Theorem hmopmt 9861

Description: The scalar product of a Hermitian operator with a real is Hermitian.

Assertion

Ref	Expression
hmopmt	|- ((A e. RR /\ T e. HrmOp) -> (A .op T) e. HrmOp)

Proof of Theorem hmopmt

Step	Hyp	Ref	Expression
1		homulclt 9602	. . . 4 |- ((A e. CC /\ T:H~-->H~) -> (A .op T):H~-->H~)
2		recnt 5285	. . . 4 |- (A e. RR -> A e. CC)
3		hmopft 9718	. . . 4 |- (T e. HrmOp -> T:H~-->H~)
4	1, 2, 3	syl2an 454	. . 3 |- ((A e. RR /\ T e. HrmOp) -> (A .op T):H~-->H~)
5		cjret 6745	. . . . . . . 8 |- (A e. RR -> (*` A) = A)
6		hmopt 9762	. . . . . . . . 9 |- ((T e. HrmOp /\ x e. H~ /\ y e. H~) -> (x .ih (T` y)) = ((T` x) .ih y))
7	6	3expb 832	. . . . . . . 8 |- ((T e. HrmOp /\ (x e. H~ /\ y e. H~)) -> (x .ih (T` y)) = ((T` x) .ih y))
8	5, 7	opreqan12d 3964	. . . . . . 7 |- ((A e. RR /\ (T e. HrmOp /\ (x e. H~ /\ y e. H~))) -> ((*` A) x. (x .ih (T` y))) = (A x. ((T` x) .ih y)))
9	8	anassrs 441	. . . . . 6 |- (((A e. RR /\ T e. HrmOp) /\ (x e. H~ /\ y e. H~)) -> ((*` A) x. (x .ih (T` y))) = (A x. ((T` x) .ih y)))
10		homvalt 9435	. . . . . . . . . . 11 |- ((A e. CC /\ T:H~-->H~ /\ y e. H~) -> ((A .op T)` y) = (A .h (T` y)))
11	10	3expa 831	. . . . . . . . . 10 |- (((A e. CC /\ T:H~-->H~) /\ y e. H~) -> ((A .op T)` y) = (A .h (T` y)))
12	11	adantrl 394	. . . . . . . . 9 |- (((A e. CC /\ T:H~-->H~) /\ (x e. H~ /\ y e. H~)) -> ((A .op T)` y) = (A .h (T` y)))
13	12	opreq2d 3961	. . . . . . . 8 |- (((A e. CC /\ T:H~-->H~) /\ (x e. H~ /\ y e. H~)) -> (x .ih ((A .op T)` y)) = (x .ih (A .h (T` y))))
14		his5t 8874	. . . . . . . . 9 |- ((A e. CC /\ x e. H~ /\ (T` y) e. H~) -> (x .ih (A .h (T` y))) = ((*` A) x. (x .ih (T` y))))
15		simpll 412	. . . . . . . . 9 |- (((A e. CC /\ T:H~-->H~) /\ (x e. H~ /\ y e. H~)) -> A e. CC)
16		simprl 414	. . . . . . . . 9 |- (((A e. CC /\ T:H~-->H~) /\ (x e. H~ /\ y e. H~)) -> x e. H~)
17		ffvelrn 3799	. . . . . . . . . 10 |- ((T:H~-->H~ /\ y e. H~) -> (T` y) e. H~)
18	17	ad2ant2l 408	. . . . . . . . 9 |- (((A e. CC /\ T:H~-->H~) /\ (x e. H~ /\ y e. H~)) -> (T` y) e. H~)
19	14, 15, 16, 18	syl3anc 856	. . . . . . . 8 |- (((A e. CC /\ T:H~-->H~) /\ (x e. H~ /\ y e. H~)) -> (x .ih (A .h (T` y))) = ((*` A) x. (x .ih (T` y))))
20	13, 19	eqtrd 1499	. . . . . . 7 |- (((A e. CC /\ T:H~-->H~) /\ (x e. H~ /\ y e. H~)) -> (x .ih ((A .op T)` y)) = ((*` A) x. (x .ih (T` y))))
21	2, 3	anim12i 333	. . . . . . 7 |- ((A e. RR /\ T e. HrmOp) -> (A e. CC /\ T:H~-->H~))
22	20, 21	sylan 448	. . . . . 6 |- (((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		homvalt 9435	. . . . . . . . . . 11 |- ((A e. CC /\ T:H~-->H~ /\ x e. H~) -> ((A .op T)` x) = (A .h (T` x)))
