Theorem eigposi 28695

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 e. ~H
eigpos.2	|- B e. CC

Assertion

Ref	Expression
eigposi	|- ( ( ( ( A .ih ( T ` A ) ) e. RR /\ 0 <_ ( A .ih ( T ` A ) ) ) /\ ( ( T ` A ) = ( B .h A ) /\ A =/= 0h ) ) -> ( B e. RR /\ 0 <_ B ) )

Proof of Theorem eigposi

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . . . . 8 |- ( ( T ` A ) = ( B .h A ) -> ( A .ih ( T ` A ) ) = ( A .ih ( B .h A ) ) )
2	1	eleq1d 2686	. . . . . . 7 |- ( ( T ` A ) = ( B .h A ) -> ( ( A .ih ( T ` A ) ) e. RR <-> ( A .ih ( B .h A ) ) e. RR ) )
3		oveq1 6657	. . . . . . . . 9 |- ( ( T ` A ) = ( B .h A ) -> ( ( T ` A ) .ih A ) = ( ( B .h A ) .ih A ) )
4	1, 3	eqeq12d 2637	. . . . . . . 8 |- ( ( T ` A ) = ( B .h A ) -> ( ( A .ih ( T ` A ) ) = ( ( T ` A ) .ih A ) <-> ( A .ih ( B .h A ) ) = ( ( B .h A ) .ih A ) ) )
5		eigpos.1	. . . . . . . . 9 |- A e. ~H
6		eigpos.2	. . . . . . . . . 10 |- B e. CC
7	6, 5	hvmulcli 27871	. . . . . . . . 9 |- ( B .h A ) e. ~H
8		hire 27951	. . . . . . . . 9 |- ( ( A e. ~H /\ ( B .h A ) e. ~H ) -> ( ( A .ih ( B .h A ) ) e. RR <-> ( A .ih ( B .h A ) ) = ( ( B .h A ) .ih A ) ) )
9	5, 7, 8	mp2an 708	. . . . . . . 8 |- ( ( A .ih ( B .h A ) ) e. RR <-> ( A .ih ( B .h A ) ) = ( ( B .h A ) .ih A ) )
10	4, 9	syl6rbbr 279	. . . . . . 7 |- ( ( T ` A ) = ( B .h A ) -> ( ( A .ih ( B .h A ) ) e. RR <-> ( A .ih ( T ` A ) ) = ( ( T ` A ) .ih A ) ) )
11	2, 10	bitrd 268	. . . . . 6 |- ( ( T ` A ) = ( B .h A ) -> ( ( A .ih ( T ` A ) ) e. RR <-> ( A .ih ( T ` A ) ) = ( ( T ` A ) .ih A ) ) )
12	11	adantr 481	. . . . 5 |- ( ( ( T ` A ) = ( B .h A ) /\ A =/= 0h ) -> ( ( A .ih ( T ` A ) ) e. RR <-> ( A .ih ( T ` A ) ) = ( ( T ` A ) .ih A ) ) )
13	5, 6	eigrei 28693	. . . . 5 |- ( ( ( T ` A ) = ( B .h A ) /\ A =/= 0h ) -> ( ( A .ih ( T ` A ) ) = ( ( T ` A ) .ih A ) <-> B e. RR ) )
14	12, 13	bitrd 268	. . . 4 |- ( ( ( T ` A ) = ( B .h A ) /\ A =/= 0h ) -> ( ( A .ih ( T ` A ) ) e. RR <-> B e. RR ) )
15	14	biimpac 503	. . 3 |- ( ( ( A .ih ( T ` A ) ) e. RR /\ ( ( T ` A ) = ( B .h A ) /\ A =/= 0h ) ) -> B e. RR )
16	15	adantlr 751	. 2 |- ( ( ( ( A .ih ( T ` A ) ) e. RR /\ 0 <_ ( A .ih ( T ` A ) ) ) /\ ( ( T ` A ) = ( B .h A ) /\ A =/= 0h ) ) -> B e. RR )
17		ax-his4 27942	. . . . 5 |- ( ( A e. ~H /\ A =/= 0h ) -> 0 < ( A .ih A ) )
18	5, 17	mpan 706	. . . 4 |- ( A =/= 0h -> 0 < ( A .ih A ) )
19	18	ad2antll 765	. . 3 |- ( ( ( ( A .ih ( T ` A ) ) e. RR /\ 0 <_ ( A .ih ( T ` A ) ) ) /\ ( ( T ` A ) = ( B .h A ) /\ A =/= 0h ) ) -> 0 < ( A .ih A ) )
