Theorem hmops 28879

Description: The sum of two Hermitian operators is Hermitian. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
hmops	|- ( ( T e. HrmOp /\ U e. HrmOp ) -> ( T +op U ) e. HrmOp )

Proof of Theorem hmops

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hmopf 28733	. . 3 |- ( T e. HrmOp -> T : ~H --> ~H )
2		hmopf 28733	. . 3 |- ( U e. HrmOp -> U : ~H --> ~H )
3		hoaddcl 28617	. . 3 |- ( ( T : ~H --> ~H /\ U : ~H --> ~H ) -> ( T +op U ) : ~H --> ~H )
4	1, 2, 3	syl2an 494	. 2 |- ( ( T e. HrmOp /\ U e. HrmOp ) -> ( T +op U ) : ~H --> ~H )
5		hmop 28781	. . . . . . 7 |- ( ( T e. HrmOp /\ x e. ~H /\ y e. ~H ) -> ( x .ih ( T ` y ) ) = ( ( T ` x ) .ih y ) )
6	5	3expb 1266	. . . . . 6 |- ( ( T e. HrmOp /\ ( x e. ~H /\ y e. ~H ) ) -> ( x .ih ( T ` y ) ) = ( ( T ` x ) .ih y ) )
7		hmop 28781	. . . . . . 7 |- ( ( U e. HrmOp /\ x e. ~H /\ y e. ~H ) -> ( x .ih ( U ` y ) ) = ( ( U ` x ) .ih y ) )
8	7	3expb 1266	. . . . . 6 |- ( ( U e. HrmOp /\ ( x e. ~H /\ y e. ~H ) ) -> ( x .ih ( U ` y ) ) = ( ( U ` x ) .ih y ) )
9	6, 8	oveqan12d 6669	. . . . 5 |- ( ( ( T e. HrmOp /\ ( x e. ~H /\ y e. ~H ) ) /\ ( U e. HrmOp /\ ( x e. ~H /\ y e. ~H ) ) ) -> ( ( x .ih ( T ` y ) ) + ( x .ih ( U ` y ) ) ) = ( ( ( T ` x ) .ih y ) + ( ( U ` x ) .ih y ) ) )
10	9	anandirs 874	. . . 4 |- ( ( ( T e. HrmOp /\ U e. HrmOp ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( ( x .ih ( T ` y ) ) + ( x .ih ( U ` y ) ) ) = ( ( ( T ` x ) .ih y ) + ( ( U ` x ) .ih y ) ) )
11	1, 2	anim12i 590	. . . . 5 |- ( ( T e. HrmOp /\ U e. HrmOp ) -> ( T : ~H --> ~H /\ U : ~H --> ~H ) )
12		hosval 28599	. . . . . . . . 9 |- ( ( T : ~H --> ~H /\ U : ~H --> ~H /\ y e. ~H ) -> ( ( T +op U ) ` y ) = ( ( T ` y ) +h ( U ` y ) ) )
13	12	oveq2d 6666	. . . . . . . 8 |- ( ( T : ~H --> ~H /\ U : ~H --> ~H /\ y e. ~H ) -> ( x .ih ( ( T +op U ) ` y ) ) = ( x .ih ( ( T ` y ) +h ( U ` y ) ) ) )
14	13	3expa 1265	. . . . . . 7 |- ( ( ( T : ~H --> ~H /\ U : ~H --> ~H ) /\ y e. ~H ) -> ( x .ih ( ( T +op U ) ` y ) ) = ( x .ih ( ( T ` y ) +h ( U ` y ) ) ) )
15	14	adantrl 752	. . . . . 6 |- ( ( ( T : ~H --> ~H /\ U : ~H --> ~H ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( x .ih ( ( T +op U ) ` y ) ) = ( x .ih ( ( T ` y ) +h ( U ` y ) ) ) )
16		simprl 794	. . . . . . 7 |- ( ( ( T : ~H --> ~H /\ U : ~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 )
18	17	ad2ant2rl 785	. . . . . . 7 |- ( ( ( T : ~H --> ~H /\ U : ~H --> ~H ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( T ` y ) e. ~H )
19		ffvelrn 6357	. . . . . . . 8 |- ( ( U : ~H --> ~H /\ y e. ~H ) -> ( U ` y ) e. ~H )
20	19	ad2ant2l 782	. . . . . . 7 |- ( ( ( T : ~H --> ~H /\ U : ~H --> ~H ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( U ` y ) e. ~H )
21		his7 27947	. . . . . . 7 |- ( ( x e. ~H /\ ( T ` y ) e. ~H /\ ( U ` y ) e. ~H ) -> ( x .ih ( ( T ` y ) +h ( U ` y ) ) ) = ( ( x .ih ( T ` y ) ) + ( x .ih ( U ` y ) ) ) )
22	16, 18, 20, 21	syl3anc 1326	. . . . . 6 |- ( ( ( T : ~H --> ~H /\ U : ~H --> ~H ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( x .ih ( ( T ` y ) +h ( U ` y ) ) ) = ( ( x .ih ( T ` y ) ) + ( x .ih ( U ` y ) ) ) )
23	15, 22	eqtrd 2656	. . . . 5 |- ( ( ( T : ~H --> ~H /\ U : ~H --> ~H ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( x .ih ( ( T +op U ) ` y ) ) = ( ( x .ih ( T ` y ) ) + ( x .ih ( U ` y ) ) ) )
