Theorem kbass2 28976

Description: Dirac bra-ket associative law ( <. A | B >. ) <. C | = <. A | ( | B >. <. C | ) i.e. the juxtaposition of an inner product with a bra equals a ket juxtaposed with an outer product. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
kbass2	|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( ( bra ` A ) ` B ) .fn ( bra ` C ) ) = ( ( bra ` A ) o. ( B ketbra C ) ) )

Proof of Theorem kbass2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 6678	. . . 4 |- ( ( ( bra ` A ) ` B ) x. ( ( bra ` C ) ` x ) ) e. _V
2		eqid 2622	. . . 4 |- ( x e. ~H |-> ( ( ( bra ` A ) ` B ) x. ( ( bra ` C ) ` x ) ) ) = ( x e. ~H |-> ( ( ( bra ` A ) ` B ) x. ( ( bra ` C ) ` x ) ) )
3	1, 2	fnmpti 6022	. . 3 |- ( x e. ~H |-> ( ( ( bra ` A ) ` B ) x. ( ( bra ` C ) ` x ) ) ) Fn ~H
4		bracl 28808	. . . . . 6 |- ( ( A e. ~H /\ B e. ~H ) -> ( ( bra ` A ) ` B ) e. CC )
5		brafn 28806	. . . . . 6 |- ( C e. ~H -> ( bra ` C ) : ~H --> CC )
6		hfmmval 28598	. . . . . 6 |- ( ( ( ( bra ` A ) ` B ) e. CC /\ ( bra ` C ) : ~H --> CC ) -> ( ( ( bra ` A ) ` B ) .fn ( bra ` C ) ) = ( x e. ~H |-> ( ( ( bra ` A ) ` B ) x. ( ( bra ` C ) ` x ) ) ) )
7	4, 5, 6	syl2an 494	. . . . 5 |- ( ( ( A e. ~H /\ B e. ~H ) /\ C e. ~H ) -> ( ( ( bra ` A ) ` B ) .fn ( bra ` C ) ) = ( x e. ~H |-> ( ( ( bra ` A ) ` B ) x. ( ( bra ` C ) ` x ) ) ) )
8	7	3impa 1259	. . . 4 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( ( bra ` A ) ` B ) .fn ( bra ` C ) ) = ( x e. ~H |-> ( ( ( bra ` A ) ` B ) x. ( ( bra ` C ) ` x ) ) ) )
9	8	fneq1d 5981	. . 3 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( ( ( bra ` A ) ` B ) .fn ( bra ` C ) ) Fn ~H <-> ( x e. ~H |-> ( ( ( bra ` A ) ` B ) x. ( ( bra ` C ) ` x ) ) ) Fn ~H ) )
10	3, 9	mpbiri 248	. 2 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( ( bra ` A ) ` B ) .fn ( bra ` C ) ) Fn ~H )
11		brafn 28806	. . . . 5 |- ( A e. ~H -> ( bra ` A ) : ~H --> CC )
12		kbop 28812	. . . . 5 |- ( ( B e. ~H /\ C e. ~H ) -> ( B ketbra C ) : ~H --> ~H )
13		fco 6058	. . . . 5 |- ( ( ( bra ` A ) : ~H --> CC /\ ( B ketbra C ) : ~H --> ~H ) -> ( ( bra ` A ) o. ( B ketbra C ) ) : ~H --> CC )
14	11, 12, 13	syl2an 494	. . . 4 |- ( ( A e. ~H /\ ( B e. ~H /\ C e. ~H ) ) -> ( ( bra ` A ) o. ( B ketbra C ) ) : ~H --> CC )
15	14	3impb 1260	. . 3 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( bra ` A ) o. ( B ketbra C ) ) : ~H --> CC )
16		ffn 6045	. . 3 |- ( ( ( bra ` A ) o. ( B ketbra C ) ) : ~H --> CC -> ( ( bra ` A ) o. ( B ketbra C ) ) Fn ~H )
17	15, 16	syl 17	. 2 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( bra ` A ) o. ( B ketbra C ) ) Fn ~H )
18		simpl1 1064	. . . . 5 |- ( ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> A e. ~H )
19		simpl2 1065	. . . . 5 |- ( ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> B e. ~H )
20		braval 28803	. . . . 5 |- ( ( A e. ~H /\ B e. ~H ) -> ( ( bra ` A ) ` B ) = ( B .ih A ) )
21	18, 19, 20	syl2anc 693	. . . 4 |- ( ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( ( bra ` A ) ` B ) = ( B .ih A ) )
22		simpl3 1066	. . . . 5 |- ( ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> C e. ~H )
23		simpr 477	. . . . 5 |- ( ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> x e. ~H )
24		braval 28803	. . . . 5 |- ( ( C e. ~H /\ x e. ~H ) -> ( ( bra ` C ) ` x ) = ( x .ih C ) )
25	22, 23, 24	syl2anc 693	. . . 4 |- ( ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( ( bra ` C ) ` x ) = ( x .ih C ) )
26	21, 25	oveq12d 6668	. . 3 |- ( ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( ( ( bra ` A ) ` B ) x. ( ( bra ` C ) ` x ) ) = ( ( B .ih A ) x. ( x .ih C ) ) )
27		hicl 27937	. . . . . 6 |- ( ( B e. ~H /\ A e. ~H ) -> ( B .ih A ) e. CC )
28	19, 18, 27	syl2anc 693	. . . . 5 |- ( ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( B .ih A ) e. CC )
29	21, 28	eqeltrd 2701	. . . 4 |- ( ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( ( bra ` A ) ` B ) e. CC )
