ 𝐴 ≠ 0_ℎ)) → 𝐵 ∈ ℝ)
16	15	adantlr 713	. 2 ⊢ ((((𝐴 ·_ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·_ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·_ℎ 𝐴) ∧ 𝐴 ≠ 0_ℎ)) → 𝐵 ∈ ℝ)
17		hiidrcl 30961	. . . . 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 ≠ 0_ℎ)) → (𝐴 ·_ih 𝐴) ∈ ℝ)
19		ax-his4 30951	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0_ℎ) → 0 < (𝐴 ·_ih 𝐴))
20	3, 19	mpan 688	. . . . 5 ⊢ (𝐴 ≠ 0_ℎ → 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 ≠ 0_ℎ)) → 0 < (𝐴 ·_ih 𝐴))
22	18, 21	elrpd 13045	. . 3 ⊢ ((((𝐴 ·_ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·_ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·_ℎ 𝐴) ∧ 𝐴 ≠ 0_ℎ)) → (𝐴 ·_ih 𝐴) ∈ ℝ⁺)
23		simplr 767	. . . 4 ⊢ ((((𝐴 ·_ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·_ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·_ℎ 𝐴) ∧ 𝐴 ≠ 0_ℎ)) → 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 ≠ 0_ℎ)) → (𝐴 ·_ih (𝑇‘𝐴)) = (𝐴 ·_ih (𝐵 ·_ℎ 𝐴)))
25		his5 30952	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐴 ·_ih (𝐵 ·_ℎ 𝐴)) = ((∗‘𝐵) · (𝐴 ·_ih 𝐴)))
26	4, 3, 3, 25	mp3an 1457	. . . . . 6 ⊢ (𝐴 ·_ih (𝐵 ·_ℎ 𝐴)) = ((∗‘𝐵) · (𝐴 ·_ih 𝐴))
27	16	cjred 15205	. . . . . . 7 ⊢ ((((𝐴 ·_ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·_ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·_ℎ 𝐴) ∧ 𝐴 ≠ 0_ℎ)) → (∗‘𝐵) = 𝐵)
28	27	oveq1d 7432	. . . . . 6 ⊢ ((((𝐴 ·_ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·_ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·_ℎ 𝐴) ∧ 𝐴 ≠ 0_ℎ)) → ((∗‘𝐵) · (𝐴 ·_ih 𝐴)) = (𝐵 · (𝐴 ·_ih 𝐴)))
29	26, 28	eqtrid 2777	. . . . 5 ⊢ ((((𝐴 ·_ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·_ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·_ℎ 𝐴) ∧ 𝐴 ≠ 0_ℎ)) → (𝐴 ·_ih (𝐵 ·_ℎ 𝐴)) = (𝐵 · (𝐴 ·_ih 𝐴)))
30	24, 29	eqtrd 2765	. . . 4 ⊢ ((((𝐴 ·_ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·_ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·_ℎ 𝐴) ∧ 𝐴 ≠ 0_ℎ)) → (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 ≠ 0_ℎ)) → 0 ≤ (𝐵 · (𝐴 ·_ih 𝐴)))
32	16, 22, 31	prodge0ld 13114	. 2 ⊢ ((((𝐴 ·_ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·_ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·_ℎ 𝐴) ∧ 𝐴 ≠ 0_ℎ)) → 0 ≤ 𝐵)
33	16, 32	jca 510	1 ⊢ ((((𝐴 ·_ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·_ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·_ℎ 𝐴) ∧ 𝐴 ≠ 0_ℎ)) → (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 class class class wbr 5148 ‘cfv 6547 (class class class)co 7417 ℂcc 11136 ℝcr 11137 0cc0 11138 · cmul 11143 < clt 11278 ≤ cle 11279 ∗ccj 15075 ℋchba 30785 ·_ℎ csm 30787 ·_ih csp 30788 0_ℎc0v 30790

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-hfvmul 30871 ax-hfi 30945 ax-his1 30948 ax-his3 30950 ax-his4 30951

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-2 12305 df-rp 13007 df-cj 15078 df-re 15079 df-im 15080

This theorem is referenced by: (None)

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