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Theorem eigposi 22341
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
StepHypRef Expression
1 oveq2 5765 . . . . . . . 8  |-  ( ( T `  A )  =  ( B  .h  A )  ->  ( A  .ih  ( T `  A ) )  =  ( A  .ih  ( B  .h  A )
) )
21eleq1d 2322 . . . . . . 7  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( A  .ih  ( T `  A )
)  e.  RR  <->  ( A  .ih  ( B  .h  A
) )  e.  RR ) )
3 oveq1 5764 . . . . . . . . 9  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( T `  A
)  .ih  A )  =  ( ( B  .h  A )  .ih  A ) )
41, 3eqeq12d 2270 . . . . . . . 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
76, 5hvmulcli 21519 . . . . . . . . 9  |-  ( B  .h  A )  e. 
~H
8 hire 21598 . . . . . . . . 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 ) ) )
95, 7, 8mp2an 656 . . . . . . . 8  |-  ( ( A  .ih  ( B  .h  A ) )  e.  RR  <->  ( A  .ih  ( B  .h  A
) )  =  ( ( B  .h  A
)  .ih  A )
)
104, 9syl6rbbr 257 . . . . . . 7  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( A  .ih  ( B  .h  A )
)  e.  RR  <->  ( A  .ih  ( T `  A
) )  =  ( ( T `  A
)  .ih  A )
) )
112, 10bitrd 246 . . . . . 6  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( A  .ih  ( T `  A )
)  e.  RR  <->  ( A  .ih  ( T `  A
) )  =  ( ( T `  A
)  .ih  A )
) )
1211adantr 453 . . . . 5  |-  ( ( ( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h )  -> 
( ( A  .ih  ( T `  A ) )  e.  RR  <->  ( A  .ih  ( T `  A
) )  =  ( ( T `  A
)  .ih  A )
) )
135, 6eigrei 22339 . . . . 5  |-  ( ( ( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h )  -> 
( ( A  .ih  ( T `  A ) )  =  ( ( T `  A ) 
.ih  A )  <->  B  e.  RR ) )
1412, 13bitrd 246 . . . 4  |-  ( ( ( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h )  -> 
( ( A  .ih  ( T `  A ) )  e.  RR  <->  B  e.  RR ) )
1514biimpac 474 . . 3  |-  ( ( ( A  .ih  ( T `  A )
)  e.  RR  /\  ( ( T `  A )  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  B  e.  RR )
1615adantlr 698 . 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 21589 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( A  .ih  A ) )
185, 17mpan 654 . . . 4  |-  ( A  =/=  0h  ->  0  <  ( A  .ih  A
) )
1918ad2antll 712 . . 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 734 . . . 4  |-  ( ( ( ( A  .ih  ( T `  A ) )  e.  RR  /\  0  <_  ( A  .ih  ( T `  A ) ) )  /\  (
( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  0  <_  ( A  .ih  ( T `  A ) ) )
211ad2antrl 711 . . . . 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 21590 . . . . . . 7  |-  ( ( B  e.  CC  /\  A  e.  ~H  /\  A  e.  ~H )  ->  ( A  .ih  ( B  .h  A ) )  =  ( ( * `  B )  x.  ( A  .ih  A ) ) )
236, 5, 5, 22mp3an 1282 . . . . . 6  |-  ( A 
.ih  ( B  .h  A ) )  =  ( ( * `  B )  x.  ( A  .ih  A ) )
2416cjred 11641 . . . . . . 7  |-  ( ( ( ( A  .ih  ( T `  A ) )  e.  RR  /\  0  <_  ( A  .ih  ( T `  A ) ) )  /\  (
( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  ( * `  B )  =  B )
2524oveq1d 5772 . . . . . 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 ) ) )
2623, 25syl5eq 2300 . . . . 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 ) ) )
2721, 26eqtrd 2288 . . . 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 ) ) )
2820, 27breqtrd 3987 . . 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 21599 . . . . 5  |-  ( A  e.  ~H  ->  ( A  .ih  A )  e.  RR )
305, 29ax-mp 10 . . . 4  |-  ( A 
.ih  A )  e.  RR
31 prodge02 9537 . . . 4  |-  ( ( ( B  e.  RR  /\  ( A  .ih  A
)  e.  RR )  /\  ( 0  < 
( A  .ih  A
)  /\  0  <_  ( B  x.  ( A 
.ih  A ) ) ) )  ->  0  <_  B )
3230, 31mpanl2 665 . . 3  |-  ( ( B  e.  RR  /\  ( 0  <  ( A  .ih  A )  /\  0  <_  ( B  x.  ( A  .ih  A ) ) ) )  -> 
0  <_  B )
3316, 19, 28, 32syl12anc 1185 . 2  |-  ( ( ( ( A  .ih  ( T `  A ) )  e.  RR  /\  0  <_  ( A  .ih  ( T `  A ) ) )  /\  (
( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  0  <_  B
)
3416, 33jca 520 1  |-  ( ( ( ( A  .ih  ( T `  A ) )  e.  RR  /\  0  <_  ( A  .ih  ( T `  A ) ) )  /\  (
( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  ( B  e.  RR  /\  0  <_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   CCcc 8668   RRcr 8669   0cc0 8670    x. cmul 8675    < clt 8800    <_ cle 8801   *ccj 11511   ~Hchil 21424    .h csm 21426    .ih csp 21427   0hc0v 21429
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-hfvmul 21510  ax-hfi 21583  ax-his1 21586  ax-his3 21588  ax-his4 21589
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-po 4251  df-so 4252  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-iota 6190  df-riota 6237  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-2 9737  df-cj 11514  df-re 11515  df-im 11516
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