20		simplr 792	. . . 4 |- ( ( ( ( A .ih ( T ` A ) ) e. RR /\ 0 <_ ( A .ih ( T ` A ) ) ) /\ ( ( T ` A ) = ( B .h A ) /\ A =/= 0h ) ) -> 0 <_ ( A .ih ( T ` A ) ) )
21	1	ad2antrl 764	. . . . 5 |- ( ( ( ( A .ih ( T ` A ) ) e. RR /\ 0 <_ ( A .ih ( T ` A ) ) ) /\ ( ( T ` A ) = ( B .h A ) /\ A =/= 0h ) ) -> ( A .ih ( T ` A ) ) = ( A .ih ( B .h A ) ) )
22		his5 27943	. . . . . . 7 |- ( ( B e. CC /\ A e. ~H /\ A e. ~H ) -> ( A .ih ( B .h A ) ) = ( ( * ` B ) x. ( A .ih A ) ) )
23	6, 5, 5, 22	mp3an 1424	. . . . . 6 |- ( A .ih ( B .h A ) ) = ( ( * ` B ) x. ( A .ih A ) )
24	16	cjred 13966	. . . . . . 7 |- ( ( ( ( A .ih ( T ` A ) ) e. RR /\ 0 <_ ( A .ih ( T ` A ) ) ) /\ ( ( T ` A ) = ( B .h A ) /\ A =/= 0h ) ) -> ( * ` B ) = B )
25	24	oveq1d 6665	. . . . . 6 |- ( ( ( ( A .ih ( T ` A ) ) e. RR /\ 0 <_ ( A .ih ( T ` A ) ) ) /\ ( ( T ` A ) = ( B .h A ) /\ A =/= 0h ) ) -> ( ( * ` B ) x. ( A .ih A ) ) = ( B x. ( A .ih A ) ) )
26	23, 25	syl5eq 2668	. . . . 5 |- ( ( ( ( A .ih ( T ` A ) ) e. RR /\ 0 <_ ( A .ih ( T ` A ) ) ) /\ ( ( T ` A ) = ( B .h A ) /\ A =/= 0h ) ) -> ( A .ih ( B .h A ) ) = ( B x. ( A .ih A ) ) )
27	21, 26	eqtrd 2656	. . . 4 |- ( ( ( ( A .ih ( T ` A ) ) e. RR /\ 0 <_ ( A .ih ( T ` A ) ) ) /\ ( ( T ` A ) = ( B .h A ) /\ A =/= 0h ) ) -> ( A .ih ( T ` A ) ) = ( B x. ( A .ih A ) ) )
28	20, 27	breqtrd 4679	. . 3 |- ( ( ( ( A .ih ( T ` A ) ) e. RR /\ 0 <_ ( A .ih ( T ` A ) ) ) /\ ( ( T ` A ) = ( B .h A ) /\ A =/= 0h ) ) -> 0 <_ ( B x. ( A .ih A ) ) )
29		hiidrcl 27952	. . . . 5 |- ( A e. ~H -> ( A .ih A ) e. RR )
30	5, 29	ax-mp 5	. . . 4 |- ( A .ih A ) e. RR
31		prodge02 10871	. . . 4 |- ( ( ( B e. RR /\ ( A .ih A ) e. RR ) /\ ( 0 < ( A .ih A ) /\ 0 <_ ( B x. ( A .ih A ) ) ) ) -> 0 <_ B )
32	30, 31	mpanl2 717	. . 3 |- ( ( B e. RR /\ ( 0 < ( A .ih A ) /\ 0 <_ ( B x. ( A .ih A ) ) ) ) -> 0 <_ B )
33	16, 19, 28, 32	syl12anc 1324	. 2 |- ( ( ( ( A .ih ( T ` A ) ) e. RR /\ 0 <_ ( A .ih ( T ` A ) ) ) /\ ( ( T ` A ) = ( B .h A ) /\ A =/= 0h ) ) -> 0 <_ B )
34	16, 33	jca 554	1 |- ( ( ( ( A .ih ( T ` A ) ) e. RR /\ 0 <_ ( A .ih ( T ` A ) ) ) /\ ( ( T ` A ) = ( B .h A ) /\ A =/= 0h ) ) -> ( B e. RR /\ 0 <_ B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   x. cmul 9941   < clt 10074   <_ cle 10075   *ccj 13836   ~Hchil 27776   .h csm 27778   .ih csp 27779   0hc0v 27781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-hfvmul 27862  ax-hfi 27936  ax-his1 27939  ax-his3 27941  ax-his4 27942

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-2 11079  df-cj 13839  df-re 13840  df-im 13841

This theorem is referenced by: (None)