30	22, 5	syl 17	. . . 4 |- ( ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( bra ` C ) : ~H --> CC )
31		hfmval 28603	. . . 4 |- ( ( ( ( bra ` A ) ` B ) e. CC /\ ( bra ` C ) : ~H --> CC /\ x e. ~H ) -> ( ( ( ( bra ` A ) ` B ) .fn ( bra ` C ) ) ` x ) = ( ( ( bra ` A ) ` B ) x. ( ( bra ` C ) ` x ) ) )
32	29, 30, 23, 31	syl3anc 1326	. . 3 |- ( ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( ( ( ( bra ` A ) ` B ) .fn ( bra ` C ) ) ` x ) = ( ( ( bra ` A ) ` B ) x. ( ( bra ` C ) ` x ) ) )
33		hicl 27937	. . . . . 6 |- ( ( x e. ~H /\ C e. ~H ) -> ( x .ih C ) e. CC )
34	23, 22, 33	syl2anc 693	. . . . 5 |- ( ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( x .ih C ) e. CC )
35		ax-his3 27941	. . . . 5 |- ( ( ( x .ih C ) e. CC /\ B e. ~H /\ A e. ~H ) -> ( ( ( x .ih C ) .h B ) .ih A ) = ( ( x .ih C ) x. ( B .ih A ) ) )
36	34, 19, 18, 35	syl3anc 1326	. . . 4 |- ( ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( ( ( x .ih C ) .h B ) .ih A ) = ( ( x .ih C ) x. ( B .ih A ) ) )
37	12	3adant1 1079	. . . . . 6 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( B ketbra C ) : ~H --> ~H )
38		fvco3 6275	. . . . . 6 |- ( ( ( B ketbra C ) : ~H --> ~H /\ x e. ~H ) -> ( ( ( bra ` A ) o. ( B ketbra C ) ) ` x ) = ( ( bra ` A ) ` ( ( B ketbra C ) ` x ) ) )
39	37, 38	sylan 488	. . . . 5 |- ( ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( ( ( bra ` A ) o. ( B ketbra C ) ) ` x ) = ( ( bra ` A ) ` ( ( B ketbra C ) ` x ) ) )
40		kbval 28813	. . . . . . 7 |- ( ( B e. ~H /\ C e. ~H /\ x e. ~H ) -> ( ( B ketbra C ) ` x ) = ( ( x .ih C ) .h B ) )
41	19, 22, 23, 40	syl3anc 1326	. . . . . 6 |- ( ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( ( B ketbra C ) ` x ) = ( ( x .ih C ) .h B ) )
42	41	fveq2d 6195	. . . . 5 |- ( ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( ( bra ` A ) ` ( ( B ketbra C ) ` x ) ) = ( ( bra ` A ) ` ( ( x .ih C ) .h B ) ) )
43		hvmulcl 27870	. . . . . . 7 |- ( ( ( x .ih C ) e. CC /\ B e. ~H ) -> ( ( x .ih C ) .h B ) e. ~H )
44	34, 19, 43	syl2anc 693	. . . . . 6 |- ( ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( ( x .ih C ) .h B ) e. ~H )
45		braval 28803	. . . . . 6 |- ( ( A e. ~H /\ ( ( x .ih C ) .h B ) e. ~H ) -> ( ( bra ` A ) ` ( ( x .ih C ) .h B ) ) = ( ( ( x .ih C ) .h B ) .ih A ) )
46	18, 44, 45	syl2anc 693	. . . . 5 |- ( ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( ( bra ` A ) ` ( ( x .ih C ) .h B ) ) = ( ( ( x .ih C ) .h B ) .ih A ) )
47	39, 42, 46	3eqtrd 2660	. . . 4 |- ( ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( ( ( bra ` A ) o. ( B ketbra C ) ) ` x ) = ( ( ( x .ih C ) .h B ) .ih A ) )
48	28, 34	mulcomd 10061	. . . 4 |- ( ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( ( B .ih A ) x. ( x .ih C ) ) = ( ( x .ih C ) x. ( B .ih A ) ) )
49	36, 47, 48	3eqtr4d 2666	. . 3 |- ( ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( ( ( bra ` A ) o. ( B ketbra C ) ) ` x ) = ( ( B .ih A ) x. ( x .ih C ) ) )
50	26, 32, 49	3eqtr4d 2666	. 2 |- ( ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( ( ( ( bra ` A ) ` B ) .fn ( bra ` C ) ) ` x ) = ( ( ( bra ` A ) o. ( B ketbra C ) ) ` x ) )
51	10, 17, 50	eqfnfvd 6314	1 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( ( bra ` A ) ` B ) .fn ( bra ` C ) ) = ( ( bra ` A ) o. ( B ketbra C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   |-> cmpt 4729   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   x. cmul 9941   ~Hchil 27776   .h csm 27778   .ih csp 27779   .fn chft 27799   bracbr 27813   ketbra ck 27814

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-cnex 9992  ax-mulcom 10000  ax-hilex 27856  ax-hfvmul 27862  ax-hfi 27936  ax-his3 27941

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-reu 2919  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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-hfmul 28593  df-bra 28709  df-kb 28710

This theorem is referenced by:  kbass6  28980