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Theorem hoddii 22569
Description: Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 22360 does not require linearity.) (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
hoddi.1  |-  R  e. 
LinOp
hoddi.2  |-  S : ~H
--> ~H
hoddi.3  |-  T : ~H
--> ~H
Assertion
Ref Expression
hoddii  |-  ( R  o.  ( S  -op  T ) )  =  ( ( R  o.  S
)  -op  ( R  o.  T ) )

Proof of Theorem hoddii
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 hoddi.2 . . . . . . 7  |-  S : ~H
--> ~H
21ffvelrni 5664 . . . . . 6  |-  ( x  e.  ~H  ->  ( S `  x )  e.  ~H )
3 hoddi.3 . . . . . . 7  |-  T : ~H
--> ~H
43ffvelrni 5664 . . . . . 6  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
5 hoddi.1 . . . . . . 7  |-  R  e. 
LinOp
65lnopsubi 22554 . . . . . 6  |-  ( ( ( S `  x
)  e.  ~H  /\  ( T `  x )  e.  ~H )  -> 
( R `  (
( S `  x
)  -h  ( T `
 x ) ) )  =  ( ( R `  ( S `
 x ) )  -h  ( R `  ( T `  x ) ) ) )
72, 4, 6syl2anc 642 . . . . 5  |-  ( x  e.  ~H  ->  ( R `  ( ( S `  x )  -h  ( T `  x
) ) )  =  ( ( R `  ( S `  x ) )  -h  ( R `
 ( T `  x ) ) ) )
8 hodval 22322 . . . . . . 7  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( S  -op  T ) `  x )  =  ( ( S `
 x )  -h  ( T `  x
) ) )
91, 3, 8mp3an12 1267 . . . . . 6  |-  ( x  e.  ~H  ->  (
( S  -op  T
) `  x )  =  ( ( S `
 x )  -h  ( T `  x
) ) )
109fveq2d 5529 . . . . 5  |-  ( x  e.  ~H  ->  ( R `  ( ( S  -op  T ) `  x ) )  =  ( R `  (
( S `  x
)  -h  ( T `
 x ) ) ) )
115lnopfi 22549 . . . . . . 7  |-  R : ~H
--> ~H
1211, 1hocoi 22344 . . . . . 6  |-  ( x  e.  ~H  ->  (
( R  o.  S
) `  x )  =  ( R `  ( S `  x ) ) )
1311, 3hocoi 22344 . . . . . 6  |-  ( x  e.  ~H  ->  (
( R  o.  T
) `  x )  =  ( R `  ( T `  x ) ) )
1412, 13oveq12d 5876 . . . . 5  |-  ( x  e.  ~H  ->  (
( ( R  o.  S ) `  x
)  -h  ( ( R  o.  T ) `
 x ) )  =  ( ( R `
 ( S `  x ) )  -h  ( R `  ( T `  x )
) ) )
157, 10, 143eqtr4d 2325 . . . 4  |-  ( x  e.  ~H  ->  ( R `  ( ( S  -op  T ) `  x ) )  =  ( ( ( R  o.  S ) `  x )  -h  (
( R  o.  T
) `  x )
) )
161, 3hosubcli 22349 . . . . 5  |-  ( S  -op  T ) : ~H --> ~H
1711, 16hocoi 22344 . . . 4  |-  ( x  e.  ~H  ->  (
( R  o.  ( S  -op  T ) ) `
 x )  =  ( R `  (
( S  -op  T
) `  x )
) )
1811, 1hocofi 22346 . . . . 5  |-  ( R  o.  S ) : ~H --> ~H
1911, 3hocofi 22346 . . . . 5  |-  ( R  o.  T ) : ~H --> ~H
20 hodval 22322 . . . . 5  |-  ( ( ( R  o.  S
) : ~H --> ~H  /\  ( R  o.  T
) : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( ( R  o.  S )  -op  ( R  o.  T
) ) `  x
)  =  ( ( ( R  o.  S
) `  x )  -h  ( ( R  o.  T ) `  x
) ) )
2118, 19, 20mp3an12 1267 . . . 4  |-  ( x  e.  ~H  ->  (
( ( R  o.  S )  -op  ( R  o.  T )
) `  x )  =  ( ( ( R  o.  S ) `
 x )  -h  ( ( R  o.  T ) `  x
) ) )
2215, 17, 213eqtr4d 2325 . . 3  |-  ( x  e.  ~H  ->  (
( R  o.  ( S  -op  T ) ) `
 x )  =  ( ( ( R  o.  S )  -op  ( R  o.  T
) ) `  x
) )
2322rgen 2608 . 2  |-  A. x  e.  ~H  ( ( R  o.  ( S  -op  T ) ) `  x
)  =  ( ( ( R  o.  S
)  -op  ( R  o.  T ) ) `  x )
2411, 16hocofi 22346 . . 3  |-  ( R  o.  ( S  -op  T ) ) : ~H --> ~H
2518, 19hosubcli 22349 . . 3  |-  ( ( R  o.  S )  -op  ( R  o.  T ) ) : ~H --> ~H
2624, 25hoeqi 22341 . 2  |-  ( A. x  e.  ~H  (
( R  o.  ( S  -op  T ) ) `
 x )  =  ( ( ( R  o.  S )  -op  ( R  o.  T
) ) `  x
)  <->  ( R  o.  ( S  -op  T ) )  =  ( ( R  o.  S )  -op  ( R  o.  T ) ) )
2723, 26mpbi 199 1  |-  ( R  o.  ( S  -op  T ) )  =  ( ( R  o.  S
)  -op  ( R  o.  T ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   A.wral 2543    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   ~Hchil 21499    -h cmv 21505    -op chod 21520   LinOpclo 21527
This theorem is referenced by:  hoddi  22570  unierri  22684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-hilex 21579  ax-hfvadd 21580  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvdistr2 21589  ax-hvmul0 21590
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039  df-neg 9040  df-hvsub 21551  df-hodif 22312  df-lnop 22421
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