24	11, 23	sylan 488	. . . 4 |- ( ( ( T e. HrmOp /\ U e. HrmOp ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( x .ih ( ( T +op U ) ` y ) ) = ( ( x .ih ( T ` y ) ) + ( x .ih ( U ` y ) ) ) )
25		hosval 28599	. . . . . . . . 9 |- ( ( T : ~H --> ~H /\ U : ~H --> ~H /\ x e. ~H ) -> ( ( T +op U ) ` x ) = ( ( T ` x ) +h ( U ` x ) ) )
26	25	oveq1d 6665	. . . . . . . 8 |- ( ( T : ~H --> ~H /\ U : ~H --> ~H /\ x e. ~H ) -> ( ( ( T +op U ) ` x ) .ih y ) = ( ( ( T ` x ) +h ( U ` x ) ) .ih y ) )
27	26	3expa 1265	. . . . . . 7 |- ( ( ( T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( ( T +op U ) ` x ) .ih y ) = ( ( ( T ` x ) +h ( U ` x ) ) .ih y ) )
28	27	adantrr 753	. . . . . 6 |- ( ( ( T : ~H --> ~H /\ U : ~H --> ~H ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( ( ( T +op U ) ` x ) .ih y ) = ( ( ( T ` x ) +h ( U ` x ) ) .ih y ) )
29		ffvelrn 6357	. . . . . . . 8 |- ( ( T : ~H --> ~H /\ x e. ~H ) -> ( T ` x ) e. ~H )
30	29	ad2ant2r 783	. . . . . . 7 |- ( ( ( T : ~H --> ~H /\ U : ~H --> ~H ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( T ` x ) e. ~H )
31		ffvelrn 6357	. . . . . . . 8 |- ( ( U : ~H --> ~H /\ x e. ~H ) -> ( U ` x ) e. ~H )
32	31	ad2ant2lr 784	. . . . . . 7 |- ( ( ( T : ~H --> ~H /\ U : ~H --> ~H ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( U ` x ) e. ~H )
33		simprr 796	. . . . . . 7 |- ( ( ( T : ~H --> ~H /\ U : ~H --> ~H ) /\ ( x e. ~H /\ y e. ~H ) ) -> y e. ~H )
34		ax-his2 27940	. . . . . . 7 |- ( ( ( T ` x ) e. ~H /\ ( U ` x ) e. ~H /\ y e. ~H ) -> ( ( ( T ` x ) +h ( U ` x ) ) .ih y ) = ( ( ( T ` x ) .ih y ) + ( ( U ` x ) .ih y ) ) )
35	30, 32, 33, 34	syl3anc 1326	. . . . . 6 |- ( ( ( T : ~H --> ~H /\ U : ~H --> ~H ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( ( ( T ` x ) +h ( U ` x ) ) .ih y ) = ( ( ( T ` x ) .ih y ) + ( ( U ` x ) .ih y ) ) )
36	28, 35	eqtrd 2656	. . . . 5 |- ( ( ( T : ~H --> ~H /\ U : ~H --> ~H ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( ( ( T +op U ) ` x ) .ih y ) = ( ( ( T ` x ) .ih y ) + ( ( U ` x ) .ih y ) ) )
37	11, 36	sylan 488	. . . 4 |- ( ( ( T e. HrmOp /\ U e. HrmOp ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( ( ( T +op U ) ` x ) .ih y ) = ( ( ( T ` x ) .ih y ) + ( ( U ` x ) .ih y ) ) )
38	10, 24, 37	3eqtr4d 2666	. . 3 |- ( ( ( T e. HrmOp /\ U e. HrmOp ) /\ ( x e. ~H /\ y e. ~H ) ) -> ( x .ih ( ( T +op U ) ` y ) ) = ( ( ( T +op U ) ` x ) .ih y ) )
39	38	ralrimivva 2971	. 2 |- ( ( T e. HrmOp /\ U e. HrmOp ) -> A. x e. ~H A. y e. ~H ( x .ih ( ( T +op U ) ` y ) ) = ( ( ( T +op U ) ` x ) .ih y ) )
40		elhmop 28732	. 2 |- ( ( T +op U ) e. HrmOp <-> ( ( T +op U ) : ~H --> ~H /\ A. x e. ~H A. y e. ~H ( x .ih ( ( T +op U ) ` y ) ) = ( ( ( T +op U ) ` x ) .ih y ) ) )
41	4, 39, 40	sylanbrc 698	1 |- ( ( T e. HrmOp /\ U e. HrmOp ) -> ( T +op U ) e. HrmOp )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   -->wf 5884   ` cfv 5888  (class class class)co 6650   + caddc 9939   ~Hchil 27776   +h cva 27777   .ih csp 27779   +op chos 27795   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-hfvadd 27857  ax-hfi 27936  ax-his1 27939  ax-his2 27940

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-hosum 28589  df-hmop 28703

This theorem is referenced by:  hmopd  28881  leopadd  28991  opsqrlem4  29002