Theorem hmopmt 9884

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 9625	. . . 4 |- ((A e. CC /\ T:H~-->H~) -> (A .op T):H~-->H~)
2		recnt 5293	. . . 4 |- (A e. RR -> A e. CC)
3		hmopft 9741	. . . 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 6753	. . . . . . . 8 |- (A e. RR -> (*` A) = A)
6		hmopt 9785	. . . . . . . . 9 |- ((T e. HrmOp /\ x e. H~ /\ y e. H~) -> (x .ih (T` y)) = ((T` x) .ih y))
7	6	3expb 833	. . . . . . . 8 |- ((T e. HrmOp /\ (x e. H~ /\ y e. H~)) -> (x .ih (T` y)) = ((T` x) .ih y))
8	5, 7	opreqan12d 3970	. . . . . . 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 9458	. . . . . . . . . . 11 |- ((A e. CC /\ T:H~-->H~ /\ y e. H~) -> ((A .op T)` y) = (A .h (T` y)))
11	10	3expa 832	. . . . . . . . . 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 3967	. . . . . . . 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 8892	. . . . . . . . 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 3805	. . . . . . . . . 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 857	. . . . . . . 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 1504	. . . . . . 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 9458	. . . . . . . . . . 11 |- ((A e. CC /\ T:H~-->H~ /\ x e. H~) -> ((A .op T)` x) = (A .h (T` x)))
24	23	3expa 832	. . . . . . . . . 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 3966	. . . . . . . 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 8890	. . . . . . . . 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 3805	. . . . . . . . . 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 857	. . . . . . . 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 1504	. . . . . . 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 1514	. . . . 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 1717	. . 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 9740	. 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 954   e. wcel 956  A.wral 1642  -->wf 3173  ` cfv 3177  (class class class)co 3954  CCcc 5212  RRcr 5213   x. cmul 5219  *ccj 6688  H~chil 8727   .h csm 8729   .ih csp 8732   .op chot 8747  HrmOpcho 8758

This theorem is referenced by:  hmopdt 9885  leopmulit 10004  leopmult 10005  leopmul2it 10006  leopnmidt 10009  nmopleidt 10010

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605  ax-hilex 8808  ax-hfvmul 8814  ax-hfi 8885  ax-his1 8888  ax-his3 8890

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-map 4314  df-en 4357  df-dom 4358  df-sdom 4359  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp 5069  df-ltp 5070  df-plpr 5144  df-mpr 5145  df-enr 5146  df-nr 5147  df-plr 5148  df-mr 5149  df-ltr 5150  df-0r 5151  df-1r 5152  df-m1r 5153  df-c 5220  df-0 5221  df-1 5222  df-i 5223  df-r 5224  df-plus 5225  df-mul 5226  df-lt 5227  df-sub 5336  df-neg 5338  df-pnf 5467  df-mnf 5468  df-xr 5469  df-ltxr 5470  df-le 5471  df-div 5680  df-re 6690  df-im 6691  df-cj 6692  df-homul 9447  df-hmop 9710

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