24	23	3expa 831	. . . . . . . . . 10 |- (((A e. CC /\ T:H~-->H~) /\ x e. H~) -> ((A .op T)` x) = (A .h (T` x)))
25	24	adantrr 395	. . . . . . . . 9 |- (((A e. CC /\ T:H~-->H~) /\ (x e. H~ /\ y e. H~)) -> ((A .op T)` x) = (A .h (T` x)))
26	25	opreq1d 3960	. . . . . . . 8 |- (((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		ax-his3 8872	. . . . . . . . 9 |- ((A e. CC /\ (T` x) e. H~ /\ y e. H~) -> ((A .h (T` x)) .ih y) = (A x. ((T` x) .ih y)))
28		ffvelrn 3799	. . . . . . . . . 10 |- ((T:H~-->H~ /\ x e. H~) -> (T` x) e. H~)
29	28	ad2ant2lr 410	. . . . . . . . 9 |- (((A e. CC /\ T:H~-->H~) /\ (x e. H~ /\ y e. H~)) -> (T` x) e. H~)
30		simprr 415	. . . . . . . . 9 |- (((A e. CC /\ T:H~-->H~) /\ (x e. H~ /\ y e. H~)) -> y e. H~)
31	27, 15, 29, 30	syl3anc 856	. . . . . . . 8 |- (((A e. CC /\ T:H~-->H~) /\ (x e. H~ /\ y e. H~)) -> ((A .h (T` x)) .ih y) = (A x. ((T` x) .ih y)))
32	26, 31	eqtrd 1499	. . . . . . 7 |- (((A e. CC /\ T:H~-->H~) /\ (x e. H~ /\ y e. H~)) -> (((A .op T)` x) .ih y) = (A x. ((T` x) .ih y)))
33	32, 21	sylan 448	. . . . . 6 |- (((A e. RR /\ T e. HrmOp) /\ (x e. H~ /\ y e. H~)) -> (((A .op T)` x) .ih y) = (A x. ((T` x) .ih y)))
34	9, 22, 33	3eqtr4d 1509	. . . . 5 |- (((A e. RR /\ T e. HrmOp) /\ (x e. H~ /\ y e. H~)) -> (x .ih ((A .op T)` y)) = (((A .op T)` x) .ih y))
35	34	ex 373	. . . 4 |- ((A e. RR /\ T e. HrmOp) -> ((x e. H~ /\ y e. H~) -> (x .ih ((A .op T)` y)) = (((A .op T)` x) .ih y)))
36	35	r19.21aivv 1712	. . 3 |- ((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))
37	4, 36	jca 288	. 2 |- ((A e. RR /\ 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)))
38		elhmopt 9717	. 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)))
39	37, 38	sylibr 200	1 |- ((A e. RR /\ T e. HrmOp) -> (A .op T) e. HrmOp)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  A.wral 1637  -->wf 3168  ` cfv 3172  (class class class)co 3948  CCcc 5204  RRcr 5205   x. cmul 5211  *ccj 6680  H~chil 8727   .h csm 8729   .ih csp 8732   .op chot 8747  HrmOpcho 8758

This theorem is referenced by:  hmopdt 9862  leopmulit 9978  leopmult 9979  leopmul2it 9980  leopnmidt 9982  nmopleidt 9983

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597  ax-hilex 8790  ax-hfvmul 8796  ax-hfi 8867  ax-his1 8870  ax-his3 8872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-map 4308  df-en 4351  df-dom 4352  df-sdom 4353  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-div 5672  df-re 6682  df-im 6683  df-cj 6684  df-homul 9424  df-hmop 9687

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