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Theorem homulass 28661
Description: Scalar product associative law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
Assertion
Ref Expression
homulass |- ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) -> ( ( A x. B ) .op T ) = ( A .op ( B .op T ) ) )
Proof of Theorem homulass
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 mulcl 10020 . . . . . . . . 9 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
2 homval 28600 . . . . . . . . 9 |- ( ( ( A x. B ) e. CC /\ T : ~H --> ~H /\ x e. ~H ) -> ( ( ( A x. B ) .op T ) ` x ) = ( ( A x. B ) .h ( T ` x ) ) )
31, 2syl3an1 1359 . . . . . . . 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 ) ) )
433expia 1267 . . . . . . 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 ) ) ) )
543impa 1259 . . . . . 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 ) ) ) )
65imp 445 . . . . 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 ) ) )
7 homval 28600 . . . . . . . . 9 |- ( ( B e. CC /\ T : ~H --> ~H /\ x e. ~H ) -> ( ( B .op T ) ` x ) = ( B .h ( T ` x ) ) )
87oveq2d 6666 . . . . . . . 8 |- ( ( B e. CC /\ T : ~H --> ~H /\ x e. ~H ) -> ( A .h ( ( B .op T ) ` x ) ) = ( A .h ( B .h ( T ` x ) ) ) )
983expa 1265 . . . . . . 7 |- ( ( ( B e. CC /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( A .h ( ( B .op T ) ` x ) ) = ( A .h ( B .h ( T ` x ) ) ) )
1093adantl1 1217 . . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( A .h ( ( B .op T ) ` x ) ) = ( A .h ( B .h ( T ` x ) ) ) )
11 ffvelrn 6357 . . . . . . . . . 10 |- ( ( T : ~H --> ~H /\ x e. ~H ) -> ( T ` x ) e. ~H )
12 ax-hvmulass 27864 . . . . . . . . . 10 |- ( ( A e. CC /\ B e. CC /\ ( T ` x ) e. ~H ) -> ( ( A x. B ) .h ( T ` x ) ) = ( A .h ( B .h ( T ` x ) ) ) )
1311, 12syl3an3 1361 . . . . . . . . 9 |- ( ( A e. CC /\ B e. CC /\ ( T : ~H --> ~H /\ x e. ~H ) ) -> ( ( A x. B ) .h ( T ` x ) ) = ( A .h ( B .h ( T ` x ) ) ) )
14133expa 1265 . . . . . . . 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 ) ) ) )
1514exp43 640 . . . . . . 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 ) ) ) ) ) ) )
16153imp1 1280 . . . . . 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 ) ) ) )
1710, 16eqtr4d 2659 . . . . 5 |- ( ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( A .h ( ( B .op T ) ` x ) ) = ( ( A x. B ) .h ( T ` x ) ) )
186, 17eqtr4d 2659 . . . 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 ) ) )
19 homulcl 28618 . . . . . . . 8 |- ( ( B e. CC /\ T : ~H --> ~H ) -> ( B .op T ) : ~H --> ~H )
20 homval 28600 . . . . . . . 8 |- ( ( A e. CC /\ ( B .op T ) : ~H --> ~H /\ x e. ~H ) -> ( ( A .op ( B .op T ) ) ` x ) = ( A .h ( ( B .op T ) ` x ) ) )
2119, 20syl3an2 1360 . . . . . . 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 ) ) )
22213expia 1267 . . . . . 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 ) ) ) )
23223impb 1260 . . . . 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 ) ) ) )
2423imp 445 . . . 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 ) ) )
2518, 24eqtr4d 2659 . . 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 ) )
2625ralrimiva 2966 . 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 ) )
27 homulcl 28618 . . . 4 |- ( ( ( A x. B ) e. CC /\ T : ~H --> ~H ) -> ( ( A x. B ) .op T ) : ~H --> ~H )
281, 27stoic3 1701 . . 3 |- ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) -> ( ( A x. B ) .op T ) : ~H --> ~H )
29 homulcl 28618 . . . . 5 |- ( ( A e. CC /\ ( B .op T ) : ~H --> ~H ) -> ( A .op ( B .op T ) ) : ~H --> ~H )
3019, 29sylan2 491 . . . 4 |- ( ( A e. CC /\ ( B e. CC /\ T : ~H --> ~H ) ) -> ( A .op ( B .op T ) ) : ~H --> ~H )
31303impb 1260 . . 3 |- ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) -> ( A .op ( B .op T ) ) : ~H --> ~H )
32 hoeq 28619 . . 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 ) ) ) )
3328, 31, 32syl2anc 693 . 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 ) ) ) )
3426, 33mpbid 222 1 |- ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) -> ( ( A x. B ) .op T ) = ( A .op ( B .op T ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   x. cmul 9941   ~Hchil 27776   .h csm 27778   .op chot 27796
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-mulcl 9998  ax-hilex 27856  ax-hfvmul 27862  ax-hvmulass 27864
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-homul 28590
This theorem is referenced by:  homul12  28664  honegneg  28665  leopmul  28993  nmopleid  28998  opsqrlem1  28999  opsqrlem6  29004
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