Theorem homulasst 9645

Description: Scalar product associative law for Hilbert space operators.

Assertion

Ref	Expression
homulasst	|- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> ((A x. B) .op T) = (A .op (B .op T)))

Proof of Theorem homulasst

Step	Hyp	Ref	Expression
1		ax-hvmulass 8798	. . . . . . . . 9 |- ((A e. CC /\ B e. CC /\ (T` x) e. H~) -> ((A x. B) .h (T` x)) = (A .h (B .h (T` x))))
2		ffvelrn 3799	. . . . . . . . 9 |- ((T:H~-->H~ /\ x e. H~) -> (T` x) e. H~)
3	1, 2	syl3an3 859	. . . . . . . 8 |- ((A e. CC /\ B e. CC /\ (T:H~-->H~ /\ x e. H~)) -> ((A x. B) .h (T` x)) = (A .h (B .h (T` x))))
4	3	3expa 831	. . . . . . 7 |- (((A e. CC /\ B e. CC) /\ (T:H~-->H~ /\ x e. H~)) -> ((A x. B) .h (T` x)) = (A .h (B .h (T` x))))
5	4	exp43 384	. . . . . 6 |- (A e. CC -> (B e. CC -> (T:H~-->H~ -> (x e. H~ -> ((A x. B) .h (T` x)) = (A .h (B .h (T` x)))))))
6	5	3imp1 844	. . . . 5 |- (((A e. CC /\ B e. CC /\ T:H~-->H~) /\ x e. H~) -> ((A x. B) .h (T` x)) = (A .h (B .h (T` x))))
7		homvalt 9435	. . . . . . . . 9 |- (((A x. B) e. CC /\ T:H~-->H~ /\ x e. H~) -> (((A x. B) .op T)` x) = ((A x. B) .h (T` x)))
8		axmulcl 5245	. . . . . . . . 9 |- ((A e. CC /\ B e. CC) -> (A x. B) e. CC)
9	7, 8	syl3an1 857	. . . . . . . 8 |- (((A e. CC /\ B e. CC) /\ T:H~-->H~ /\ x e. H~) -> (((A x. B) .op T)` x) = ((A x. B) .h (T` x)))
10	9	3expia 833	. . . . . . 7 |- (((A e. CC /\ B e. CC) /\ T:H~-->H~) -> (x e. H~ -> (((A x. B) .op T)` x) = ((A x. B) .h (T` x))))
11	10	3impa 826	. . . . . 6 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> (x e. H~ -> (((A x. B) .op T)` x) = ((A x. B) .h (T` x))))
12	11	imp 350	. . . . 5 |- (((A e. CC /\ B e. CC /\ T:H~-->H~) /\ x e. H~) -> (((A x. B) .op T)` x) = ((A x. B) .h (T` x)))
13		homvalt 9435	. . . . . . . 8 |- ((B e. CC /\ T:H~-->H~ /\ x e. H~) -> ((B .op T)` x) = (B .h (T` x)))
14	13	opreq2d 3961	. . . . . . 7 |- ((B e. CC /\ T:H~-->H~ /\ x e. H~) -> (A .h ((B .op T)` x)) = (A .h (B .h (T` x))))
15	14	3expa 831	. . . . . 6 |- (((B e. CC /\ T:H~-->H~) /\ x e. H~) -> (A .h ((B .op T)` x)) = (A .h (B .h (T` x))))
16	15	3adantl1 801	. . . . 5 |- (((A e. CC /\ B e. CC /\ T:H~-->H~) /\ x e. H~) -> (A .h ((B .op T)` x)) = (A .h (B .h (T` x))))
17	6, 12, 16	3eqtr4d 1509	. . . 4 |- (((A e. CC /\ B e. CC /\ T:H~-->H~) /\ x e. H~) -> (((A x. B) .op T)` x) = (A .h ((B .op T)` x)))
18		homvalt 9435	. . . . . . . 8 |- ((A e. CC /\ (B .op T):H~-->H~ /\ x e. H~) -> ((A .op (B .op T))` x) = (A .h ((B .op T)` x)))
19		homulclt 9602	. . . . . . . 8 |- ((B e. CC /\ T:H~-->H~) -> (B .op T):H~-->H~)
20	18, 19	syl3an2 858	. . . . . . 7 |- ((A e. CC /\ (B e. CC /\ T:H~-->H~) /\ x e. H~) -> ((A .op (B .op T))` x) = (A .h ((B .op T)` x)))
21	20	3expia 833	. . . . . 6 |- ((A e. CC /\ (B e. CC /\ T:H~-->H~)) -> (x e. H~ -> ((A .op (B .op T))` x) = (A .h ((B .op T)` x))))
22	21	3impb 827	. . . . 5 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> (x e. H~ -> ((A .op (B .op T))` x) = (A .h ((B .op T)` x))))
23	22	imp 350	. . . 4 |- (((A e. CC /\ B e. CC /\ T:H~-->H~) /\ x e. H~) -> ((A .op (B .op T))` x) = (A .h ((B .op T)` x)))
24	17, 23	eqtr4d 1502	. . 3 |- (((A e. CC /\ B e. CC /\ T:H~-->H~) /\ x e. H~) -> (((A x. B) .op T)` x) = ((A .op (B .op T))` x))
25	24	r19.21aiva 1706	. 2 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> A.x e. H~ (((A x. B) .op T)` x) = ((A .op (B .op T))` x))
26		hoeqt 9603	. . 3 |- ((((A x. B) .op T):H~-->H~ /\ (A .op (B .op T)):H~-->H~) -> (A.x e. H~ (((A x. B) .op T)` x) = ((A .op (B .op T))` x) <-> ((A x. B) .op T) = (A .op (B .op T))))
27		homulclt 9602	. . . . 5 |- (((A x. B) e. CC /\ T:H~-->H~) -> ((A x. B) .op T):H~-->H~)
28	27, 8	sylan 448	. . . 4 |- (((A e. CC /\ B e. CC) /\ T:H~-->H~) -> ((A x. B) .op T):H~-->H~)
29	28	3impa 826	. . 3 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> ((A x. B) .op T):H~-->H~)
30		homulclt 9602	. . . . 5 |- ((A e. CC /\ (B .op T):H~-->H~) -> (A .op (B .op T)):H~-->H~)
31	30, 19	sylan2 451	. . . 4 |- ((A e. CC /\ (B e. CC /\ T:H~-->H~)) -> (A .op (B .op T)):H~-->H~)
32	31	3impb 827	. . 3 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> (A .op (B .op T)):H~-->H~)
33	26, 29, 32	sylanc 471	. 2 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> (A.x e. H~ (((A x. B) .op T)` x) = ((A .op (B .op T))` x) <-> ((A x. B) .op T) = (A .op (B .op T))))
34	25, 33	mpbid 195	1 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> ((A x. B) .op T) = (A .op (B .op T)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955  A.wral 1637  -->wf 3168  ` cfv 3172  (class class class)co 3948  CCcc 5204   x. cmul 5211  H~chil 8727   .h csm 8729   .op chot 8747

This theorem is referenced by:  homul12t 9648  honegnegt 9649  leopmult 9979  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-hvmulass 8798

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-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-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-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-m1r 5145  df-c 5212  df-mul 5218  df-homul 9424

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