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Theorem hmopm 28880
Description: The scalar product of a Hermitian operator with a real is Hermitian. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hmopm |- ( ( A e. RR /\ T e. HrmOp ) -> ( A .op T ) e. HrmOp )
Proof of Theorem hmopm
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 recn 10026 . . 3 |- ( A e. RR -> A e. CC )
2 hmopf 28733 . . 3 |- ( T e. HrmOp -> T : ~H --> ~H )
3 homulcl 28618 . . 3 |- ( ( A e. CC /\ T : ~H --> ~H ) -> ( A .op T ) : ~H --> ~H )
41, 2, 3syl2an 494 . 2 |- ( ( A e. RR /\ T e. HrmOp ) -> ( A .op T ) : ~H --> ~H )
5 cjre 13879 . . . . . 6 |- ( A e. RR -> ( * ` A ) = A )
6 hmop 28781 . . . . . . 7 |- ( ( T e. HrmOp /\ x e. ~H /\ y e. ~H ) -> ( x .ih ( T ` y ) ) = ( ( T ` x ) .ih y ) )
763expb 1266 . . . . . 6 |- ( ( T e. HrmOp /\ ( x e. ~H /\ y e. ~H ) ) -> ( x .ih ( T ` y ) ) = ( ( T ` x ) .ih y ) )
85, 7oveqan12d 6669 . . . . 5 |- ( ( A e. RR /\ ( T e. HrmOp /\ ( x e. ~H /\ y e. ~H ) ) ) -> ( ( * ` A ) x. ( x .ih ( T ` y ) ) ) = ( A x. ( ( T ` x ) .ih y ) ) )
98anassrs 680 . . . 4 |- ( ( ( A e. RR /\ T e. HrmOp ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( ( * ` A ) x. ( x .ih ( T ` y ) ) ) = ( A x. ( ( T ` x ) .ih y ) ) )
101, 2anim12i 590 . . . . 5 |- ( ( A e. RR /\ T e. HrmOp ) -> ( A e. CC /\ T : ~H --> ~H ) )
11 homval 28600 . . . . . . . . 9 |- ( ( A e. CC /\ T : ~H --> ~H /\ y e. ~H ) -> ( ( A .op T ) ` y ) = ( A .h ( T ` y ) ) )
12113expa 1265 . . . . . . . 8 |- ( ( ( A e. CC /\ T : ~H --> ~H ) /\ y e. ~H ) -> ( ( A .op T ) ` y ) = ( A .h ( T ` y ) ) )
1312adantrl 752 . . . . . . 7 |- ( ( ( A e. CC /\ T : ~H --> ~H ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( ( A .op T ) ` y ) = ( A .h ( T ` y ) ) )
1413oveq2d 6666 . . . . . 6 |- ( ( ( A e. CC /\ T : ~H --> ~H ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( x .ih ( ( A .op T ) ` y ) ) = ( x .ih ( A .h ( T ` y ) ) ) )
15 simpll 790 . . . . . . 7 |- ( ( ( A e. CC /\ T : ~H --> ~H ) /\ ( x e. ~H /\ y e. ~H ) ) -> A e. CC )
16 simprl 794 . . . . . . 7 |- ( ( ( A e. CC /\ T : ~H --> ~H ) /\ ( x e. ~H /\ y e. ~H ) ) -> x e. ~H )
17 ffvelrn 6357 . . . . . . . 8 |- ( ( T : ~H --> ~H /\ y e. ~H ) -> ( T ` y ) e. ~H )
1817ad2ant2l 782 . . . . . . 7 |- ( ( ( A e. CC /\ T : ~H --> ~H ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( T ` y ) e. ~H )
19 his5 27943 . . . . . . 7 |- ( ( A e. CC /\ x e. ~H /\ ( T ` y ) e. ~H ) -> ( x .ih ( A .h ( T ` y ) ) ) = ( ( * ` A ) x. ( x .ih ( T ` y ) ) ) )
2015, 16, 18, 19syl3anc 1326 . . . . . 6 |- ( ( ( A e. CC /\ T : ~H --> ~H ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( x .ih ( A .h ( T ` y ) ) ) = ( ( * ` A ) x. ( x .ih ( T ` y ) ) ) )
2114, 20eqtrd 2656 . . . . 5 |- ( ( ( A e. CC /\ T : ~H --> ~H ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( x .ih ( ( A .op T ) ` y ) ) = ( ( * ` A ) x. ( x .ih ( T ` y ) ) ) )
2210, 21sylan 488 . . . 4 |- ( ( ( A e. RR /\ T e. HrmOp ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( x .ih ( ( A .op T ) ` y ) ) = ( ( * ` A ) x. ( x .ih ( T ` y ) ) ) )
