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Theorem hoddii 22585
Description: Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 22376 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 5680 . . . . . 6  |-  ( x  e.  ~H  ->  ( S `  x )  e.  ~H )
3 hoddi.3 . . . . . . 7  |-  T : ~H
--> ~H
43ffvelrni 5680 . . . . . 6  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
5 hoddi.1 . . . . . . 7  |-  R  e. 
LinOp
65lnopsubi 22570 . . . . . 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 22338 . . . . . . 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 5545 . . . . 5  |-  ( x  e.  ~H  ->  ( R `  ( ( S  -op  T ) `  x ) )  =  ( R `  (
( S `  x
)  -h  ( T `
 x ) ) ) )
115lnopfi 22565 . . . . . . 7  |-  R : ~H
--> ~H
1211, 1hocoi 22360 . . . . . 6  |-  ( x  e.  ~H  ->  (
( R  o.  S
) `  x )  =  ( R `  ( S `  x ) ) )
1311, 3hocoi 22360 . . . . . 6  |-  ( x  e.  ~H  ->  (
( R  o.  T
) `  x )  =  ( R `  ( T `  x ) ) )
1412, 13oveq12d 5892 . . . . 5  |-  ( x  e.  ~H  ->  (
( ( R  o.  S ) `  x
)  -h  ( ( R  o.  T ) `
 x ) )  =  ( ( R `
 ( S `  x ) )  -h  ( R `  ( T `  x )
) ) )
157, 10, 143eqtr4d 2338 . . . 4  |-  ( x  e.  ~H  ->  ( R `  ( ( S  -op  T ) `  x ) )  =  ( ( ( R  o.  S ) `  x )  -h  (
( R  o.  T
) `  x )
) )
161, 3hosubcli 22365 . . . . 5  |-  ( S  -op  T ) : ~H --> ~H
1711, 16hocoi 22360 . . . 4  |-  ( x  e.  ~H  ->  (
( R  o.  ( S  -op  T ) ) `
 x )  =  ( R `  (
( S  -op  T
) `  x )
) )
1811, 1hocofi 22362 . . . . 5  |-  ( R  o.  S ) : ~H --> ~H
1911, 3hocofi 22362 . . . . 5  |-  ( R  o.  T ) : ~H --> ~H
20 hodval 22338 . . . . 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 2338 . . 3  |-  ( x  e.  ~H  ->  (
( R  o.  ( S  -op  T ) ) `
 x )  =  ( ( ( R  o.  S )  -op  ( R  o.  T
) ) `  x
) )
2322rgen 2621 . 2  |-  A. x  e.  ~H  ( ( R  o.  ( S  -op  T ) ) `  x
)  =  ( ( ( R  o.  S
)  -op  ( R  o.  T ) ) `  x )
2411, 16hocofi 22362 . . 3  |-  ( R  o.  ( S  -op  T ) ) : ~H --> ~H
2518, 19hosubcli 22365 . . 3  |-  ( ( R  o.  S )  -op  ( R  o.  T ) ) : ~H --> ~H
2624, 25hoeqi 22357 . 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 1632    e. wcel 1696   A.wral 2556    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   ~Hchil 21515    -h cmv 21521    -op chod 21536   LinOpclo 21543
This theorem is referenced by:  hoddi  22586  unierri  22700
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-hilex 21595  ax-hfvadd 21596  ax-hvass 21598  ax-hv0cl 21599  ax-hvaddid 21600  ax-hfvmul 21601  ax-hvmulid 21602  ax-hvdistr2 21605  ax-hvmul0 21606
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-sub 9055  df-neg 9056  df-hvsub 21567  df-hodif 22328  df-lnop 22437
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