23 homval 28600 . . . . . . . . 9 |- ( ( A e. CC /\ T : ~H --> ~H /\ x e. ~H ) -> ( ( A .op T ) ` x ) = ( A .h ( T ` x ) ) )
24233expa 1265 . . . . . . . 8 |- ( ( ( A e. CC /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( ( A .op T ) ` x ) = ( A .h ( T ` x ) ) )
2524adantrr 753 . . . . . . 7 |- ( ( ( A e. CC /\ T : ~H --> ~H ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( ( A .op T ) ` x ) = ( A .h ( T ` x ) ) )
2625oveq1d 6665 . . . . . 6 |- ( ( ( A e. CC /\ T : ~H --> ~H ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( ( ( A .op T ) ` x ) .ih y ) = ( ( A .h ( T ` x ) ) .ih y ) )
27 ffvelrn 6357 . . . . . . . 8 |- ( ( T : ~H --> ~H /\ x e. ~H ) -> ( T ` x ) e. ~H )
2827ad2ant2lr 784 . . . . . . 7 |- ( ( ( A e. CC /\ T : ~H --> ~H ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( T ` x ) e. ~H )
29 simprr 796 . . . . . . 7 |- ( ( ( A e. CC /\ T : ~H --> ~H ) /\ ( x e. ~H /\ y e. ~H ) ) -> y e. ~H )
30 ax-his3 27941 . . . . . . 7 |- ( ( A e. CC /\ ( T ` x ) e. ~H /\ y e. ~H ) -> ( ( A .h ( T ` x ) ) .ih y ) = ( A x. ( ( T ` x ) .ih y ) ) )
3115, 28, 29, 30syl3anc 1326 . . . . . 6 |- ( ( ( A e. CC /\ T : ~H --> ~H ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( ( A .h ( T ` x ) ) .ih y ) = ( A x. ( ( T ` x ) .ih y ) ) )
3226, 31eqtrd 2656 . . . . 5 |- ( ( ( A e. CC /\ T : ~H --> ~H ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( ( ( A .op T ) ` x ) .ih y ) = ( A x. ( ( T ` x ) .ih y ) ) )
3310, 32sylan 488 . . . 4 |- ( ( ( A e. RR /\ T e. HrmOp ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( ( ( A .op T ) ` x ) .ih y ) = ( A x. ( ( T ` x ) .ih y ) ) )
349, 22, 333eqtr4d 2666 . . 3 |- ( ( ( A e. RR /\ T e. HrmOp ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( x .ih ( ( A .op T ) ` y ) ) = ( ( ( A .op T ) ` x ) .ih y ) )
3534ralrimivva 2971 . 2 |- ( ( A e. RR /\ T e. HrmOp ) -> A. x e. ~H A. y e. ~H ( x .ih ( ( A .op T ) ` y ) ) = ( ( ( A .op T ) ` x ) .ih y ) )
36 elhmop 28732 . 2 |- ( ( A .op T ) e. HrmOp <-> ( ( A .op T ) : ~H --> ~H /\ A. x e. ~H A. y e. ~H ( x .ih ( ( A .op T ) ` y ) ) = ( ( ( A .op T ) ` x ) .ih y ) ) )
374, 35, 36sylanbrc 698 1 |- ( ( A e. RR /\ T e. HrmOp ) -> ( A .op T ) e. HrmOp )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   x. cmul 9941   *ccj 13836   ~Hchil 27776   .h csm 27778   .ih csp 27779   .op chot 27796   HrmOpcho 27807
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-rep 4771  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-hilex 27856  ax-hfvmul 27862  ax-hfi 27936  ax-his1 27939  ax-his3 27941
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-map 7859  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  df-homul 28590  df-hmop 28703
This theorem is referenced by:  hmopd  28881  leopmuli  28992  leopmul  28993  leopmul2i  28994  leopnmid  28997  nmopleid  28998  opsqrlem1  28999  opsqrlem4  29002
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